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%\journal{Applied Soft Computing}


\begin{document}

\section*{HIGHLIGHTS}
\begin{itemize}
%\item{} Interactive Granular Computing (IGrC) is proposed as a basis for the design and analysis of intelligent systems (IS's) dealing with complex phenomena.
%
%\item{}  IS's in IGrC are examples of complex granules (c-granules)  that connect abstract and physical objects.
%
%\item{} An extension of rough sets to adaptive rough sets in IGrC is developed as a basis for approximate reasoning of c-granules.
%
%\item{} The control of c-granules responsible for the generation of granular computations along which approximate solutions are constructed, is specified.
%
%\item{} Insightful reasoning of the control module of c-granules supporting their behavior is characterized.

\item{} Further development of the foundations of Intelligent Systems (IS's) based on IGrC.

\item{} Generalization of Rough Sets (RS) to Adaptive RS in IGrC as a Basis for Approximate Reasoning.

\item{} Presentation the evolution of rough set approaches to concept approximation.

\item{}  Elaboration of the reasoning module of the control module of c-granules.

\item{} RS approach to discovery of the high quality solutions to problems by  IS's.


\end{itemize}


\begin{frontmatter}

%% Title, authors and addresses

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\title{Adaptive Rough Sets\\ based on Interactive Granular Computing:\\ Toward Intelligent Systems\\ Dealing with Complex Phenomena}

\author[a]{Andrzej Skowron}
\affiliation[a]{organization={Systems Research Institute, Polish Academy of Sciences},
            addressline={\\Newelska 6},
            city={Warsaw},
            postcode={01-447},
            country={Poland}}
\author[b,c]{Dominik \'{S}l\k{e}zak}
\affiliation[b]{organization={Institute of Informatics, University of Warsaw},
            addressline={Banacha 2},
            city={Warszawa},
            postcode={02-097},
            country={Poland}}
\affiliation[c]{organization={QED Software Sp. z o.o.},
            addressline={Mazowiecka 11/49},
            city={Warsaw},
            postcode={00-052},
            country={Poland}}
%\begin{document}


%\maketitle

\begin{abstract}
This paper presents the adaptive rough set approach based on the Interactive Granular Computing (IGrC) framework.  The evolution of rough set approaches to this adaptive approach is discussed. The IGrC model is the basis for the development of foundations of Intelligent Systems (IS's) dealing with complex phenomena. IS's are special cases of the basic objects of IGrC, called complex granules (c-grules for short). They enable the perception of situations in the physical world through interactions. Adaptation and approximation are performed by a control module of c-granules not only on the basis of optimization of parameters of data models, but also on the basis of continuous interaction of IS's with the real physical world and advanced approximate reasoning techniques supporting the perception of situations or objects in this world. The role of approximate reasoning techniques in supporting the control of the behavior of C-granules is emphasized. 

%1.	Approach to adaptive rough sets in the framework of IGrC for Intelligent Systems dealing with complex phenomena.
%
%2.	Adaptation and approximation not only on the basis of optimization of parameters of models but also on the basis of continuous interaction of intelligent system with the real physical world supporting perception of situations in the physical world.
%
%3.	Interactions are controlled by special mechanisms of control that use the basic objects called c-granules on which IGrC is based.
%
%4.	The proposed approach is grounded in the IGrC model. Issues related to adaptation are further elaborated in IGrC.


\end{abstract}

\begin{keyword}
Intelligent system\sep (Adaptive) rough sets\sep  (Interactive) granular computing\sep Complex granule (c-granule) \sep Network of approximation spaces \sep Network of c-granules \sep Interactive granular computations of c-granule \sep Approximate reasoning over computations of c-granules.
\end{keyword}

\end{frontmatter}

\linenumbers

\section{Introduction}\label{sec:intro}

The aim of the approach presented in this paper is to show that the  extension of the rough set approach to the rough sets over complex granules (c-granules) in Interactive Granular Computing (IGrC) can be treated as a basis for developing foundations for IS's dealing with complex phenomena. According to opinions of different authors (see, {\em e.g.}, \cite{ch12-brooks1995mythical}), using in such cases of IS's classical mathematical modeling does not work. Moreover, it is necessary for such IS's to have abilities for perception of situations in the real physical world \cite{ch12-Turing-test-mod}. Basing the design of IS's on the IGrC model allows to continuously perceive the real world situations and acquire new relevant data, allowing to adjust the current temporal data models accordingly.

We begin by discussing the motivations that indicate the need to develop a new computational model relevant for the design and analysis of IS's dealing with complex phenomena. We propose to choose as such a model the IGrC model. This model uses as basic objects complex granules (c-granules) connecting abstract and physical objects. It allows to perceive interactions of both kinds of objects and to synchronize issues of language and reasoning as well as perception and action. 

We further develop the modeling of the control of c-granules, especially with respect to the reasoning mechanisms that support the control behavior.  In particular, we show that the evolving foundations for the design and analysis of ISs require relevant approximate reasoning techniques to support the search by IS's for high quality approximate solutions to the problems to be solved by IS's. We propose to perform this approximate reasoning over granular computations based on the IGrC model. Approximate solutions to the problems to be solved by IS's are constructed along computations over the IGrC model.

The approximate reasoning techniques are based on the far-reaching generalization of the current rough set approach presented in the paper. To better explain this, we discuss the evolution of approaches in rough sets to approximate concepts as well as problems related to approximate reasoning about them. We start with the basic Pawlak model and then present several examples of generalizations that have been developed over the years. In the new approach, we emphasize the importance of networks of granular spaces. Roughly speaking, these are paramatereized relational systems over objects of different types connected by interfaces that allow to express relationships between granular spaces that are components of the networks. Furthermore, such networks should be dynamic. Their dynamics cannot be defined a priori, but depend on the control of the c-granules and their interactions with the physical world (the environment). It is worth noting that the models considered so far, other than those based on IGrC, have been static or their dynamics have been defined a priori.

The above-mentioned networks of approximation spaces are used to define networks of c-granules (granular networks), which are constructed from these networks of approximation spaces by control of c-granules and realized in interaction with the environment. Granular computations of c-granules are defined over such granular networks. The control of c-granules aims at constructing computations of high quality with respect to the requirements represented by given specifications of problems to be solved by c-granules, in particular by IS's. The quality of the computations is measured by relevant quality measures.

The behavior of c-granules is determined by adaptive complex games, which are constructed and modified by control and possible interaction with humans and physical objects. These complex games are finite sets of pairs in which classifiers for compound complex concepts are labeled by actions or plans represented by so-called transformations. The concepts suggest the activation of the realization of so-called transformations when they are satisfied to a satisfactory degree. The realization in the physical world of these transformations (called as physical semantics) is performed by the implementational module of control. One of the important tasks for approximation techniques used by c-granules is to support the discovery of complex games. There are many other tasks that should be performed by the reasoning module of the control of c-granules, in particular IS's dealing with complex phenomena. Some of them are typical for machine learning, others may be related to the discovery of complex entities (objects or computations with high quality). Examples of such entities are presented in the paper. This can be considered as the discovery of complex entities (objects or computations), {\em e.g.}, from the lower approximation of complex concepts defined in the universe of granular computations, in cases where no such positive entities are available a priori.

The novelty of the paper lies in 
\begin{itemize} 

\item{} Further development of the Interactive Granular Computing Model (IGrC) as a basis for the design and analysis of Intelligent Systems (IS) dealing with complex phenomena.

\item{} Generalization of rough sets to adaptive rough sets in IGrC as a basis for approximate reasoning over c-granules.

\item{} Presenting the evolution of rough set approaches for approximating concepts.

\item{} Elaboration of the reasoning module of the control of c-granules..

\item{} Presentation of the rough-set approach to finding high-quality solutions of problems to be solved by IS's.

\end{itemize}


The paper is structured as follows ...

\vspace{2cm}

%%%%%%%%%%%%%%%%%%
Evolution of approximation spaces:

static rough sets (over static granular spaces)

   approximation space is a relational structure (partition and partial inclusion of sets):

   Pawlak's rough set model

   Tolerance rough set model and VPRS

   multirelational rough sets

   approximation space is a relevant relational structure selected in optimization from large family of candidates forming calculus of approximation spaces (partitions):


   rough sets over calculus of partitions -- optimization

It is worth adding some comments on the definition of relations between granules. The first approach is based on the syntax of granules. For example, a granule with syntax $a=v \wedge b=v^{\prime}$ is included in the granule obtained from this granule by dropping a descriptor, {\em e.g.}, $sem(a=v \wedge b=v^{\prime})$ is included in $sem(a=v )$ on any universe of objects $U$. The second approach, often used in ML in inducing decision rules, is based on inductive reasoning. For example, knowing that the following inclusion $sem(a=v \wedge b=v^{\prime}) \subseteq sem(d=0)$ holds on a sample of objects from a given decision system with decision attribute $d$ and conditional attributes $a,b$, it is inferred that this inclusion also holds on the extension $U^* \supset U$. Thus, some relations between granules are defined with some risk, i.e., they may not hold on the currently considered universes of objects which are extensions of universes on which they have been estimated.


[adaptive (dynamic) rough sets over dynamic granular spaces and networks in GrC]

adaptive (dynamic) rough sets over  dynamic granular spaces and networks in IGrC
%%%%%%%%%%%%%

\section{The Pawlak rough set model}\label{sec:evol-approx-basic}

\subsection{Approximation of concepts in basic model}\label{subsec:basic}

\subsection{Approximation of concepts in information systems}\label{subsec:inf-sys}

def

beginning of optimization



\section{Extensions of approximation spaces and their optimization}\label{sec:evol-approx-extension-optimization}


\subsection{Rough sets and granular computing}

elementary granules

aggregations of granules

aggregations of granules under constraints

inducing new granules

relations over granules

calculi of granules

\subsection{Illustrative examples of extensions of approximation spaces}\label{subsec:extensions}

Inclusion and quality measures
    Tolerance approximation spaces

From partitions to coverings

Networks of approximation spaces and networks of granules over networks of approximation spaces


\subsection{Beginning of adaptive approximation of concepts}\label{subsec:adaptive}


\section{Rough sets and Interactive granular computing: Introduction}\label{sec:RS-IGrC-Intro}

%%%%%%%%%%%%%%%%

\section{From Granular Computing to Interactive Granular Computing}\label{sec:GrC-to-IGrC}
In this section we presents a preliminary discussion of the Granular Computing (GrC) model and its extension, the Interactive Granular Computing (IGrC) model. Additionally, we add some comments on the rough set approach within these frameworks.

% and provide a roadmap for the further development of rough sets based on IGrC.

This paper builds upon previous works related to IGrC (see, {\em e.g.}, \cite{ch12-Pawlak-in-Mem-2012,ch12-SON-AS-Memoriambook2013,ch12-Rodos2021,ch12-CSP2021}
as well as \cite{ch5-InfSci-sub-2024,ch5-ASC-sub-AS-DS-2024}
and papers referred at \url{https://dblp.org/pid/s/AndrzejSkowron.html}).

This paper presents a discussion of the potential for the IGrC model to serve as a foundation for the design of Intelligent Systems (IS's) capable of addressing complex phenomena. IS's are regarded as a specific category of basic objects in IGrC, designated as complex granules (c-granules) with control. The linking of abstract and physical objects is a crucial aspect of modeling the perception of situations in the physical world by IS's. The control of c-granules is responsible for steering the computations generated over granular networks in a manner that achieves the desired target goals ({\em i.e.}, satisfies the specified requirements).
It is essential that the control of c-granules has a comprehensive understanding of the perceived situations in the physical world. This enables the relevant transformations of configurations of physical objects to be realized, which in turn facilitates the generation of high-quality computations over granular networks by c-granule with control (or Intelligent System (IS)).

%For example, the optimization of traffic lights can facilitate the smooth passage of cars through urban areas (see \cite{ch12-pawel-gora2024,ch5-AS-HSN-in-memoriam-book2013})

We present a generalization of the rough set approach in this context.

In essence, the control of c-granules is based on an efficient and online judgment mechanism of c-granules, thereby enabling c-granules to respond to the following queries:
\begin{itemize}
\item{} How to construct a list of the most pressing issues that require resolution in the context of the current situation?
\item{} How might one identify high-quality approximate solutions to the most pressing problems, given the constraints of real-world scenarios?
\item{} What is the optimal subsequent action, that is, which transformations should be activated in the current situation ({\em e.g.}, facilitating a more comprehensive understanding of the situation and/or taking a step towards achieving the desired outcome)?
\end{itemize}

One of the primary challenges in developing IS's that address complex phenomena is the creation of satisfactory computational models that implement adaptive judgment. This judgment is based on the granular computations generated by control mechanisms. The following sections will provide a more detailed discussion of these computations. In particular, we will examine how the approximations of concepts in IGrC are anchored in  c-granule with control, rather than relying on the a priori defined approximation spaces delineated by relational systems, as is typical in the current rough set approach.

The objective of adaptive judgment is to develop the capacity for sound judgment and the ability to select the optimal course of action based on knowledge, experience, and understanding (see \cite{ch12-wisdom-webseter}).


%%%%%%%%%%%%%%dodatek perception
%Another challenge in developing ISs is
%discovering methods for approximate reasoning from measurements to
%perception, {\em i.e.}, deriving from concepts resulting from sensor
%measurements concepts to expressions enunciated in natural language that express
%perception understanding. Nowadays, new emerging computing paradigms are investigated
%attempting to make progress in solving problems related to this
%challenge related to Perception Based Computing (PBC).
%
%There is a huge literature dedicated to perception (see, {\em e.g.}, \cite{ch12-perception-book2024,ch12-Barsalou-perceptual-symbol-sys}). Here, we would like only to mention that the essence of perception offered by IGrC is closely related to the approach presented in the book by No\"{e} \cite{ch12-Noe-book} (p. 1):
%%%%%%%%%%%%%%%%DeepL
Another challenge in the development of IS's is
in the discovery of methods for approximate reasoning from measurements to
perception, {\em i.e.}, from concepts derived from sensor
concepts derived from sensor measurements to expressions expressed in natural language that express
perception understanding. Today, new emerging computing paradigms are being explored to
to make progress in solving problems related to this challenge in
Perception Based Computing (PBC).

There is a huge literature dedicated to perception (see, {\em e.g.}, \cite{ch12-perception-book2024,ch12-Barsalou-perceptual-symbol-sys}). Here, we will only mention that the essence of perception offered by the IGrC is closely related to the approach presented in the book by No\"{e} \cite{ch12-Noe-book} (p. 1):

\begin{quotation}
 [...] {\em perceiving is a way of acting. Perception is not something that happens to us, or in us. It is something we do. Think of blind person tap-tapping his or her way around a cluttered space, perceiving the space by touch, not all at once, but through time, by skillful probing and movement. This is, or at least ought to be, our paradigm of what perceiving is. The world makes itself available to the perceiver through physical movement and interaction.}
\end{quotation}

%PBC provides capability to compute and reason with perception based
%information as humans do to perform a wide variety of physical and mental
%tasks without any measurement and computation. Reflecting the finite
%ability of the sensory organs and (finally the brain) to resolve details,
%perceptions are inherently granular.
%Boundaries of perceived granules ({\em e.g.}, classes) are unsharp and the values
%of the attributes they can take are granulated. In general, perception may be considered as
%understanding of sensory information. This point of view is, {\em e.g.}, presented in
%Computing with Words and Perception  (CWP) \cite{ch12-zadeh01new} which
%%%%%%DeepL
PBC provides the ability to compute and reason with perceptual information just as humans do, to perform a wide variety of physical and mental tasks without measurement and calculation. Given the finite
organs and (ultimately the brain) to resolve details, perceptions are inherently granular.
The boundaries of perceived granules ({\em e.g.}, classes) are fuzzy, and the values of the
of the attributes they can take are granular. In general, perception can be seen as
understanding of sensory information. This point of view is discussed, for example, in
Computing with Words and Perception (CWP) \cite{ch12-zadeh01new}:
\begin{quotation}
{\em derives from the fact that
it opens the door to computation and reasoning with information
which is perception -- rather than measurement-based. Perceptions
play a key role in human cognition, and underlie the remarkable
human capability to perform a wide variety of physical and mental
tasks without any measurements and any computations. Everyday
examples of such tasks are driving a car in city traffic, playing
tennis and summarizing a story}.
\end{quotation}
IGrC enhanced by dialogues with experts becomes  closely related to CWP.

It should be noted that for the design of IS's addressing complex phenomena, applications of rough sets or fuzzy sets in the framework of PBC can be very useful. However, this requires changing the approach from one based on objects closed in only abstract space to one based on interacting
objects that combine abstract and physical objects. This is the goal of IGrC.
This topic is discussed in more detail here.
%%%%%%%%%%%%%%%%%%

We emphasize the need for special reasoning techniques, called adaptive reasoning, performed over interactive granular computations aimed at generating high-quality approximate solutions to problems to be solved by IS's. This reasoning is also important for the discovery of complex patterns as computational building blocks for cognition (see \cite{ch5-Valiant}) in these computations.

One should note that there are several 'white spots' here. This means that we do not have satisfactory formal techniques supporting reasoning necessary for understanding the perceived situation to a degree making it possible to make the right decisions by IS's. Among them are reasoning techniques based on commonsense reasoning, experience based reasoning or analogy based reasoning. It is worthwhile to cite here the following opinion
\cite{ch12-Mitchell-analogy,ch12-Mitchell-intelligence} about analogy based reasoning belonging to the experience based reasoning domain:
\begin{quotation}
{\em The quest for machines that can make abstractions and analogies is as old as the AI field itself, but the problem remains almost completely open.}
\end{quotation}
This causes that dialogues of IS's with human experts are unavoidable.

We also outline an approach to dynamic information systems considered in the context of the control of c-granules. This approach differs significantly from dynamic information systems reported in the literature
(see, {\em e.g.}, \cite{ch12-CampagnerCD22, ch12-DongWC23, ch12-Wolski10}), where the dynamics are given a priori.

We hope to convince the reader that there are many challenges requiring development of new reasoning techniques leading from data perceived in interaction with the physical world to perception making it possible to understand the perceived situations, e.g., by constructing high quality approximations of complex vague concepts responsible for lunching the right decisions by IS's. Such complex vague concepts are components of complex games, which are crucial for the control of c-granules in the control of granular computations.

It should also be noted that this paper leads to opening new research perspectives for rough sets and fuzzy sets. Approximation methods are based on the above-mentioned reasoning approaches, not limited to the existing ones, often based, {\em e.g.}, on partial inclusion of sets only. In fact, the proposed approach to rough sets and fuzzy sets based on IGrC creates a step towards the development of the foundations of Artificial Intelligence (AI).

Our approach is influenced by the book \cite{ch12-smart-mach-think}, which stimulated our work on the foundations for the design and analysis of IS's not burdened with unnecessary formalisms of partial approaches.



\section{Preliminaries on Adaptive Rough Sets}\label{subsec:RS-future-adaptive-RS}

%\section{Higher Order Vagueness and Rough Sets: Toward Adaptive Rough Sets}\label{sec:HOV-RS}
In the definition presented in
this chapter, we have observed that the notion of boundary region is
defined as a crisp set. However, in \cite{ch10-keefe00book}, it is stressed that boundaries of vague concepts are not crisp. In this context, let us notice the crucial point in the definition of boundary region, as given above, lies in the fact that
this definition is relative to the subjective knowledge expressed
with respect to a set of attributes and sample of objects (universe of decision system). Different sources of information, may be called agents, may use
different sets of attributes for approximating a concept with the basis of the same set of objects. Hence, the
boundary region can change when we consider these different views.
Another reason behind the change of boundary region could be because of incomplete information about concepts at some point of time.
At some point of time, we only have information available for some samples of objects \cite{ch12-friedman01elements}.
Hence, when new objects appear again the boundary region may change.
Following the already existing literature it is well known that vague concepts cannot be approximated with a satisfactory quality by
{\it static} constructs such as induced membership/inclusion
functions, or models that are derived from a sample.
Understanding of vague concepts can be only realized in a process
in which the induced models are adaptively matching the concepts
in a dynamically changing environment. This conclusion seems to
have important consequences for further development of rough set
theory, in combination with fuzzy sets and other soft computing
paradigms, towards adaptive approximate reasoning. For further details readers are referred to
\cite{ch12-skowron05Marcus,ch12-SkowronSwiniarski05Regina}.

So, because of different aspects of changes, an agent's perception about a vague concept gets adapted with
dynamically changing environment and time. Thus, we obtain a family of lower approximations, upper approximations and boundary
regions of a considered vague concept in accordance with changing time (see Figure~\ref{fig:adaptive-rs}).

\begin{figure}[h!]
    \centering
    \includegraphics[width=0.7\textwidth]{adaptive-rs.pdf}
     \caption{Adaptive rough sets.}\label{fig:adaptive-rs}
    \end{figure}


From the above considerations it follows that for dealing with higher order vagueness one should consider all the above possibilities  in the formal definition of rough sets, and hence in the definition of information systems. In the following sequel we take an attempt to throw light on the issues regarding how to extend the present notion for information systems focusing on the aspects that different perspectives for a concept may come due to viewing a set of objects with respect to different sets of attributes, or perceiving the same concept with respect to different objects as well as attributes, or having change of perception with the appearance of new objects along the progress of time. Though we admit the phenomenon of higher order vagueness, we need to still fix the approximate understanding of a concept at some level, and for that we need to have different strategies for aggregating different perspectives of a vague concept. In order to aggregate information
available at different information system, a notion of interaction between information systems, mathematically which may be called \textit{infomorphism} following \cite{ch12-Barwise97book}, would play an important role.


The control of an agent is responsible for predicting values of parameters necessary for constructing the relevant current information system. This prediction is performed on the basis of knowledge accumulated in the memory of control.
The aim  of the control of an agent is to satisfy the needs of the agent by controlling computations on c-granules. The algorithms, called classifiers (or regressors), for predicting the values of parameters are induced on the basis of information dynamically accumulated by the agent in the form of interactive information (decision) systems. These systems are dynamically changing with time due to interactions of the control with the environment. The  process of inducing
classifiers (or regressors) is often supported using hierarchical learning ({\em e.g.}, \cite{ch12-WisIGrC17jaw,ch12-Skow:Szcz09,ch5-Bazan08TRSIX}).

Moreover, we would like to emphasize on the necessity of developing adaptive strategies on the basis of the history/memory of control, which can guide  how the information is gathered in such interactive systems, as well as how the structures of classifiers (regressors) are used in the past for predicting values of parameters. All these are inducing the high quality classifiers (regressors) for predicting values of the parameters for the current situation. The challenge is to develop methods for learning classifiers (regressors) for predicting adaptation of parameters following the agent's available learning about the already perceived changes in situations and in the classifiers (regressors). The induced classifiers (regressors) can be treated as the temporary approximations of decision functions (see  \cite{ch10-SON-AS-Memoriambook2013,ch10-Sorites16}).

\section{Introduction to Granular Computing} \label{sec:GrC}

Granular Computing (GrC) is an emerging paradigm that focuses on processing and managing information at different levels of granularity. The basic concepts of GrC, such as information granularity and information granules, have emerged in various contexts and numerous areas of AI, such as data analytics or computer vision.

%For example in Artificial Intelligence (AI) information granularity is central to a way of problem solving through problem decomposition. Information granules are represented and manipulated by intelligent systems communicating with the humans using information granules.

%GrC is evolving rapidly, with ongoing research exploring its theoretical foundations and practical applications. Its ability to manage complexity and uncertainty positions it as a valuable tool in the data-driven landscape of today. GrC effectively manages uncertainty by utilizing multi-level representations, approximate reasoning, and integrating theories like rough sets and fuzzy logic. These mechanisms enhance the ability to make informed decisions in complex, uncertain environments, making GrC a powerful tool in various applications where uncertainty is prevalent.
%Granular Computing (GrC) is directly associated with the pioneering studies by Zadeh \cite{ch12-Zadeh-InfoGran97}.
%He coined an informal, yet highly descriptive and compelling concept of information granule as
%a clump of objects (or points) which are drawn
%together by indistinguishability, similarity, proximity or functionality.

GrC is rapidly evolving, with ongoing research exploring its theoretical foundations and practical applications. Its ability to manage complexity and uncertainty positions it as a valuable tool in today's data-driven landscape. GrC effectively manages uncertainty through the use of multi-level representations, approximate reasoning, and the integration of theories such as rough sets and fuzzy logic. These mechanisms enhance the ability to make informed decisions in complex, uncertain environments, making GrC a powerful tool in various applications where uncertainty is prevalent. GrC is directly related to the pioneering work of Zadeh \cite{ch12-Zadeh-InfoGran97}.
He coined an informal but highly descriptive and compelling concept of an information granule as
a cluster of objects (or dots) that are bound together by indistinguishability, similarity, proximity, or functionality.


Information granulation can be viewed as a human way of achieving
data compression and it plays a key role in the implementation of
the strategy of divide-and-conquer in human problem-solving
\cite{ch12-zadeh01new}.
%\begin{quotation}
%{\em Information granulation can be viewed as a human way of achieving
%data compression and it plays a key role in implementation of the
%strategy of divide-and-conquer in human problem-solving.}
%\end{quotation}
Objects obtained as the result of information granulation or de-granulation are
information granules \cite{ch12-zadeh01new,ch12-Pedrycz-AS-VK-handGC08,ch10-Pedrycz-book-Intro-fuzzy2021}.

Granulation is inherent in human thinking and reasoning processes.
Information granules play an important role in human cognition, system modeling and decision-making activities.
GrC provides an information processing framework where computation
and operations are performed on information granules, and it is based on the
realization that precision is sometimes expensive and not much meaningful in modeling
and controlling complex systems.
GrC is about representing, constructing, processing, and communicating information granules. The concept of information granules is omnipresent and this becomes well documented through a
series of applications \cite{ch12-Pedrycz-GrC-Manifesto2018,ch12-bargiela03}.
Information granules can be of different type of abstraction and they can be treated as hints for the control of IS's (or control of c-granules we will consider later on) to carry over them reasoning leading to the right decisions.
Examples of information granules include indiscernibility or tolerance (similarity) classes, information or decision systems, clusters, decision rules, sets of decision rules, classifiers, and time series along with their various components  (see, {\em e.g.},~\cite{ch12-JarekBook08,ch10-skowron03information,ch12-JSAS2023}).
In reasoning about data
and knowledge under uncertainty and imprecision many compound information granules are used (see, {\em e.g.},
\cite{ch12-polkowski01rough,ch12-polkowski99towards,ch12-skowron01toward,ch12-skowron01information,ch12-skowron03information}).
%More compound information granules are defined by
%means of less compound ones. Note that inclusion or closeness
%measures between information granules should be considered rather
%than their strict equality. Such measures are also defined
%recursively for information granules.

The challenges arise in discovering the relevant information granules from the computations generated by IS's in relation to the problems they are aiming to solve. We will discuss this in more detail in the following sections.


In GrC are investigated information granules embedded in the abstract space. There is a large literature dedicated to GrC  (see, {\em e.g.},  \cite{ch12-bargiela-pedrycz-2003,ch12-Pedrycz-AS-VK-handGC08,ch12-pedrycz-book-2013,ch10-Pedrycz-book-Intro-fuzzy2021,ch12-Pedrycz-justifable2022}). However, in \cite{ch12-Pedrycz-overview-2023}
it is mentioned the lack of theoretical foundations of GrC. In the following sections we discuss some basic issues related to the foundations of GrC, in particular some issues related to questions like:
\begin{itemize}
\item{} How are information granules defined?
\item{} What are the objects over which information granules are defined?
\item{} How are different types of objects discovered over which information granules are defined?
\item{} How is a language for expressing information granules chosen?
\item{} How to choose measures to estimate the quality of information granules?
\item{} How to control the generation of information granules in order to efficiently obtain high quality granules?
\item{} Are information granules satisfactory for the design and analysis of intelligent systems (IS) dealing with complex phenomena?
\end{itemize}


\section[Granular spaces and granular networks] {From approximation spaces to networks of granular spaces and granular networks (networks of granules) over them}
\label{sec:approx-spac-to-networks-granular-spaces}


%In this section, we introduce one of the fundamental concepts essential for developing the foundations of GrC. This concept pertains to the generalization of the approximation space, as considered in the rough set approach.
%To recall, in Section~\ref{sec:mrel-learn} of Chapter~\ref{chap5}, we discussed the multirelational approach to rough sets. Any approximation space of this kind consists of a universe of objects and a set of relations defined over this universe. Structures formed by a set $S$ and an indexed family of relations $R_{\alpha}$ over
%$S$ are referred to as relational structures. Furthermore, the generalizations of approximation spaces discussed, such as those in \cite{ch5-ASJS1996}, are special cases of relational structures.
%In GrC, it is necessary to consider a generalization of this concept to granular spaces, which consist of objects known as granules. This generalization addresses the following issues:

In this section, we introduce one of the fundamental concepts that is essential for developing the foundations of GrC. This concept concerns the generalization of the approximation space as it is considered in the rough set approach.
In the multirelational approach to rough sets any such approximation space consists of a universe of objects and a set of relations defined over that universe. Structures formed by a set $S$ and an indexed family of relations $R_{\alpha}$ over
$S$ are called relational structures. Furthermore, the generalizations of approximation spaces discussed, {\em e.g.}, in \cite{ch5-ASJS1996} are special cases of relational structures.
In GrC, it is necessary to consider a generalization of this concept to granular spaces, which consist of objects called granules.
 This generalization addresses the following issues:
  \begin{itemize}
  \item{} Granules in the universe vary in type, understood as properties of sets of granules constructed (or discovered) from basic (generic) granules.
  \item{} These types are developed through the control of IS's (or the control of granules, which will be discussed later). This construction is supported by reasoning based on data sets, domain knowledge databases, and interactions with humans.
   \item{} The construction of types begins with basic granules, from which more complex types of granules are formed using aggregation operations applied to the basic objects or to granules already generated by these operations.
   \item{} Granules of different types are connected to each other through interfaces consisting of relations (functions) that specify the relationships between objects of different types. These interfaces also contain tools for evaluating the quality of granules, such as inference rules that allow the properties of higher-level granules to be inferred from those of the types from which they were constructed, as well as inference rules that justify the robustness of granule construction.
   \item{} Certain relationships between objects of a given type are identified, such as inclusion, partial inclusion, or proximity of granules within the considered space.
   \item{} The constructions described above require the discovery of appropriate languages to express granules and their properties.
   \item{} A granular space is a dynamic structure that is modified by control strategies (of IS's or granules with control). In particular, besides the ability to construct new types of granules, it is essential for control to include operations for decomposing previously used granules. This is necessary, {\em e.g.}, when control determines that the current granular space is ineffective in finding solutions to specific problems to be solved.
\end{itemize}


One can observe some analogies in the discussed issues to feature extraction (feature engineering) in Machine Learning (ML), where it is necessary to discover a language for expressing features and then to select the relevant features expressed in that language \cite{ch5-RussellNorvig-bookAI,ch5-friedman01elements}.
The discovery of relevant granular spaces that allow for the generation of high-quality computations over granular networks is an important challenge.

Let us now look at some concepts in more detail.


First, we introduce generic granular spaces. Each generic granular space consists of information granules defined by indiscernibility or similarity classes of indiscernibility or similarity relations, respectively, from given multi-relational approximation spaces or information systems together with relations of (partial) inclusion between such granules.
More formally, any generic granular space $GS$ is a pair \[(G, Rel),\] where $G$ is a set of granules and $Rel$ -- a set of relations between granules -- is defined relative to a given multi-relational approximation space $AS=(U,R)$ (where $U$ is a finite set of objects and $R$ is a set of indiscernibility or similarity relations over $U$) as follows.


\begin{itemize}
\item{} $G$ is a set of granules, {\em i.e.}, pairs $g=(syn(g), sem(g),$ where
\begin{enumerate}
\item[(i)] $syn(g)$ is a syntax of granule $g$ corresponding to $x \in U$ in the form of descriptors defining the signature of $x$;
\item[(ii)] $sem(g)$ denotes the semantics of $syn(g)$ in the form $[x]_r,$ where $x\in U,$ $r\in R$ and $[x]_r=\{y\in U: xry\}$ is the indiscernibility or similarity class defined by $x.$
\end{enumerate}
\item{} $Rel$ is a set of relations over granules from  $G$ representing (partial) inclusion of indiscernibility or similarity classes of granules.
\end{itemize}



%%%%%%%%%%%%%%%%%%%%
In the sequel we assume that for different moments $t$ different approximation spaces $AS_t=(U_t,R_t)$ are provided and related to them families $F_{gen,t}$ of generic granular spaces and strategies of generation of new granular spaces and interfaces between them.

We assume that a family $F_{gen}$ of generic granular spaces and strategies for generating new granular spaces and interfaces between them are given.
Networks of granular spaces are elements of $F_{gen}$ or are generated from $F_{gen}$ by applying a finite number of times to the granular spaces of $F_{gen}$ and already generated networks given transformations for generating new granular spaces and interfaces between them. Thus, any network of granular spaces is a generic granular space or a network generated from generic granular spaces.

%In the sequel we assume that for different moments of time $t$ are provided different information systems $AS_t=(U_t,R_t)$ and related to them families $F_{gen,t}$ of generic granular spaces and strategies of generation of new granular spaces and interfaces between them.

Any strategy of generation of new granular network $GN^{\prime}$ from a given one $GN$  is defined relatively to a distinguished granular space $GS_{0}=(G_0, Rel_0)$ already generated from $F_{gen}.$ The granular network $GN$ is transformed to $GN^{\prime}$  using an extension of $GS_{0}$ to a new granular space $GS^{\prime}$ with granules defined  by means of aggregation operations transforming granules from $GS_{0}$ (or Cartesian product of $GS_{0}$) to granules from $GS^{\prime}$ with possible use of constraints defined by relations from $Rel_0$. The interface created by these operations and constraints helps to establish relationships of properties of granules from these two granular networks.
In the consequence, the semantics of information granules in the new granular space $GS^{\prime}$ is defined relative to information granules from the granular space $GS_{0}.$ Hence, we have

\begin{definition}\label{def:gran-space}
Granular space $GS$ constructed from a given already constructed granular space $GS_{0}=(G_0, Rel_0)$ is a tuple
\[GS =(G, Rel),\]
where
\begin{itemize}
\item{} $G$ is a set of information granules of the form $g=(syn(g), sem(g)),$
\item{} $Rel$ is a set of relations over $G$ and
\item{} the semantics $sem$ of information granules $G$ is defined in terms of granules from $G_0,$ {\em i.e.}, $sem(g) \subseteq G_0$ for $g\in G.$
\end{itemize}
\end{definition}

Let us now explain the concept of interfaces between granular spaces. These interfaces provide tools for reasoning from sensory measurements through successive levels of granularity, ultimately reaching the highest level of granularity. This process allows the approximation of complex, vague concepts and facilitates the realization of transformations related to actions and plans.

To simplify the reasoning, we consider two granular spaces $GS, GS^{\prime}.$ An interface between them is defined by a set of relations and aggregation operations over these two granular spaces, as well as some reasoning rules. More formally, and interface
$Inter(GS, GS^{\prime})$ (or, $Inter$, in short) between $GS, GS^{\prime}$ is a tuple
\[(Rela, Fun, Rul), \]
where
\begin{itemize}
\item{} $Rela$ is a set of relations over granules, {\em i.e.}, subsets of Cartesian product of granules from $GS$ or $GS^{\prime}.$
\item{} $Fun$ is a set of aggregation operations from the set of granules of $GS$ (or Cartesian product of such sets of granules) into the set of granules of $GS^{\prime}.$
\item{} $Rul$ is a set of rules for inferring properties of granules obtained by aggregation from properties of  granules being aggregated. These properties are expressed by granules on the corresponding granulation layers.
\end{itemize}
Hence, this interface defines how  granules from $GS$ are related to  granules of $GS^{\prime}.$ In particular, the interface specifies how some granules from $GS^{\prime}$ are constructed from granules from $GS$ and some properties of these constructed granules from $GS^{\prime}$ are related to properties of granules from which they have been constructed.

One should note that $syn(g)$ represents a structure of granule $g$ consisting of, {\em e.g.}, description of granular components of $g$ as well as relations between them. For example, granule $g$ may be constructed from some granules of $G_0$ consisting of granules representing training signatures of objects together with binary decisions for them and relevant conjunctions of descriptors of the form $a=v,$ where $a$ denotes  an attribute and $v$ its value as well as relations of inclusion of these granules into decision classes of a given binary decision system. These conjunctions of descriptors and heir links with decisions correspond to decision rules.The constructed new granule $g$ may represent a rule based classifier with binary decision. Semantics  $sem$ is usually specified by a procedure for checking if granules from (a given subset of) $G_0$ (or cartesian product of $G_0$) belong to $sem(g).$ For the considered example $sem(g)$ represents the set of objects represented by signatures accepted by the classifier, {\em i.e.}, signatures of objects for which the classifier returns the value 1.

Relations from $Rela$ are used as constraints in aggregation of granules by aggregation operations from $Fun.$ The rules from $Rul$ are making it possible to reason about constructed granules by aggregation operations. Let us consider an illustrative example of such rule:
\[\frac{r_1(g_1),\ldots, r_k(g_k)}{r(f(g_1,\ldots,g_k))}.\]
This rule has the following meaning: if granules $g_1,\ldots,g_k$  from $GS$ have properties $r_1,\ldots,r_k,$ respectively then their aggregation $f(g_1,\ldots,g_k)$ by operation $f\in Fun$ has property $r.$ Granules $g_1,\ldots,g_k$ may represent, {\em e.g.}, clusters and $f(g_1,\ldots,g_k)$ may represent their aggregation.
This rule expresses a robustness of the construction performed by $f: G^k \rightarrow G^{\prime}.$ This may be explained formally  by
\[f(sem(r_1)\times \ldots \times sem(r_k))\subseteq sem(r),\]
where $sem$ defines semantics of relations, {\em i.e.}, $sem(r_i)\subseteq G, $ for $i=1,\ldots, k,$  $sem(r) \subseteq G^{\prime}$ and $f(sem(r_1)\times \ldots \times sem(r_k))$ is the image of the Cartesian product $sem(r_1)\times \ldots \times sem(r_k)$ under the operation $f.$

It is worthwhile to mention that the relations $r_1,\ldots, r_k,r$ may be defined by aggregation operations over
granules from $G$ and $G^{\prime}.$ For example, they may be defined using similarity or tolerance over basic granules from $G$ and $G^{\prime}$ by experts or discovered from data. If $sim \subseteq G\times G$, and $sim \subseteq G^{\prime}\times G^{\prime}$ are similarity relations in $G,G^{\prime},$ respectively and $g_1^0,\ldots,g_k^0\in G$, $g^0\in G^{\prime}$ are distinguished granules then
\[sem(r_i)= \{g\in G: sim(g_i^0,g)\}\]
and \[sem(r)= \{g\in G^{\prime}: sim(g^0,g)\}.\]
This means that $r_i(g)$ holds iff $sim(g_i^0,g)$ holds for $i=1,\ldots,k$  and $g\in G$ as well as $r(g)$ holds iff $sim(g^0,g)$ holds for $g\in G^{\prime}.$

Rules as described above can be composed under natural constraints. For example (see Fig.~\ref{fig-composition-of-rules}), from rules
\[\frac{r_1(g_1),\ldots, r_k(g_k)}{r(f(g_1,\ldots,g_k))} \mbox{\ and \ } \frac{r_1^{\prime}(g_1^{\prime}),\ldots, r_l^{\prime}(g_l^{\prime})}{r^{\prime}(h(g_1^{\prime},\ldots,g_l^{\prime}))}\]
and a constraint $r(g) \rightarrow r_1^{\prime}(g)$ one can derive a rule
\[
\frac
{r_1(g_1),\ldots, r_k(g_k), r_2^{\prime}(g_2^{\prime}),\ldots, r_l^{\prime}(g_l^{\prime})}
{r^{\prime}(h(f(g_1,\ldots,g_k),g_2^{\prime},\ldots, g_l^{\prime}))}.
\]

\begin{figure}[htb!]
\centering
\includegraphics[width=4.5in]{fig-composition-of-rules.pdf}
\caption{Composition of rules.}
\label{fig-composition-of-rules}
\end{figure}

%%%%%%%%%%%%%%%%%%%%hierarchical 20-02
%%%%%%%%%%%z pracy z JS
%\begin{example}
The composition of rules discussed above is related to hierarchical granule networks.To make the concept of hierarchical granule networks clearer, let us consider an illustrative example of hierarchical granules, as shown in Fig.~\ref{fig-hierarch-granule}.
 \begin{figure}[htb!]
\begin{center}
  \includegraphics[scale=.4]{fig-hierarch-granule.pdf}
 \caption{Illustrative hierarchical network of granules. }
 \label{fig-hierarch-granule}
\end{center}
\end{figure}
In Fig.~\ref{fig-hierarch-granule}, at different levels of the hierarchy are decision systems ($DS_i$) and their corresponding granular networks $G(DS_i)$, as discussed earlier (see Fig.~\ref{fig-new}). Granules from these networks can be used, {\em e.g.}, to connect different hierarchical levels by proposing new features at higher levels based on approximated regions at lower levels, or by granularizing objects according to their discovered similarities at lower levels. Links between granules at different levels are replaced by procedures that identify generalizations at the higher level. These procedures determine whether a lower-level object belongs to (or is part of) the identified generalization. At the lower level of the hierarchy, the decision system is updated through sensory measurement by the implementational module (IM) based on a given specification of the relevant association.

%%%%%%%%%%%dodatek JA 21-02-2025
Let us consider more details about the information flow in hierarchical networks of granules shown in Fig.~\ref{fig-hierarch}. On different hierarchical levels, granules represent algorithms ($Al$), which transform their input concepts,
represented by algorithms ($Al^{in}_i$) together with their current degrees of satisfiability ($\varepsilon_i$), into output concepts (also represented by relevant algorithms ($Al^{out}_i$)) together with their degrees of satisfiability ($\varepsilon_i^{\prime}$). Generally, the presented construction scheme can be extended by c-granules representing interfaces between $Al$ and $Al_i$ (and/or $Al_i$) responsible for encoding or decoding information transferred between algorithms.

 \begin{figure}[htb!]
\begin{center}
  \includegraphics[scale=.35]{fig-hierarch.pdf}
 \caption{Granule $g$ responsible for transformation of its input algorithms defining concepts together with their satisfiability degrees into its output algorithms defining concepts with their satisfiability degrees. }
 \label{fig-hierarch}
\end{center}
\end{figure}

One can consider many different schemes of algorithmic transformation shown in Fig.~\ref{fig-hierarch}. For example
\begin{itemize}
\item{} $Al$ transforming approximate regions (in rough sets \cite{ch5-pawlak81information,ch5-pawlakSkowron07InfSci}) with their membership degrees of input concepts into approximate regions of output concepts with their membership degrees;
\item{} $Al$ (represented by a rule of fuzzy logic \cite{ch10-Pedrycz-book-Intro-fuzzy2021}) transforming input fuzzy concepts with their membership degrees into output fuzzy concepts with their membership degrees;
\item{} $Al$ transforming value vectors of conditional attributes from a given decision system into left sides of decision rules \cite{ch10-pawlakSkowron07InfSciC};
\item{} $Al$ transforming a collection of decision rules into a rule-based classifier \cite{ch10-pawlakSkowron07InfSciC};
\item{} $Al$ transforming value vectors of conditional attributes from a given decision system into clusters of similar objects \cite{PolkArtj-2015};
\item{} $Al$ transforming an input collection of classifiers into their ensemble \cite{ch5-ensemble}.
\end{itemize}

From the above discussion, it can be seen that the satisfiability degrees of concepts represented by algorithms
can be generated on the basis of relevant degrees of some concepts from the previous hierarchical levels.
It is worth mentioning here that satisfiability degrees of concepts can also be obtained by
by associations, which are realizations of algorithms representing concepts in the real physical world.
This topic will be discussed in more detail later.
Moreover, in applications it may be relevant to have algorithmic schemes not with exact satisfiability
degrees of concepts but with, {\em e.g.}, inequalities such as {\em at least to a degree} $\varepsilon_i$ instead of numerical values of $\varepsilon$ \cite{ch10-Pedrycz-book-Intro-fuzzy2021}. In advanced applications, satisfiability degrees can also be granules with compound structure. As an example one can consider linguistic variables \cite{ch10-Pedrycz-book-Intro-fuzzy2021} or collections of arguments {\em for} and {\em against} expressed in natural language \cite{argumentation2025}.

As previously mentioned, the ability to discover composable rules can be used to derive explainable conclusions from deep derivation trees. First, it is crucial to identify a relatively small number of interpretable levels within the derivation process. This should be supported by dialogue with humans. Next, one can "compress" the deep derivation trees by keeping only those levels that contain these interpretable concepts and the results of the derivations between them, provided by the relevant composition of derivation subtrees to rules between these levels.
The efficacy of this approach in enhancing explainability in practical, real-life scenarios will necessitate
 extensive experimental research, which is planned as part of our future work.
%%%%%%%%%%%%%%%


One may refer to the distinction between reactive and deliberative control in agents or robots \cite{ch5-RussellNorvig-bookAI}. Reactive control relies on rules that provide decisions based on sensory measurements, while deliberative control involves reasoning that leads to decisions. In our case, reasoning is based on searching for rules that, through multiple compositions, yield a decision. This may require the control system to engage in additional reasoning to determine what new measurements are necessary to obtain. The approximate reasoning schemes related to the composition of reasoning rules over granules are discussed in \cite{ch10-skowron03information}.


Networks of granular spaces are constructed from a family of granular spaces and interfaces between them.

The existing networks of granular spaces may be extended using strategies making it possible to define a new granular spaces, establishing new interfaces of them  with the already defined granular spaces and constructing on this basis new granular networks.

Networks of granules over a given network of granular spaces $Net$ (or network of granules, in short) consists of selected instances of granules from $Net$ together with a set of instances of facts expressed  by relations from $Net$ on some selected instances of granules from $Net.$ Facts have the following form $r(g_1,\ldots, g_k),$ where $r$ is a relation form $Net,$  $k$ denotes the arity of $r$ and $g_1,\ldots, g_k$ are instances of granules from $Net.$


%%%%%%%%\ pracy AS-JS
\subsection{Illustrative example of granular spaces and networks of granules for the Pawlak rough set model and its generalizations}\label{sec:GN-Pawlak-model}


In this section, we recall the Pawlak model \cite{ch5-pawlak82rough,ch5-pawlak91rough,ch5-pawlakSkowron07InfSci} and using this model we distinguish some important components of approximation process related to it. In the abstract setting the approximation space was defined as a pair $(U,R),$ where $U$ is a finite set of objects and $R\subseteq U\times U$ is an equivalence (indiscernibility) relation. This relation defines a partition of $U$ into equivalence classes called indiscernibility classes. These classes  $[x]_R$ (for $x\in U$) are elementary granules. The aggregation operation on granules is defined producing new granules from families of elementary granules by taking  the union of them. One should note that in the Pawlak model the universe $U$ is treated as given set and the issues how objects form this set are perceived in the physical world are outside of this model.


Together with the indiscernibility relation is considered a partition of $U$ into decision classes (decision granules). Let us denote the corresponding equivalence relation to this partition by $R_d$. In this way we obtain a triple $AS=(U, R, R_d)$ called approximation space. For any decision granule $X\subseteq U$ is defined its lower approximation relative to $R$ by $LOW(AS,X)=\{x\in U: [x]_R\subseteq X\}$\footnote{We will write also $LOW(R,X)$ instead of $LOW(AS,X).$} and the upper approximation of $X$ by $UPP(AS,X)=\{x\in U: [x]_R\cap X \neq \emptyset\}.$
The set $Bd(AS,X)= UPP(AS,X)\setminus LOW(AS,X)$ is called the boundary region of $X$ relative to $R$. If $Bd(AS,X)\neq \emptyset$ then $X$ is called rough (relative to $R$), otherwise $X$ is crisp (relative to $R$).

Hence, one can observe that in this model we deal with two granular spaces $G_R$ and $G_{R_d}$ corresponding to the partitions defined by relations $R$ and $R_d$:  $G_R=(\{[x]_R\}_{x\in U}, agg),$ where $agg: P(\{[x]_R\}_{x\in U})\rightarrow P(U)$ and  $agg(Z)= \bigcup Z$ for $Z\subseteq \{[x]_R\}_{x\in U}$ and $G_{R_d}=U/R_d$ where $U/R_d= \{[x]_{R_d}\}_{x\in U}$. The first component of $G_R$ contains a family of elementary granules defined by $R$ and the second component is an aggregation operation generating definable granules from elementary granules. In the case of $G_{R_d},$ we have only one component with the family of decision granules.

The granular spaces $G_R, G_{R_d}$ are linked by an interface $Inter(G_R, G_{R_d})$ including some relations between granules from these two granular spaces as well as some reasoning methods making it possible to find relevant relationships between granules from these spaces. In the case of the discussed model this interface has a simple form and is defined by
\begin{itemize}
\item{} the set theoretical inclusion relation between definable granules from $G_R$ and granules from $G_{R_d},$

\item{} The set  theoretical 'non-empty intersection' relation between  definable granules from $G_R$ and granules from $G_{R_d}$  and

\item{} $Res$ -- reasoning module (RM) consisting of reasoning tools supporting, {\em e.g.}, definition of approximation regions such as lower approximations of decision classes or boundary regions; construction of searching heuristic for the maximal (relative to set inclusion) definable granules included into decision granules for the lower approximation of decision granules and minimal (relative to set inclusion) definable granules including decision granules for the upper approximation of decision granules.
\end{itemize}

In Fig.~\ref{fig-RS-GS} is presented an example of network of granular spaces for the basic Pawlak rough set model.

\begin{figure}[htb!]
\centering
\includegraphics[width=5in]{fig-RS-GS.pdf}
\caption{Example of network of granular spaces for the Pawlak rough set model.}
\label{fig-RS-GS}
\end{figure}


In the example, two granular spaces $G_R$ and $G_{d}$ are shown. For simplicity only elementary granules defined by indiscernibility classes are considered. For granules $g_x=(syn(g_x),sem(g_x)),$ where $x\in U,$  $syn(x)$ is the name of the indiscernibility class defined by $x$ (using any bijection $f: U/R \rightarrow \{1,\ldots, \mid U/R\mid \}$) and $sem(x)$ is the indiscernibility class defined by $.$. In the interface two relations are, namely the inclusion relation between semantics of granules as well as the hitting relation making it possible to check if semantics of two granules have non-empty intersection. $Res$ consists mechanisms for reasoning leading to construction of approximation regions, corresponding to the lower approximation and upper approximation of  decision classes and the boundary region.

From the granular space presented in Fig.~\ref{fig-RS-GS} granular networks are generated. In the considered example, this is realized by selection of some granules from these two granular spaces $G_R$ and $G_d$ and by adding links between these granules pointed out by reasoning mechanism related to construction of approximation regions.

%On the basis of the reasoning mechanism mentioned above, the current configuration (based on  granular spaces and interface) is changing by adding new links between granules from $G_R$ to $G_{R_d}$ recording  discovered by the used heuristic new inclusions and non-empty intersections of elementary granules from $G_R$ into decision granules $G_{R_d}$ (see Fig.~\ref{fig-new}).

\begin{figure}[htb!]
\centering
\includegraphics[width=5in]{fig-RS-GN.pdf}
\caption{Example of granular space for the Pawlak rough set model generated from the network of granular spaces presented in Figure~\ref{fig-RS-GS}.}
\label{fig-new}
\end{figure}

The illustrative example in Fig.~\ref{fig-new} showcases a simple granular network summarizing our considerations.  The network is built from two granular spaces, $G_{R}$ and $G_{R_d}$, which consist of elementary information granules (equivalence classes). These granules are defined by conditional attributes that determine the relation $R$ in the universe of objects $U.$ Additionally, a decision attribute determines the relation $R_d$ in $U.$ Both the description of these elementary information granules and their meaning  within the universe $U$ are defined in an abstract (mathematical) space. The network itself is constructed from two sets of elementary granules, one from each space. Arrows connect these granules, indicating two types of relationships: inclusion or partial inclusion. In general, these links can represent more complex relationships between granules from different, more advanced granular spaces.
For instance, information granules might contain details about the location and time of a measurement stored in their informational layers. They could also concern properties of various segments of multi-time series data or the aggregation of this data into clusters or more complex structural objects. Notably, these granules can also include specifications for so-called associations, which allow the control of IS's (c-granules) to generate their meaning in the physical world (what will be discussed later in more detail). Links can even represent relationships between different granules or their parts that are dependent on time.

Specifications  may be in the form like in the case of information granules but also in the form of specification of association. In the case of association the predicted results of physical realization may be different from the real ones obtained in perception of the physical world what causes that control of c-granule should be aware of this and adapt of the currently used complex game or modify accordingly the target goals, if modification can't lead to possibility that the generated computation will be of a satisfactory quality.

We have already seen that in networks of granular spaces, different universes can consist of different objects (granules). An example shown in Fig~\ref{fig-RS-GS-induced} illustrates that one universe of granular spaces can be an extension of another.

\begin{figure}[htb!]
\centering
\includegraphics[width=5.4in]{fig-RS-GS-induced.pdf}
\caption{Example of granular space induced from the granular space of the Pawlak rough set model.}
\label{fig-RS-GS-induced}
\end{figure}

At the bottom of this figure is a (simplified) network of granular spaces related to Pawlak's rough set model. In $Res$ there is a relation $r$ connecting elementary granules defined by indiscernibility classes of conditional attributes with decision classes in which they are contained. The upper part of this figure shows another network of granular spaces in which the universe $U^*$ is an extension of the universe $U$. This time in $Res^*$ consists of an inductive extension $r^*$ of the inclusion relation  $r$ of granules based in our simple illustrative example on the idea of 1-kNN \cite{ch5-RussellNorvig-bookAI}.

%%%%%%%%%%%%%%%


In this way, granular computations have states representing granular networks and the discovered links.
The IS control specifies the realized transitions between states. In the final state, all elementary granules are linked to decision granules and approximations of decision granules are computed. This simple example illustrates an important aspect of approximation processes running over structures defined by granular spaces and interfaces between them along  which are constructed approximations of specified target granules. In the discussed example decision granules play the role of target granules. In different applications different complex objects of the high quality required to be constructed, {\em e.g.}, learning algorithms, classifiers, clusters, complex physical or abstract objects, or the whole granular computations with the required properties.

%\begin{figure}[htb!]
%\centering
%\includegraphics[width=4.5in]{fig-GS.pdf}
%\caption{Illustrative example of granular space.}
%\label{fig-GN}
%\end{figure}


The current state of approximation process is represented by granules from $G_R$ and $G_{R_d}$ which have been used in reasoning (and not yet forgotten) and already established links between granules from $G_R$ to granules from $G_{R_d}$ representing inclusion. For example, granules from $G_R$ linked to at least two decision granules from $G_{R_d}$ belong to the boundary region.

In applications, the relation  $R$ is defined by a finite set of attributes $A,$ where any $a\in A$ is a function from $U$ into a value set $V_a.$
For any $x\in A$, $Inf_A(x)$ denotes the signature of $x$ relative to $A$ defined by $\{(a,a(x)): a\in A\}.$ For any $x,y\in U$ it is assumed that    $xRy$ if and only if  $Inf_A(x)=Inf_A(y)$. Then $R$ is denoted by $R_A.$
 In the considered case, the definition of granules is modified accordingly. Elementary granules are equivalence classes of $R_A$ labeled by signatures of objects defining them, {\em i.e.}, for any $x\in U$ the equivalence class $[x]_{R_A}$ is labeled by $Inf_A(x).$
 Instead of signature $Inf_A(x)$ one can consider the conjunction $\bigwedge_{a\in A} a=a(x).$  The semantics of $\bigwedge_{a\in A} a=a(x)$ is defined by
 $\parallel \bigwedge_{a\in A} (a=a(x)) \parallel_U = \{y\in U: a(y)=a(x) \mbox{\ for\ } a\in A \}.$
  Obviously, \[[x]_{R_A}=  \parallel \bigwedge_{a\in A} (a=a(x)) \parallel_U.\]

  Hence, the granule $g_x$ corresponding to $x\in U$ can be treated as a pair $(\bigwedge_{a\in A} (a=a(x)), [x]_{R_A}),$ with its syntax  $\bigwedge_{a\in A} (a=a(x))$  and  the semantics $[x]_{R_A}.$
It is worthwhile mentioning that this approach is important when one would like to consider inductive extensions of approximation spaces. In this case the aggregation operation in $G_{R_A}$ is restricted to unions of elementary granules defined by formulas obtained by dropping some conditions from $Inf_A.$ Among such formulas the computational building blocks for approximation of decision concepts from $G_{R_d}$ are selected. Any equivalence class $[x]_{R_A}$ is linked to a formula $\alpha_x$ obtained from $\bigwedge_{a\in A} (a=a(x))$ by dropping the maximal number of conditions under assumption that the semantics of $\alpha_x$ has non-empty intersection with exactly the same decision granules from $G_d$ as the semantics of $\bigwedge_{a\in A} a=a(x).$ Then, {\em e.g.} for $X\in U/{R_d}$ its lower approximation is obtained by the union of semantics of all formulas $\alpha_x$ included in $X.$ Hence, in this case we consider reasoning supporting linking the elementary granules $[x]_{R_A}$ from $G_{R_A}$ with $\alpha_x$  and $\alpha_x$ with decision classes for $x\in U$.

The process of inducing approximations of decision granules can be  realized on tolerance decision systems (for more information about tolerance rough sets see e.g. \cite{ch5-ASJS1996,ch12-JS2008}). The three-way  rough set based approach for approximation of decision granules in Intelligent Systems is discussed in  \cite{ch12-JSAS2023}.


An important aspect of approximation is related to the quality of resulting approximation of decision granules from $G_{R_d}$. In the discussed model, the positive region of $G_{R_d}$ relative to $G_R,$ i.e., \[POS_{G_R}(G_{R_d})= \bigcup_{X\in G_{R_d}} LOW(R,X)\] is used to define the quality of approximation of $G_{R_d}$ relative to $G_R$ by \[\gamma=\frac{|POS_{G_R}(G_{R_d})|}{|U|},\] where $|U|$ denotes the cardinality of $U.$

However, in applications it may be important also the description length of representation of the constructed approximations. Shorter descriptions may be preferred over longer from the point of view of readability or explainability of approximations. In this case, reasoning methods based on reduction of the length of representation may be applied. In particular, Boolean reasoning can be applied to support search for the minimal description of approximations of decision granules or classification defined by them. Efficient methods for generation of different kinds of reducts have been developed and used for this purpose \cite{ch10-brown90boolean,ch10-skowron92discernibility,ch5-Son06TRSV,ch10-pawlakSkowron07InfSciC}. In the discussed case, one can add a second component to the quality of approximation of decision granules, namely the component related to the description length.

The quality of approximation in the case of Pawlak's model is related to the given universe of objects $U$. This approach can be extended to the case when $U$ is only a training sample and the derived approximations should be also extended on testing objects (cases) not seen so far, like in Machine Learning. This issue will be discussed shortly in one of the following sections.  We would like to only stress here that in such a case one more component related to testing the derived models of approximation should be developed.

Let us illustrate the ideas of constructing granular spaces and granular networks over the by one more example illustrated in Fig.~\ref{fig-GS} and Fig.~\ref{fig-GN}, respectively.

\begin{figure}[htb!]
\centering
\includegraphics[width=5in]{fig-GS.pdf}
\caption{Illustrative example of granular space.}
\label{fig-GS}
\end{figure}

The granular space $G_{A}$ consists elementary granules defined by indiscernibility classes of the indiscernibility relation $IND(A).$ In the granular space $G_{P(A)}$ we have all indiscernibility classes of indiscernibility relations $IND(A^{\prime}),$ where $A^{\prime}\subset A.$ Hence, the number of granules in
$G_{P(A)}$ is exponential relative to the cardinality of $A.$ The interface between the granular spaces $G_A$ and $G_{P(A)}$ consists relation of inclusion of semantics of granules. In the granular space $G_d$ granules are decision classes defined by the indiscernibility relation $IND(d).$ The interface between the granular spaces $G_{P(A)}$ and $G_{d}$ consists relation of inclusion of semantics of granules. One of the rules of reasoning of the considered network of granular spaces is related to transitivity of inclusion.

\begin{figure}[htb!]
\centering
\includegraphics[width=5in]{fig-GN.pdf}
\caption{Illustrative example of granular network over granular space presented in Fig.~\ref{fig-GS}.}
\label{fig-GN}
\end{figure}

In the granular network presented in Fig.~\ref{fig-GN}, the second component corresponding to $G_{P(A)}^{\prime}$ is obtained by reducing the number of granules to such indiscernibility classes which are maximal relative to set inclusion (minimal with respect to the number of descriptors describing them) and also preserve  'hitting' property expressed by non-empty intersection with decision classes of granules from $G_A.$ Granules from $G_{P(A)}^{\prime}$ are obtained by generalization (by dropping some descriptors over attributes from $A$ describing them). The precise definition of hitting' property is presented in Fig.~\ref{fig-GN}. To the relation of inclusion is added the relation of non-empty intersection of granules. Let  us observe that the selection process of granules in the second component can be supported by Boolean reasoning and the discussed example is related to minimal decision rules (see, {\em e.g.}, \cite{ch5-Son06TRSV,ch10-SON-AS-Memoriambook2013}).  Obtained by Boolean reasoning heuristics may be parts of interfaces in the considered example.





%%%%%%%%%%%%


It should be noted that in networks of granular spaces and granular networks in different components may appear granules with semantics in different universes of objects or granules. To explain this we present illustrative examples.

Our first example concerns a role of interface in communication of different granular spaces or networks. Let us consider an information system $IS=(U,A)$ with $a: U \rightarrow V_a,$ where $V_a$ is a value set of attribute $a.$ We also assume that for any attribute $a$ is given a relational structure $Rel_a=(V_a, R_a),$ where $R_a$ is a set of relations over $V_a,$ {\em i.e.}, for $r\in R_a,$ $r\subseteq V_a^{ar(r)},$ where $ar(r)$ is called the arity of relation $r.$ For example, $V_a$ can be a set of reals and $R$ can consists the relation $r$ of linear order over $V_a.$ Moreover, together with the relational structure $Rel_a$ we consider a set of formulas $F_a$ and its semantics $sem: F_a \rightarrow P(U)$ assigning to any $\alpha\in F$ its meaning $sem(\alpha)subset U.$ In this way is defined a granular space $GS$ with the set of granules $G$ equal to a family of granules $(\alpha, sem(\alpha)$ for all attributes $a\in A,$ families $F_a$ and semantics $sem.$ As relations between granules from $G$ we consider inclusion relation. One can extend this example by considering instead of a set of formulas  $F_a$ a family of such sets. Moreover, one can consider another extension by considering relational structures over Cartesian products of different value sets of attributes and relations over such products. The new defined granular space  can be treated as a large potential set of sensory granules from which the next granular space should  select attributes used for perceiving objects objects. The next granular space corresponds to a decision system $DS=(U,Atr,d),$ where $Atr$ is a set of attributes selected from the granular space $GS.$ In this new granular space $GS^{\prime}$ the set of granules $Atr$ is a finite set of formulas selected from granules in $GS$ together with its semantics. In this granular space, one can consider relation inclusion between granules defined by these granules and granules defined by the decision $d.$
From the described granular network consisting $GS, GS^{\prime}$ granular networks are created by selecting some granules from $GS$ and linking them to granules in $GS^{\prime},$ in particular to granules defined by the decision attribute $d.$

Our previous example pertained to sensory granules. The example we are going  to present is related to discovery of new attributes from already defined ones. Let us consider a granular space related to an information system $IS=(U,A)$ with the set of granules $G$ defined by indiscernibility classes of $IND(A)$ and the empty set of relations between granules.
We assume that there is a one distinguished attribute $t\in A$ representing time with the values set $V_t=\{1,\ldots,n\}$.
We define the second granular space $GS^{\prime},$ in which the set of granules $G^{\prime}$ consists of granules from $G$ as well as granules generated by  operations from a given set $Op$  applied to $IS.$  For our example, we assume that $Op$ has only one operation. First of all, any object of the semantic space $U_{time}$ of these granules is a time window of the length $T$, {\em i.e.}, a sequence of the length $T$ of terms being value vectors of attributes obtained from signatures of objects from $U$ such that each next term of the sequence has the value of $t$ one greater than the preceding term. Over such objects are defined granules which are sets of time windows with semantics expressed by formulas from a given set of formulas. Using these new generated granules we would like to approximate decision classes of a decision $d_{time}:U_{time}\rightarrow V_{d_{time}}.$ These two granular spaces $GS,GS^{\prime}$ are linked by interface $Inter(GS,GS^{\prime})$ consisting of relations of inclusion and hitting as well as $Res$ with mechanisms supporting reasoning for selection of the relevant granules from  $GS^{\prime}$ for approximation decision classes of the decision $d.$ Certainly, one can generalize this example by considering more than one operation or by considering new granules  obtained by application of a finite number of times operations to $IS$ or already generated granules.

The reader familiar with Machine Learning (ML) will observe that the discussed examples pertain to the main considered in ML problems: feature selection and extraction (feature engineering) \cite{ch5-friedman01elements,ch5-RussellNorvig-bookAI,ch5-Son06TRSV,ch10-SON-AS-Memoriambook2013}.
The mentioned extension of the second example is corresponding to hierarchical learning (see, {\em e.g.}, \cite{ch12-stone00layered,ch12-Bazan08TRSIX}). The main challenge is to provide reasoning mechanisms for developing efficient heuristics searching for relevant granules in granular space $GS$  making it possible to construct the high quality approximations of complex vague concepts at the highest level of the hierarchy.

Granular networks are modified by control mechanism of granules to achieve specific target goals. Based on the current granular network, a transformation is selected by control on the basis of its set of rules to change it into a new granular network that represents the next step in granular computation. This control aims to generate granular computation over granular networks that satisfy a given specification (to a satisfactory degree). This specification may pertain to the final state of the computation or encompass the entire computation, depending on the task performed by the granule.
The quality of computations is assessed using selected quality measures. Various quality measures have been developed in Machine Learning (ML) and across different application areas, such as risk management, algorithmic trading, and drug discovery \cite{ch12-Japkowicz-book-2011,ch12-RiskM-book-2019,ch12-AlgTrad2020-book,ch12-Drug-Disovery-book-2021}. For instance, in ML, the Minimum Description Length (MDL) Principle \cite{ch5-rissanen85} and various measures based on the confusion matrix \cite{ch12-Japkowicz-book-2011} are well-known for estimating the quality of learning algorithms and classifiers.
In some applications, it may be necessary to maintain certain properties ({\em e.g.}, the 'safety' of the system) represented by a granule as invariant throughout the computation over granular networks. In other cases, it may be required to enrich a granular network with properties that express, {\em e.g.}, that a required drug has just been synthesized under constraints related to the costs of executing the computation.


More details on control of granules will be presented in a subsequent section. Before that, we will provide an outline of Interactive Granular Computing (IGrC).  In IGrC, we address complex granules (shortened as c-granules) that connect objects from both abstract and physical realms. It is worthwhile to mention that IS's can be viewed as compound c-granules (with control). The extension of GrC to IGrC is essential for addressing the perception of situations in the physical world as perceived by IS's. Understanding perceived situations in the physical world is crucial for IS's, especially those dealing with complex phenomena, for making by them the right decisions pertained to these situations. These decisions concern transformations. The  implementations of specification of transformations in the physical world are aiming to accordingly control granular computations of the considered IS (or control of c-granule) for achieving the target goals of IS.

For IS's dealing with complex phenomena two kinds of granules should be considered: (i) information granules defined in GrC by  formulas over the relevant language with semantics in the space of granules at a given hierarchical level and (ii) complex granules (c-granules) in IGrC with physical semantics realized in the real-physical world ensuring perceiving some properties of objects and configurations of physical objects as well as their interactions.

It is essential to recognize that the control of IS must continuously interact with the physical environment to respond to changes in perceived situations caused by environmental interactions. Simply collecting data from these interactions once is inadequate for developing a high-quality model. The models created must be adapted to account for any perceived substantial changes in the physical space. Additionally, the control of IS should be equipped with appropriate reasoning techniques to support its behavior. These topics will be explored in greater detail in the following sections.


\section{Interactive granular computing (IGrC)}\label{sec:IGrC}

%%%%%%%%%%%praca zDS
\subsection{IGrC - motivation, basic intuition and concepts}\label{subsec:igrc}

In this section, we present an intuitive explanation of some basic concepts related to IGrC. We start from motivations for generalization of GrC to IGrC.

The GrC model is inadequate for modeling IS's that address complex phenomena. The primary reasons for introducing granules with both informational and physical layers in the IGrC framework can be summarized as follows.
\begin{itemize}
\item{} Any IS dealing with complex phenomena must engage with the physical world to accurately perceive situations within it. The GrC model operates solely in an abstract space, yet interactions with the real physical world are essential for IS's to effectively interpret situations. Relying exclusively on abstract modeling of perception is insufficient. For modeling perception, it is crucial to align language and reasoning with perception and action \cite{ch12-Turing-test-mod}. Modeling perception requires granules that integrate both informational and physical layers, ensuring these layers interact appropriately. This is the focus of IGrC.

\item{} Constructing high-quality data models, granules, and computational building blocks necessary for cognition \cite{ch5-Valiant} ({\em i.e.}, understanding perceived situations) based on a predefined information system (data set) is not adequate (see, {\em e.g.}, the opinion cited below \cite{ch12-brooks1995mythical}). Continuous interaction between IS's and complex phenomena as well as proper steering them in the physical world is essential for understanding perceived situations.

\item{} Engaging in dialogues with domain experts is vital for IS's that address complex phenomena \cite{ch12-human-centered-AI,ch12-human-in-the-loop}.
\end{itemize}

One of the most important issue related to complex systems as well as IS's dealing with complex phenomena is understanding interactions \cite{ch12-Omnici06}:
\begin{quotation}
[...] {\em interaction is a critical issue in the understanding of complex systems of any sorts: as such, it has emerged in several well-established scientific areas other than computer science, like biology, physics, social and organizational sciences.}
\end{quotation}
Interactions take place between physical objects. Hence, IS (c-granule with control) should have skills to implement in the physical world specifications of configurations of physical objects and perceive properties of these objects as well as interactions between them. These configurations are created using special transformations called associations. Control of c-granule is equipped with the implementational module (IM) for (i) constructing  physical semantics of associations  consisting of configurations of the physical objects, (ii) initializing interactions of such configurations (by encoding relevant information in the physical objects which are directly accessible by control), (iii)  perceiving some properties of objects in these configurations and their interactions (by decoding information from the physical objects directly accessible by control or by judgment (a special kind of reasoning) based on already perceived information, domain knowledge bases or physical laws (computations should be  dependent on physical laws!). Hence, information granules from GrC are generalized to complex granules (c-granules, in short) in IGrC with granules consisting of two layers: informational and physical. Control of c-granule provides possibility of interaction between these two layers. It consists of a set of rules which are used for triggering realization of transformations in the abstract layer or associations in the physical layer with the support of IM.
In the consequence,  computations over granular networks are generalized to granular networks over c-granules. From the discussion above it follows that crucial for the IGrC model is also perception model. Let us recall that in IGrC we follow the already cited idea of perception presented in \cite{ch12-Noe-book}.


%%%%%%%%%%%%%%%%%%praca z DS
The IGrC model leverages existing partial results from various fields (including multi-agent systems,
perception and action, machine learning, natural language processing, federated learning, cognitive networks,
smart cities, cyber-physical systems, complex adaptive systems etc.)
by an attempt of putting them into sync.

We emphasize the fact that IS's often are dealing with complex phenomena in the physical world what requires a new kind of modeling for solving problems with the design and analysis of IS.
Particularly, this requires grounding the approach on the relevant computing model.
%%%%%%%%%%%%

We have selected the IGrC model  as the basis for developing theoretical foundations for the design and analysis of IS's dealing with complex phenomena in the physical world.
In the considered case, according to opinions of top researchers,  classical mathematical modeling is not satisfactory. For example, in \cite{ch12-brooks1995mythical} one can find the following opinion by Frederick Brooks, Turing award winner:
\begin{quotation}
{\em Mathematics and the physical sciences made great strides for three centuries by constructing simplified models of complex phenomena, deriving, properties from the models, and verifying those properties experimentally.
This worked because the complexities ignored in the models were not the essential properties of the phenomena. It does not work when the complexities are the essence.
}
\end{quotation}
%%%%%%%%
Let us also cite another opinion by Vladimir Vapnik  expressing the need for considering the physical world as the basis for computations related to learning of problems in applications \cite{ch12-vapnik98statistical}:
\begin{quotation}
[...] \em {further study of this} [learning] \em {phenomenon requires analysis that goes beyond pure mathematical models. As does any branch of natural science, learning theory has two sides:}
\em{
 \begin{itemize}
\item{} The mathematical side that describes laws of generalization which are
valid for all possible worlds and
\item{} The physical side that describes laws which are valid for our specific
world, the world where we have to solve our applied tasks.
\end{itemize}
}
\end{quotation}
\begin{quotation}
[...] {\em {To be successful, learning machines must use structures on the set of functions that are appropriate for problems of our world.}} [...] {\em {Constructing the physical part of the theory and unifying it with the mathematical part should be considered as one of the main goals of statistical learning theory.}}
[...] {\em {In spite of all results obtained, statistical learning theory is only in its infancy...}}
\end{quotation}
Accordingly to Vapnik \cite{ch12-vapnik98statistical}, that there are many  many branches of this theory that have not yet been analyzed and that are important both for understanding the phenomenon of learning and for practical applications. Definitely, one of such area of the research should consider
the necessity of linking the abstract world of mathematics with the physical world.
This is also related to the the grounding problem investigated
in psychology \cite{ch12-harnad87perception,ch12-symbol-grounding,ch12-ACT-R-book2007,ch12-WisIGrC17jaw}.
%%%%%%%%%%


It is also worthwhile to cite here explanation of the concept of complex systems \cite{ch12-Gershenson2004} created by IS's dealing with complex phenomena:
\begin{quotation}
{\em
Etymologically: complexity -- plexus in Latin (interwoven). Complex system: the elements are difficult to separate. This difficulty arises from the interactions between elements. Without interactions, elements can be separated. But when interactions are relevant, elements co-determine their future states. Thus, the future state of an element can not be determined in isolation, as it co-depends on the states of other elements, precisely of those interacting with it.}
\end{quotation}

This definition and the opinion presented above have significant implications for the design and analysis of IS's and development of the IGrC model. It is particularly important to note that IS's must make decisions about situations in the physical world that involve complex phenomena. Therefore, it is inadequate to model such IS's based solely on single models derived from isolated discoveries, which rely on data and knowledge about fragments of physical reality disconnected from their environmental interactions. For IS's, it is essential to continuously perceive various relevant aspects of different fragments of the physical reality, enabling them to acquire the knowledge necessary for developing adaptive strategies to modify the models currently in use. This highlights the critical importance of perception in the IGrC model, especially concerning c-granules or IS's treated as a special kind of c-granules with control. This requires to develop for IS's attention mechanisms supported by reasoning techniques concerning queries related to {\em what}, {\em when}, {\em where}, {\em how} etc. to perceive.

Aristotle already was aware of importance of the role of attention (see, {\em e.g.}, \cite{ch12-Jank:Skow09b} and Fig.~\ref{fig-AT}) in interaction with the physical world.
\begin{figure}[ht]
\centering
\includegraphics[width=5in]{AT.pdf}
\caption{Aristotle tetrahedron \cite{ch12-Jank:Skow09b}}
\label{fig-AT}
\end{figure}
In particular, in \cite{ch12-Aritotle-attention}, one can read:
\begin{quotation}
{\em Spoken words are the symbols (symbola) of mental experience
(pathemata) and written words are the symbols of spoken words.
Just as all men have not the same writing, so all men have not the
same speech sounds, but the mental experiences, which these
directly symbolize, are the same for all, as also are those things
(pragmata) of which our experiences are the images (homoiomata).}
\end{quotation}


It is worthwhile to mention here opinion from \cite{ch12-Turing-test-mod}:
\begin{quotation}
{\em
The Turing test, as originally conceived, focused on language and reasoning; problems of perception and action were conspicuously absent. The proposed tests will provide an opportunity to bring four important areas of AI research (language, reasoning, perception, and action) back into sync after each has regrettably diverged into a fairly independent area of research.
}
\end{quotation}

One of the consequences of this opinion is that in the discussed IGrC model it is necessary to have objects linking abstract and physical objects. Hence, this model can't be closed only in the abstract space.

The idea of complex granules (c-granules, for short) in IGrC is very well expressed as computational building blocks for cognition by Leslie Valiant \cite{ch5-Valiant}:
\begin{quotation}
{\em
A fundamental question for Artificial Intelligence is to characterize the computational building blocks that are
necessary for cognition.
}
\end{quotation}

The computing IGrC model should enable continuous interaction between the designed on the basis of this model system and the physical environment, allowing for the collection of relevant data which can be used to infer data models temporarily characterized by the high quality.

The basic objects in IGrC are complex granules (c-granules, for short). They consists two layers: informational and physical. In the informational layer is stored information about perceived situations as well as specifications of tasks realized over them as well as information about the expected results of realization of these tasks in different parts of the physical world. This information is labeling specifications of spatio-temporal windows (addresses) describing regions of the physical space where the information is perceived.

The physical layer of c-granule consists parts like soft\_suite, link\_suit and hard\_suit. Soft\_suit consists of physical objects directly accessible, {\em i.e}, objects which properties can be decoded by measurements into the information layer or objects into which some relevant information from the information layer can be encoded. This is realized by special elementary c-granules generated by control of c-granules. Information about physical objects which are not directly accessible is inferred by reasoning tools using knowledge bases or physical laws. Hence, computations in IGrC depend on physical laws contrary to the Turing model \cite{ch12-physical-comp-Deutsch-art2000}:
\begin{quotation}
{\em
It seems that we have no choice but to recognize the dependence of our mathematical knowledge} [(...)] {\em on physics, and that being so, it is time to abandon the classical view of computations as purely logical notion independent of that of computation as a physical process.
}
\end{quotation}
In link\_suit are physical objects used for transmission of interactions from soft\_suit to hard\_suit and hard\_suit contains physical objects  to be perceived according to the specifications labeling spatio-temporal windows represented in the information layer. One should note that, c-granules are under control of other c-granules or their own control.



\subsection{Control of c-granules}\label{subsec-Control-of-c-gran}

For simplicity of reasoning, we consider here the case when c-granules are under the control of  IS  which can be treated as a higher order c-granule. This control is responsible for generating computations of IS. The computations are sequences of c-granules (or their networks including information about relationships of other c-granules which are parts of the networks). IS is aiming to generate such computations  realizing in the best way the task of IS, {\em i.e.}, they are aiming to generate computations along which the high quality approximate solutions of problems to be solved by IS are constructed. These approximate solutions of problems may concern classifiers or compound physical objects like sensors, robots or chemical components.

In each step of computation, the control module (CM) of IS verifies whether the information about the current situation is satisfactory to initiate the appropriate transformation of the current c-granule configuration in the form of network of c-granules. This may involve suspending, modifying existing c-granules, or generating new ones.
CM includes a special implementation module (IM) responsible for realization the transformation specifications in the physical world. In essence, the IM realizes the so-called physical semantics of the transformations' specifications.

Here's an idea how it works: The specifications of these transformations are included on the right-hand side of rules located in the rule module (RM) of CM. In each step, the control checks if the information about the currently perceived situation matches the left-hand side of any rules. If there's a match, the RM uses reasoning mechanisms to resolve any conflicts among these rules.
If the conflicts can be resolved, the c-granule control selects from RM the rule for execution. Otherwise, it suggests to gather more information about the perceived situation. This can be done by:
\begin{itemize}
 \item{} realization by IM of some associations in the physical world concerning, {\em e.g.}, measurement of values of some attributes in some specified regions of the physical space or
 \item{} reasoning with the use of some information granules representing a partial knowledge about the physical environment of c-granule; these information granules can be treated as domain databases and  reasoning mechanisms of control applied to these domain databases together with information about the currently perceived situation in the physical world allow the control of c-granule to improve understanding this situation ({\em e.g.}, it can be knowledge about  the properties concerning the transmission of encoded information in configurations of physical objects).
\end{itemize}

In Figure~\ref{fig-cycle} is illustrated the basic cycle of c-granule control.
\begin{figure}[htb!]
\begin{center}
  \includegraphics[scale=.42]{fig-cycle.pdf}
 \caption{Algorithmic cycle of c-granule control. }
 \label{fig-cycle}
\end{center}
\end{figure}
In this figure, {\em inf} is information about the currently perceived situation and {\em Rules} is the set of control rules from RM. Note that the control of the c-granule should be supported by relevant reasoning techniques that support the realization of the basic cycle. For example, to decide whether {\em inf} is satisfactory for selecting a rule, the control should perform reasoning aimed at resolving conflicts between rules from $Rules$ that match $inf$.

Let us consider an illustrative example concerning the mentioned above reasoning with the use of domain data bases.
%Reasoning processes realized by the IS control are based on networks of c-granules, the fundamental objects of IGrC. Let us consider an illustrative example.
Fig.~\ref{fig-kb} depicts reasoning performed on information granules, including the knowledge base (denoted by $KB$).
In the first step, based on information about the current situation ($inf$) perceived in the physical world, the control aims to extract the relevant piece ($fr$) from $KB$ that could be used to extend $inf$ in a useful way. This requires a proper knowledge representation in $KB$ to support the control's efficient extraction of $fr.$ Next, using $inf$ and $fr,$ the control reasons to return an extended version of the information about the perceived situation, denoted by $inf^{\prime}.$ This extension is expected to be helpful, {\em e.g.}, in resolving conflicts between control rules that match $inf^{\prime}.$  Due to its generality, this type of reasoning may require decomposition through several levels before the control can realize it using accessible information granules.

\begin{figure}[ht!]
\centering
\includegraphics[width=4.5in]{fig-kb1}
\caption{Reasoning performed on information granules, including the knowledge base $KB$ and information about the current situation ($inf$) perceived in the physical world.}
\label{fig-kb}
\end{figure}

The execution of the rule by CM involves realizing the transformation specification from its right-hand side. The IM is responsible for carrying out this transformation in the physical world. The set of rules in the RM can be viewed as a complex game involving intricate rules composed out of vague concepts (learned by the IS control) labeled by transformations.  The rules in RM are adaptively changed according to perceived changes by CM. For more details about CM, the reader is referred to \cite{ch12-AS-DS-IJCRS2023}.

%%%%%%%%%%%z dodatku o ip-morphism

\begin{figure}[ht]
\centering
\includegraphics[width=5in]{fig-info-assoc-morphism.pdf}
\caption{Basic cycle of c-granule control: ip-morphism.}
\label{fig-ip-morphism}
\end{figure}

Fig.~\ref{fig-ip-morphism} illustrates the concept of the basic cycle of c-granule control behavior. This c-granule consists of two layers: informational (il) and physical (pl). The symbol $gn$ represents a granular network that reflects the currently perceived situation in physical space. The projection $\pi(gn)$ denotes its representation on the informational layer (il), comprising a family of specifications for spatio-temporal windows (addresses) labeled with information gathered about physical objects in the regions of space indicated by the specifications.
Based on $\pi(gn)$, the control mechanism predicts the desired properties of the next granular network in realized  computation. This prediction is illustrated in the figure by the operation $Next$, which assigns the desired property to $\pi(gn).$ The subsequent granular network in the computation, generated by c-granule control, is expected to either satisfy this property or be 'close enough' to a network that does.

By drawing an analogy with infomorphisms in information flow (see \cite{ch12-Barwise97book}), we can define the satisfiability relation $\models_{Next}$ as follows:
\[x \models_{Next} \pi(gn) \mbox{\ iff\ } Next(\pi(gn))=x,\]
where $x$ denotes the property assigned to $\pi(gn)$ by $Next.$ This expresses the intended semantics of the transition relation over granular networks realized by c-granule control.

The c-granule control must possess the capabilities necessary to generate the next granular network following $gn$ in the physical world. Based on
$\pi(gn),$  the c-granule control generates a specification $tr$ for the association (transformation) to be realized in the physical world using the operation denoted in the figure by $spec.$
This specification $tr$ is then transformed by the implementational module (IM) of c-granule control, using the association $Assoc,$  into a configuration $c$ of the physical objects. One should note that we use the term association to emphasize that $Assoc$ establishes correspondence between abstract and physical objects.

By drawing an analogy with infomorphisms in information flow (see \cite{ch12-Barwise97book}), we can define the satisfiability association $\mid\approx_{Assoc}$ as follows:
\[c \mid\approx_{Assoc} tr \mbox{\ iff\ } Assoc(tr)=c,\]
where $c$ is the configuration of physical objects assigned by the association $tr.$

After initializing interactions in
$c,$ the behavior of $c$ is subsequently perceived by c-granule control, and the results of this perception are stored in the informational layer, represented in the figure as $percep(c).$
The information $percep(c)$ describes the real next granular network following $gn.$
The c-granule control uses a relevant distance measure ($dist$ in the figure) to check whether $percep(c)$
is 'close enough' to a granular network possessing the property $x.$
If this condition is not met, the c-granule control may choose to modify the
$spec$ operation to obtain a granular network of the desired quality, or it may adjust the target goal of the c-granule if modification is not feasible.

It is important to note that $percep$ also functions as a type of association operation on physical objects, enabling the decoding of information related to the perceived physical objects and their interactions. This decoding is achieved by accessing properties of physical objects that are 'directly' available to the c-granule control or by reasoning (judgment)  based on previously perceived information, data stored in domain knowledge databases, or established physical laws. For example, in \cite{ch12-Physics-informed-ML2021} neural networks can be trained from additional information obtained by enforcing the physical laws.


Our computing IGrC model is influenced by interactions with the physical world, in contrast to the classical  computing model that is confined to the abstract space \cite{ch12-physical-comp-Deutsch-art2000,ch12-AI-meets-physics-2024}.
As a result, the presented IGrC model is not purely mathematical; it requires engagement with both abstract and physical objects. Furthermore, one should be aware that, {\em e.g.}, the results of the association $percep$ may depend on the interactions of the configuration of physical objects $c$ with the environment.

%%%%%%%%%%%

The control consists several other important modules. More detailed description of control is included in the cited paper on IGrC (see, {\em e.g.}, \cite{ch10-IGrCSpr16,ch12-skowron2016rough,ch12-Durgapur2021,ch12-ICCCI2021,ch10-WisIGrC17jaw,ch12-AS-DS-IJCRS2023,ch12-dutta2021interactive,ch12-IJCRS24,ch5-InfSci-sub-2024}). The RM plays one of the most important role of the control of IS. In the simplest case, the rules are embedded in this module by designers. However, in many cases these rules should be learned and changed according to perceived changes. The central role in the control of IS play reasoning techniques supporting the IS control in its behavior.

It should be mentioned that in description of the c-granule control we restrict our considerations to specification explaining the intended behavior of the control. The realization of the control in the physical world aiming to satisfy a given specification is not discussed. This may be the task of  human designer or other c-granule representing {\em e.g.}, robot.

\subsection{Summary of comments pertained to realization of associations in IGrC}\label{subsec:further-comm-GrC}

IGrC goes beyond abstract concepts like information granules in GrC. It also handles granules that interact with the physical world.
The control of IS is equipped with the module IM responsible for realization so called physical semantics. The IM module takes specifications of associations (a broader term than specification of mathematical functions) and generates or uses existing configurations of physical objects. It then initializes interactions within these configurations and allows the IS control to perceive properties related to the object interactions.
Based on this perceived information, along with knowledge bases and physical laws, the IS control can infer properties of the perceived objects as results of the realized association. It's important to note that these inferred properties might differ from expectations (expressed in specifications) due to environmental interactions (see Fig.~\ref{fig-assoc}). If the differences between the expected and perceived results of realization of transformations are too large than CM is looking for adaptation of rules stored in RM \footnote{One should note that Aristotle already emphasized the necessity of using adaptation in the decision-making process \cite{ch12-Thiele-judgment-book}.}.


\begin{figure}[htbp]
\centering
\includegraphics[width=4.5in]{fig-assoc}
\caption{Associations and their physical semantics.}
\label{fig-assoc}
\end{figure}

Referring to our previous discussion on granular spaces and granular networks, it is important to note that associations can occur at different granular levels. These associations suggest that their physical realizations through IM will facilitate the perception of information represented by information granules at the relevant granular levels. For instance, the granular level corresponding to the behavior of an animal's heart can be represented by information granules derived from sensory measurements related to that heart behavior.

%%%%%%%%%%%%%%%%%z dodatku information granules vs c-granules

In the context of Granular Computing (GrC), information granules can be seen as a specific type of c-granule. This allows us to focus on the specifications of c-granules, which are represented by information granules.
The difference between information granules considered in GrC and c-granules from IGrC can be illustrated as follows.
 Let us assume that information $inf$ specifying an intended change of a given information system $IS$
 to $IS^{\prime}$ is related to updating $IS$ by adding a new attribute to the set of attributes of $IS$,
 with values computed according to a given procedure $proc$.
 If this specification is admissible for $IS$ by its type then the change of $IS$ to $IS^{\prime}$
 is realized by implementation in computer hardware of the procedure $proc$
 and taking the computed by the procedure value as the value of attribute for each considered object.
 In this case, assuming that the realization in computer hardware is correct,
 in particular, not disturbed by the environment one can restrict considerations
 about the realized transformation to the corresponding informational
 layer without referring to the physical world.
%This is because we assume that these specifications are correctly implemented in the physical world and remain unchanged by environmental external to them interactions.
Therefore, the computational building blocks needed for cognition include both information granules and c-granules. C-granules are generated by control of IS using reasoning techniques. This control aims to construct high-quality approximate solutions for problems that IS needs to solve.
These computational building blocks can take various forms, including patterns, clusters, information systems, classifiers, and physical objects such as new sensors, robots, or chemical compounds.

%%%%%%%%%%%%%IJCRS z DS 2023

The basic concepts related to c-granules and ic-granules are summarized in Table~\ref{tab:Notation} and Table~\ref{tab:Notation2}, and illustrated in Fig.~\ref{Fig-network-of-ic-gr-general} and Fig.~\ref{Fig-network-of-ic-gr3-zoom}. In general, the structure of the control may be much compound and contain many other modules (see Fig.~\ref{Fig7-control-structure}).

\begin{figure}[ht!]
\begin{center}
  {\includegraphics[scale=.5]{Fig-network-of-ic-gr-general.pdf}}
 \caption{A c-granule and a network of informational granules (ic-granules).}
 \label{Fig-network-of-ic-gr-general}
\end{center}
\end{figure}

\begin{figure}[ht!]
\begin{center}
  {\includegraphics[scale=.5]{Fig-network-of-ic-gr3-zoom.pdf}}
 \caption{Details of one of ic-granules that compose the network of ic-granules illustrated in Fig.~\ref{Fig-network-of-ic-gr-general}.}
 \label{Fig-network-of-ic-gr3-zoom}
\end{center}
\end{figure}

\begin{table}[ht!]
\centering \caption{Notation used in this chapter}\label{tab:Notation}
\resizebox{5.5in}{!}{\begin{tabular}{|c|c|}
  \hline
    Name &amp; Interpretation \\ \hline
  $g_c$ &amp;   complex granule (c-granule) \\ &amp; (sometimes called informational granule (ic-granule) \\ &amp; when it is enhanced by non-empty informational layer) \\ &amp; is a dynamic object characterized \\ &amp; (at a given moment of local time of $g_c$)  by \\ &amp;
  the control and a network of subordinate ic-granules (Fig.~\ref{Fig-network-of-ic-gr-general}); \\ &amp; the control is aiming to achieve the goals of $g_c$ transforming \\ &amp; the current network $GN$ of ic granules into a new one by selecting \\ &amp; the specification of network transformation $tr$ (relevant to the current network) \\ \hline
  $GN$ &amp;   network of ic-granules composed out of a finite number of ic-granules \\ \hline
  $tr$ &amp;   specification of network transformation realized \\ &amp; in the physical space by the implementational module (IM) of the control \\ &amp; (possibly preceded by decomposition of specification \\ &amp; to the form directly realizable by IM  in the physical space)\\ \hline
  $control$ &amp;   contains several modules such as \\ &amp; reasoning module (RM),  implementational module (IM), \\ &amp; attention module (AM), adaptation module (AdM), dialogue module (DM) and\\ &amp;  transition relation $rel$, goals and specification of family of networks $Fam\_net $\\ \hline
   $rel$  &amp; if $GN\ rel\ GN^{\prime}$ then $GN^{\prime}$ is the real result of realization\\ &amp; (in the physical world) of the selected transformation by the control at  $GN$  \\ \hline
   $comp$ &amp;   (finite) computation over $Fam\_net$:
   $GN_1 \mbox{\ rel\ } GN_2 \ldots \mbox{\ rel\ } GN_k,$\\ &amp; where $GN_i\in Fam\_net$ for $i=1,\ldots, k$ (see Fig.~\ref{Fig-transition-relation})\\ \hline
   $trace(comp)$ &amp;  information trace of $comp$: $inf\_l(GN_1),\ldots, inf\_l(GN_k),$ \\ &amp; where $inf\_l(GN_i)$ is the information layer of $GN_i$ for $i=1,\ldots, k$ \\ \hline
   $goal$ &amp;   goal of $g_c$ interpreted by the control as a quality (utility) function\\ &amp; over computations with values in $[0,1]$\\ \hline
  $w$ &amp;   specification of a spatio-temporal window (in a given language)\\ \hline
  $\|w\|$ &amp;   subset of $\Re^3,$ where $\Re$ is the set of reals, defined by $w$\\ \hline
  $o_w$ &amp; physical object, {\em i.e.}, part of the physical space corresponding to $\|w\|$ \\ \hline
  $g$ &amp; informational granule (ic-granule) composed \\ &amp; out of informational layer  $inf\_layer(g)$ and physical layer $ph\_layer(g)$\\ \hline
  \end{tabular}}
\end{table}

\begin{table}[ht!]
\centering \caption{Notation used in this chapter  (cd.)}\label{tab:Notation2}
\resizebox{5.5in}{!}{\begin{tabular}{|c|c|}
\hline
    Name &amp; Interpretation \\ \hline
    $inf\_layer(g)$ &amp; informational layer of $g$ consists of \\ &amp; family of tuples $(w, tr, inf),$ where $inf$ is the current information \\ &amp; updated by information  perceived on $o_w$ by $g$\\ &amp; during realization by IM of the transformation specification $tr$ \\ \hline
  $scope(g)$ &amp; a distinguished $w$ from $inf\_layer(g)$ where $\|w\|$ \\ &amp; is the largest among all $\|w^{\prime}\|$ from $inf\_layer(g)$\\ \hline
  $\alpha \Longrightarrow tr:\beta$ &amp;   rule, where $\alpha$ is a condition triggering rule defined \\ &amp; over  (relevant part of) $inf\_layer(g),$ \\ &amp;  $tr$ is a specification of network transformation\\ &amp; and $\beta$ in the expected property of \\ &amp; the resulting network after $tr$ implementation\\ \hline
  $Rule\_set$ &amp;   set of rules (complex game) referring to the physical world\\ \hline
  $ph\_layer(g)$ &amp; physical layer of $g$ consists of parts: \\  &amp;
  $soft\_suit,$ $link\_suit,$ $hard\_suit$ \\ &amp; creating a dynamical physical system perceived by ic-granule $g$;\\
  &amp; perceived information is recorded in $inf\_layer(g)$ \\ \hline
  $soft\_suit$ &amp; consists of $o_w$, where $w$ is from $inf\_layer(g)$\\ &amp; and $o_w$ is directly accessible for measurement by $g$\\
  &amp; {\em e.g.}, IM can directly realize in the physical space the specifications\\ &amp; $enc(inf,w), dec(w)$ of encoding $inf$ in $o_w$ and decoding information\\ &amp;  from the object after realization of $enc(inf,w)$ \\ &amp; such that the realization of $enc$ and next $dec$ ({\em i.e.}, $dec(enc(inf,w))$) by IM \\ &amp; results in $inf$ and after that $enc(dec(o_w),w)$ results in $o_w$ \\ &amp; provided that the environment is not disturbing these implementation processes \\ \hline
  %$dir_g(o_w)$ &amp; $o_w$ is directly accessible by ic-granule $g$ \\ \hline
  %$not\_dir_g(o_w)$ &amp; $o_w$ is not directly accessible by ic-granule $g$\\ \hline
  $hard\_suit$ &amp; consists the target objects $o_w$ not necessarily directly accessible; \\&amp; information about such objects recorded in information layer is\\ &amp; obtained by the control on the basis of reasoning about objects in $scope$ \\ &amp; using the   current information in this layer including \\ &amp; domain knowledge or physical laws \\ \hline
  $link\_suit$ &amp; consists objects used for transmission of interactions between \\ &amp; $soft\_suit$  and $hard\_suit,$ \\ &amp; information about such objects recorded in information layer \\ &amp; in case they are not directly accessible  is\\ &amp; obtained by the control on the basis of reasoning about objects from $scope$ using the \\ &amp; current information from informational layer \\ &amp; including domain knowledge or physical laws \\ \hline
   \end{tabular}}
\end{table}

%\begin{figure}[ht!]
%\begin{center}
%  \includegraphics[scale=.5]{Fig-transition-relation.pdf}
% \caption{Transition relation $rel$.}
% \label{Fig-transition-relation}
%\end{center}
%\end{figure}

\begin{figure}[ht!]
\begin{center}
  \includegraphics[scale=.5]{fig-computation.pdf}
 \caption{Computation over granular networks.}
 \label{Fig-transition-relation}
\end{center}
\end{figure}


\begin{figure}[t]
\begin{center}
  {\includegraphics[scale=.5]{Fig7-control-structure.pdf}}
 \caption{The control of c-granule.}
 \label{Fig7-control-structure}
\end{center}
\end{figure}

\section[Further comments on the IS control]{Further comments on  behavior of the IS control and reasoning supporting it }\label{sec:control}

%%%%%%%%%praca z JS
Our research explores new directions for applying the generalized rough set approach to IS. This work builds upon the IGrC model and leverages existing partial results from various fields (including multi-agent systems, perception and action, machine learning, natural language processing, federated learning, cognitive networks, smart cities, cyber-physical systems, complex adaptive systems etc.) by putting them into sync. We also emphasized the fact that IS are dealing with complex phenomena in the physical world what requires a new kind of modeling for solving problems with the design and analysis of IS. It was pointed out that dialogues with domain experts are unavoidable for IS due to the fact that still we do not have satisfactory formal reasoning techniques making it possible to deal to a satisfactory degree with commonsense reasoning or experience based reasoning. The discussed  comprehensive approach has the potential to establish a solid foundation for the design and analysis of IS.

%\section{Adaptive reasoning (judgment) in IGrC}\label{sec:reasoning-in-IGrC}

Among reasoning methods in RM of control are  methods based on deduction, induction or abduction \cite{ch5-RussellNorvig-bookAI,ch10-TheoriesJudgment-book-2006}. However, there are some `white spots' which require further research. Among them are reasoning (judgment) methods based on experience ({\em e.g.}, reasoning by analogy) \cite{ch12-Mitchell2019-AIbook,ch12-Mitchell-analogy} or mechanisms supporting discovery \cite{ch12-Povalej2009,ch12-Comp-Creativity-book-2019}. This causes that in some applications, especially dealing with complex phenomena ({\em e.g.}, medical) it is not possible to eliminate dialogues with domain experts \cite{ch5-human-centered-AI,ch5-human-in-the-loop,Human-in-the-Loop24} and/or chatbots \cite{ch12-chatbots-book24}.


One of the challenges for c-granules with control, in particular for IS's based on IGrC is to develop methods for reasoning over interactive computations performed on granular networks supporting c-granules in controlling such computations toward achieving the target tasks. Such reasoning techniques we sometimes call
 {\em adaptive judgment}. {\em Intuitive judgment} and {\em rational judgment} are distinguished as different kinds of judgment \cite{ch12-Kahneman-PrizeLecture2002}.
 Adaptive judgment is the basic tool in discovering of relevant patterns of different complexity used for approximation of complex vague concepts and inducing approximate reasoning schemes on such approximations, called after Leslie Valiant as computational building blocks \cite{ch5-Valiant}.
Adaptive judgement in IS's is  a mixture of reasoning  based on deduction, abduction, induction, case based or analogy based reasoning, experience, observed changes in the environment, meta-heuristics from natural computing is used (see Fig.~\ref{fig:interaction-granulation}).
\begin{figure}[hbt!]\centering
    \includegraphics[width=0.95\textwidth]{interaction-granulation.pdf}
    \caption{Interactions, adaptive judgement and granulation} \label{fig:interaction-granulation}
    \end{figure}

It is worthwhile to cite here the opinion about practical judgment from
\cite{ch12-Thiele-judgment-book}:
\begin{quotation}
{\em Practical judgment is not algebraic calculation. Prior to any deductive or inductive reckoning, the judge is involved in selecting objects
and relationships for attention and assessing their interactions. Identifying things of importance from a potentially endless pool of candidates,
assessing their relative significance, and evaluating their relationships is well beyond the jurisdiction of reason.}
\end{quotation}
This opinion is strongly related to IGrC, in particular to the role of IM and AM modules in realization in the physical world of associations.

We have already discussed some issues related to reasoning by control of IS. Fig.~\ref{fig-reasoning} illustrates a variety of components of reasoning module (RM) of IS from the point of view of different tasks of control.

\begin{figure}[htb]
\centering
\includegraphics[width=3.75in]{fig-reasoning}
\caption{Different components of RM in IS (c-granule with control).}
\label{fig-reasoning}
\end{figure}

%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%JS
These components are crucial for the IS control in supporting generation of computations over granular networks by IS's aiming to solve specified problems such as

%%%%%%%%%%%%ex tasks
\begin{itemize}
\item{} {\em Learning algorithms and Classifiers Construction} \cite{ch5-RussellNorvig-bookAI,ch5-friedman01elements}:
The IS takes training data, quality requirements for the classifier, and databases containing information relevant to classifier construction ({\em e.g.}, search strategies) as input.  The IS then returns a classifier that meets the specified quality requirements.  This can be seen as searching for a complex structure (granule) that satisfies a given specification to a satisfactory degree, as measured by the quality of the constructed classifier. In other words, the returned classifier should be an example of complex granule belonging  (with the high degree) to the lower approximation of the concept `classifiers satisfying the given specification to a satisfactory degree'.


\item{} {\em Automatic Design of Robots} \cite{ch12-Robot-design-PNAS}: The IS takes databases of robot parts and a  specification for the robot ({\em e.g.}, concerning desired tasks and performance metrics) as input. It then returns a physical robot that meets the given specification to a satisfactory degree.  Similarly to the first example, this  represents finding a complex structure (c-granule) that fulfills the desired criteria. In this case, the returned robot as a physical object should be an example of c-granule belonging  (with the high degree) to the lower approximation of the concept `robots satisfying the given specification to a satisfactory degree'.

 \item{} {\em Drug Discovery} \cite{ch12-Drug-Disovery-book-2021}: The IS takes various relevant knowledge bases (e.g., medical and chemical data) and a specification for the desired medicine ({\em e.g.}, expected effects) as input. It then returns a physical medicine that meets the specified requirements to a satisfactory degree.  Once again, this translates to searching for a complex structure (granule) that adheres to the defined criteria. Here, the returned medicine as a physical object should   belong  (with the high degree) to the lower approximation of the concept `medicine satisfying the given specification to a satisfactory degree'.
 \end{itemize}

%%%%%%%%%%%%%

One can see that in each of the above mentioned applications IS should construct objects, called here c-granules,
satisfying to a satisfactory degree a given specification. This can be expressed  by requirement that the generated granules should belong, {\em e.g.}, to the lower approximation of the concept consisting granules representing approximate solutions or granular computations along which these solutions are constructed.
Hence, IS should be equipped with advanced reasoning methods for performing  approximate reasoning processes
along which such complex granules may be constructed during generation of granular computations. In developing models of such approximate reasoning processes, one should answer several important questions. Among them are the following ones:
\begin{itemize}
\item{} How the target complex granules are specified?

\item{} How the target complex granules are processed by IS?

\item{} What are the objects (c-granules) over which approximate reasoning processes are running?

\item{} How approximations of c-granules are defined and constructed along performed reasoning by IS?
How calculi of granules on different hierarchical levels are defined and/or
discovered? On what kind of reasoning are based strategies for discovery of relevant
calculi of granules? What are the basic steps of reasoning supporting construction   of the
high quality solutions of the given problem from a given c-granule $g$ representing specification?

\item{} What are the methods of reasoning over generated by IS granular computations aiming to control them toward solutions satisfying given specifications to a satisfactory degree? Which methods can provide insightful reasoning for control of IS?

 \item{} What kind of quality measures should be used  for estimation of the quality of granules in construction of solutions ({\em e.g.}, classifiers, clusters, plans, and other complex abstract and physical objects) evaluated along the approximate reasoning processes?

  \item{} What kind of computing model should be used by IS to guarantee feasibility of construction of the high quality of approximate solutions realized in the physical world?

   \item{} What kind of testing methods IS systems should use to verify the quality of provided solutions?
\end{itemize}

Answering these question requires from the IS control reasoning techniques pertained to different components of RM. In general, reasoning techniques from RM module are supporting decision making about the currently perceived situation and are performed along granular computations over granular networks.  Information from c-granules used in reasoning may be obtained as a consequence of interactions with the physical objects ({\em e.g.}, by measurement or performing actions being consequences of realization of associations in the physical world) as well as aggregation of information granules also with the use of domain knowledge databases  or physical laws represented in informational layers of information granules.

This discussed approach focuses on IS that handle complex phenomena. When solving problems with such IS, two key issues arise:
\begin{enumerate}
    \item the appropriate computing model,
    \item the approximate reasoning processes that guide computations within this model.
\end{enumerate}
These problems involve IS generating computations that lead to solutions, {\em e.g.}, within the lower approximation of concept encompassing all solutions to a given problem. This formally expresses that the generated solution is of high quality or is sufficiently satisfactory.

%%%%%%%%%%%%%%%

In particular, we already emphasized the role of reasoning supporting optimization of different tasks performed by IS in searching for the high quality solutions of given problems to be solved, {\em e.g.}, optimization of parameters of approximation spaces, quality measures, learning algorithms. In this section, we discuss some others of these components. To summarize, let us remind that reasoning by control is performed on complex granules (c-granules) among which are granules linking abstract and physical objects. Control is performing reasoning on granular networks generated by control of IS and along performed reasoning are generated granular computations aiming to lead to the high quality of approximate solutions of considered problems.


The states of IS's are networks of granules from different networks of granular spaces. Granules from different granular spaces are linked by more compound relations than granules linked from the same granular space. Granular networks are storing  properties of single granules and properties of computations over granular networks. The links between granules represent different relationships between granules. Some granules represent domain knowledge, some other are used to represent components of IS control. There is also a distinguished granule representing the  state of perception of the currently perceived situation.

Control of IS is deciding in each state what kind of transformation of the current granular network to perform. The transformation may concern suspending some measurements, elimination of some granules from the network, updating of the granular network with new granules or updating the existing granules. The control decides which transformation to select by matching rules being at its disposal against the granule representing information about the currently perceived situation. The set of rules can be treated as a complex game (see. {\em e.g.}, \cite{ch10-IGrCSpr16,ch12-skowron2016rough,ch12-Durgapur2021,ch12-ICCCI2021,ch10-WisIGrC17jaw,ch12-AS-DS-IJCRS2023}) consisting of concepts (often, complex vague concepts) labeled by transformations which should de realized if the concept is satisfied (to a satisfactory degree) (see Fig.~\ref{fig-game}).

The complex game illustrated in Fig.~\ref{fig-game} consists of pairs $(C_j,tr_{j_i})$ for $j=1,\ldots,n,$  where each $C_j$ is a classifier for a vague concept triggering the transformation specification $t_{j_i}.$ If $C_j$ is satisfied (to a satisfactory degree) in the current perceived situation, the IS control registers this as a match. Subsequently, the control attempts to resolve conflicts among all matched classifiers to select a transformation specification for the IM to execute in the physical world. This is supported by appropriate reasoning methods. If successful, the selected specification is transmitted to the IM; otherwise, the IS control seeks additional information about the perceived situation ({\em e.g.}, through sensory measurements or knowledge derived from c-granules representing domain knowledge bases). The IM executes the transformation specification in the physical world through the following steps: (i) creating (or organizing) a configuration of physical objects defining the scope of the transformation, (ii) initiating interactions within this configuration, (iii) allowing the IS control to perceive relevant object properties and interactions over the relevant period of time and recording results in the informational layers of appropriate c-granules, and (iv) treating the appropriate recorded values as the transformation's outcomes. It's important to note that these outcomes may deviate from expectations due to environmental interactions.

Both the concept and labeling it transformation are complex granules (c-granules) in IGrC. The satisfiability of the concept to a satisfactory degree (under additional assumption that conflict resolution strategy between rules satisfiable to satisfactory degree points to the concept) initiates interaction  with granule activating transformation leading to its realization.


\begin{figure}[ht!]
\centering
\includegraphics[width=4.5in]{fig-games-new1.pdf}
\caption{Complex granule consisting of set of rules interpreted as a complex game with granules $C_1,\ldots C_n$ representing vague concepts and their classifiers  labeled by transformations $tr_{i_1},\ldots, tr_{i_n}$.}
\label{fig-game}
\end{figure}

%powtorzenie
%On the basis of the results of this matching the control is attempting to resolve conflicts between rules matching the granule with with information about the currently perceived situation. If this matching is successful, the control starts the realization of the selected transformation, otherwise is trying to get more information about the perceived situation, {\em e.g.}, by measurement or interaction with granules representing domain knowledge bases or with human experts.

One should note that the rules set of the control of IS may have a compound structure, {\em e.g.}, representing distributed control or priorities of rules. Moreover, IS should be equipped with reasoning methods for development strategies for adaptation of rules based, {\em e.g.}, on optimization of various parameters of entities leading to construction of rules including granular spaces and networks over them, constructions of granular networks and forms of transformations.
Fig.~\ref{fig-adaptation} presents a simplified example of an adaptive complex game. Classifiers for complex vague concepts, denoted as $C_1,\ldots,C_m,$ trigger complex games $G_1,\ldots G_m.$ Using these classifiers, the IS control identifies concepts $C_1,\ldots,C_m$ that match the current perceived situation and employs appropriate reasoning methods to resolve conflicts among them. If conflicts can be resolved, a complex game from $G_1,\ldots,G_m$  is selected; otherwise, the control seeks additional information about the perceived situation to facilitate conflict resolution. Challenging problems are related to learning the high quality classifiers for $C_1,\ldots,C_m.$ One should also note that the $IC$ control must determine whether the goals outlined in the problem specification can be achieved. If not, it should adjust the goals accordingly.
Hence, it is becoming important  reasoning supporting recognition on feasibility of realization of goals, otherwise their modification should be proposed ({\em e.g.}, with the use of Maslov hierarchy  of needs  \cite{ch12-Maslov-hier1943} or its modification).

\begin{figure}[ht]
\centering
\includegraphics[width=4.5in]{fig-adaptation-new1.pdf}
\caption{Adaptive complex game consisting of vague concepts with classifiers $C_1,\ldots,C_m$ labeled by complex games $G_1,\ldots,G_m.$}
\label{fig-adaptation}
\end{figure}

Notably, complex games and adaptive complex games can be viewed as more intricate c-granules. Their behavior arises from interactions between granules representing vague concepts.  These interacting granules are labeled by c-granules representing decisions ({\em e.g.}, plans or complex games relevant to the domains defined by these vague concepts). Additionally, reasoning processes influence these interactions by resolving conflicts between votes on different decisions predicted by the concepts. This reasoning ultimately establishes the interaction deciding which plan or complex game to launch.

More details about these important issues will be presented in our next works. Here, we would put attention of the reader to the importance and richness of the required reasoning methods supporting the control of IS.

Let us consider in more detail two more examples related to reasoning methods related to aggregation and decomposition of granular networks.


Aggregation of granules is widely used in Machine Learning, {\em e.g.}, in feature engineering or hierarchical learning \cite{ch5-friedman01elements,ch10-stone00layered,ch5-Bazan08TRSIX} and Granular Computing (GrC). In feature engineering, new features are defined over the existing ones. Hence, a partition of the universe of objects defined by a new feature if defined by partitions of some known already features. This may be interpreted as a transformation of granules from one granular space into a new one.

In general, the decomposition problems are challenging. According to the opinion of Judea Pearl \cite{ch10-Pearl}:
\begin{quotation}
{\em Traditional statistics is strong in devising ways of describing data and inferring distributional parameters from sample.
Causal inference requires two additional ingredients:
\begin{itemize}
\item{}a science-friendly language for articulating causal knowledge, and
\item{} a mathematical machinery for processing that knowledge, combining it with data and drawing new causal conclusions about a phenomenon.
\end{itemize}
}
\end{quotation}

This transition from `a science-friendly language for articulating causal knowledge' to `a mathematical machinery for processing that knowledge' usually requires decomposition through several layers.


Decomposition (degranulation \cite{ch12-Pedrycz2021}) of granules is an important issue in  GrC and IGrC. Here, we discuss shortly some important problems related to decomposition of granules, in particular described in natural language. Language of granules consists complex vague concepts expressed in natural language. Their semantics is defined by classifiers induced from data. We emphasize reasoning methods leading to the relevant decomposition. Let us mention two illustrative examples: decomposition of specification of tasks of IS's and decomposition of specifications of transformations in realization of associations.

In \cite{ch5-InfSci-sub-2024} we have presented examples of such tasks. In each of these examples, the description of task is presented using complex vague concepts what can be hardly understandable by IS without additional support. It is worthwhile to cite here an opinion of Lotfi Zadeh \cite{ch12-Zadeh-RNC-book,ch10-zadeh01new}:
\begin{quotation}
{\em Information granulation plays a key role in implementation of the strategy of divide-and-conquer in human problem-solving.
}
\end{quotation}
One can ask how to acquire or learn such a strategy. One possible approach is to provide dialogue methods with domain experts. This is especially important for medical applications \cite{ch5-human-in-the-loop,ch5-human-centered-AI}. Another one is related to applications of chatboots \cite{ch12-chatbots-book24} and Large Language Models (LLM) \cite{ch12-LLM-book2024,ch12-LLM-DeepDive-book2024}. It may be necessary to go through several levels of decomposition, with possible backtracking,  before the level with concepts which can be efficiently and with high quality approximated can be reached on the basis of available data. In the case of realization of specifications of associations for this last level a direct realization of this level in the physical world should be possible. This approach proved to be successful for several real-life projects where so called the rough set-based ontology approximation of concepts was applied: after decomposition to the lowest level the obtained ontology of concepts was approximated using bottom-up strategy (see,  {\em e.g.}, \cite{ch5-Bazan08TRSIX}).

IS also require reasoning mechanisms to handle situations where the current search strategy for relevant granules is unlikely to succeed. In such cases, the reasoning system of IS should guide the backtracking process by appropriately decomposing (degranulating) some already generated granules up to a suitable level. Additionally, more advanced IS should leverage information from unsuccessful trials to learn and adapt their search paths towards discovering the relevant granules.

\section{Rough sets in IgrC paradigm}\label{sec:rs-in-IGrC}

\subsection{Information systems in IGrC}\label{subsec:inf-sys-in-IGrC}

In this section, we present some comments on management of information systems in c-granules by their CM module. Information systems in IGrC, contrary to the existing approaches, are in `hands' of CM (see Fig.~\ref{fig-inf-sys-control-of-c-granule}), {\em i.e.}, their dynamics depends on CM and environmental interactions.

\begin{figure}[ht!]
\centering
\includegraphics[width=4.5in]{fig-inf-sys-control-of-c-granule.pdf}
\caption{Information system in `hands' of control of c-granule.}
\label{fig-inf-sys-control-of-c-granule}
\end{figure}

The c-granules (with control) hold specific pieces of information stored in their informational layers. Among these pieces are information systems. In our discussion, we consider IS as a c-granule with control.
IS typically manages multiple information systems. Therefore, its control requires a proper addressing mechanism (realized by spatio-temporal windows or addresses) to locate the relevant information system, considering both space and time.

Each information system, identified by its address, stores objects of a predefined type. The type of these objects is defined by a formula that allows the IS control to determine if a piece of information sent by the IS control can modify the system and how. For instance, the type might specify that the system holds objects defined by a spatio-temporal window describing where specific attributes should be measured. It could also include time information, such as the start time for the measurement and the expected duration.

Information systems can store more complex object types. These could include, {\em e.g.}, properties of segments from different multi-time series that the IS control perceives during measurements. These segments could be aggregated into clusters or even more intricate structures. Additionally, types can hold properties related to interactions with physical and abstract objects. These properties might also include conditions expressing relationships between attributes, such as specifying that certain parts of the observed objects are physically close.

Formally, these types can be represented as formulas $\alpha(x)$ in a specific language. When checking if information $inf$ is relevant for updating a particular information system $IS$, the variable $x$ in the formula is replaced with the information $inf.$ This information describes how the IS control intends to modify $IS$ by adding a new object to its collection.

Let us consider an illustrative example in which this new object $o$ is defined by information $inf$ in the format:\[ w: a_1, \ldots, a_m; spec.\] Here, $w$ is a spatio-temporal window specification, identifying a part of physical space where the values of attributes $a_1$ to $a_m$ should be measured. The expression $spec$ refers to a specification for how the module IM of control should obtain these values.

If this object satisfies the type formula $\alpha(x)$, the IS control follows these steps:
\begin{itemize}
\item{} It expands the universe of objects within $IS$ by adding the new object $o$.
\item{}The IS control sends a request to the IM to realize the specification $spec$ in the physical world.
\item{} IM initiates a process in which it perceives the values of attributes $a_1$ to $a_m$ in the specified part of the physical space (defined by $w$).
\item{} These values are then stored in the expanded information system $IS^{\prime}$ as attribute values for the newly created object $o$.
\end{itemize}

Before summarizing our considerations in this section, let us make some complementary comments.

Let us consider an illustrative example concerning types of aggregated information systems. We consider an aggregation of two information systems $IS_1,IS_2$ of type $\alpha_1(x), \alpha_2(y),$ respectively. In aggregation we take the Cartesian product of the universes of objects of $IS_1, IS_2$ and filter it with constraint $\alpha(x,y)$ defined with the use of some relations between value sets of attributes from $IS_1,IS_2$ ({\em e.g.}, specifying closeness of objects $x$ and $y$). The type of this exemplary aggregation is defined by $\alpha(x,y) \wedge \alpha_1(x) \wedge \alpha_2(y).$

One can observe that the specification of type of information system $IS$ is closely related to specification of a family of admissible changes of $IS.$.

%The difference between information granules considered in GrC and c-granules from IGrC can be illustrated as follows. Let us assume that information $inf$ specifying an intended change of a given information system $IS$ to $IS^{\prime}$ is related to updating $IS$ by adding a new attribute to the set of attributes of $IS$, with values computed according to a given procedure $proc$. If this specification is admissible for $IS$ by its type then the change of $IS$ to $IS^{\prime}$ is realized by implementation in computer hardware of the procedure $proc$ and taking the computed by the procedure value as the value of attribute for each considered object. In this case, assuming that the realization in computer hardware is correct, in particular, not disturbed by the environment one can restrict considerations about the realized transformation to the corresponding informational layer without referring to the physical world.


%%%%%%%%%%%%%
Summarizing, we propose the following  changes in modeling of information systems in comparison to the Pawlak model:
\begin{itemize}
\item{} {\em The Key to Dynamics: Open Information Systems.}
We propose a generalization of Pawlak's information systems into 'open' information systems. These systems are dynamic entities, and the IS control is responsible for their evolution during computations seeking high-quality approximate solutions for problems the IS needs to solve.  Hence, the dynamics of these systems is not defined a priori as it was proposed in papers on dynamic information systems so far (see, {\em e.g.}, \cite{ch12-CampagnerCD22, ch12-DongWC23, ch12-Wolski10}). Pawlak's information systems can be seen as starting points, or `seeds.' We need to consider huge spaces of information systems around them. Within these spaces, it is necessary to search for (semi-)optimal information systems (or approximation spaces). The relevant reasoning techniques should be developed supporting this search making it possible to induce the relevant computational building blocks for cognition, like classifiers or clusters. Furthermore, the IS control system must be aware that these vast information spaces are dynamic and change over time.
\item{} {\em Challenges and Networks.}
The intended dynamics may not always be achieved due to unforeseen interactions with the physical environment. Furthermore, the IS control often deals with multiple interconnected information systems rather than a single one, forming networks of information systems. These networks can be viewed as networks of c-granules over which computations are generated by control of IS.

\item{} {\em System Types and Objects in Information Systems.}
Each information system has a type that specifies the allowed type of objects in it, in particular those that can be added in updating.  The type is typically specified using some properties of fragments of granular computations generated by the IS.  Objects within an information system must be compatible with the system's type. It can be treated as a filter for objects to be stored in information system.
Objects in information systems are descriptions of structural objects labeled by specifications of spatio-temporal windows, like bitmaps of images, fragments of time series or their clusters, together with encoded in these descriptions procedures and/or specifications of associations used by control of IS for computing values of the relevant attributes. The control of IS is using
\begin{itemize}
\item[(i)] a procedure encoded in the object to compute necessary attribute values for an object using, {\em e.g.}, information from other information systems and/or
\item[(ii)] a specification of an association encoded in the object with the specified fragment of the physical space; a process is associated with this specification that is realized in the physical world by IM, allowing IS to perceive, {\em e.g.}, the desired attribute values and store them in the information system.
\end{itemize}
Hence, attributes considered in the paper are not necessarily abstract functions by they may be defined by specifications of associations and their realization in the physical world by IM.

The type specification can also include information on how the IS control can update the information system, {\em e.g.}, by adding or removing rows or columns. For example, a type might specify a type of spatio-temporal windows, a list of attributes to measure within the fragments of the physical space corresponding to those windows, additional information like measurement time frames and expected results. Control of IS is responsible for updating the system with the perceived in physical realization measurement values.


\item{} {\em Dynamic by Design, Not Random.}
These generalized information systems are dynamic, not randomly so. Their evolution is determined by the IS control's intended dynamics, which can be influenced by interactions with the physical environment.
This implies that the information systems in this paper are not purely mathematical objects. Their dynamics are defined by c-granules representing the systems themselves and their interactions with other c-granules, working like physical pointers to specific parts of the physical space. These pointers allow the IS control to perceive properties of physical objects in those parts and use this information to update the current state of the systems.

\end{itemize}

This type of modeling is essential for seriously considering issues related to perceiving complex situations in the physical world and making the right decisions about them. We demonstrated that our approach to information systems can be seen as a constructive way to define dynamic approximation spaces. These spaces are changing accordingly to changes of corresponding to them information systems and they can be used to search for  computational building blocks for cognition, such as patterns, clusters, or concept approximations (classifiers) a well as their hierarchical structures relevant to problem-solving by the IS.

%%%%%%%%%%%%%%%%%

Typically, control of IS deals with a family of information systems generated in the perception of situations. Moreover, the information systems from such families are linked by different relations representing relationships between objects, fragments of the whole information systems. In this way are created networks of information systems. They are examples of more compound c-granules (or information granules), corresponding according to our previous discussion, to families of approximation spaces. Here, it is worthwhile mentioning the relationships with information flow \cite{ch5-Barwise97book,ch12-InfFlow-GradedCons2019} attempting to develop logical foundations for distributed computing. One should also refer here Fuzzy(-Rough) Cognitive Networks (see, {\em e.g.}, \cite{ch12-Kosko-1986,ch12-Rugh-Fuzz-Cog-Net-2016,ch12-NAPOLES201779,ch12-CognMaps-book2024}) as well as Federated Learning (see, {\em e.g.}, \cite{ch5-IoT-book2020,ch12-FedLearn-book2023}) as examples of techniques aiming to create machine learning models with improved performance on distributed datasets (without sacrificing privacy). On the way to create such models many challenges appear concerning, {\em e.g.}, creating, designing, operationalizing or maintaining distributed systems. One of the challenges is related to developing reasoning methods supporting solving these challenges.

The proposed approach focuses on developing reasoning methods that support the construction of approximate solutions for specified tasks.  These methods must consider additional constraints during construction. These constraints can include privacy requirements, limitations on data aggregation due to resource limitations, or adherence to principles expressed in natural language standards ({\em e.g.}, ISO standards) that may contain complex and vague concepts.
Importantly, these reasoning methods should not only analyze pre-constructed solutions but also actively support the construction process itself, working alongside the granular computations generated by IS. Dialogues with human experts may play a crucial role in this process.
Furthermore, at different stages of the IS computations, solved subproblems can be treated as optimization problems within large families of approximation spaces.


Aggregation and decomposition operations as well as filtration (see, {\em e.g.}, \cite{ch5-skowron01information,ch10-skowron03information}) of information systems enable us to construct new information systems on the basis of which new relevant c-granules being computational building blocks for comprehension of the perceived situations are discovered.

One of the fundamental issue of information systems under the control of IS is that they are open to interaction. They are changed by control of IS. The changes are controlled by reasoning skills of control. We discuss this issue in more detail in the subsequent section.

\subsection{Approximate solutions generated by IS's based on the rough set approach grounded in IGrC}\label{subsec:approximate-solutions-in-IGrC}

In this section, we discuss and summarize shortly perspectives of rough sets in the framework of IS's and IGrC.

Fig.~\ref{fig-new-rs} illustrates a paradigm shift in approximation from the Pawlak model to the model presented in this paper.
Approximation in the Pawlak model involves concepts represented as subsets of a given finite universe of objects. These concepts are approximated using so-called definable sets, which are unions of indiscernibility classes derived from an equivalence relation within a specified approximation space. A fundamental goal in IS is to identify high-quality approximate solutions for problems to be addressed by IS. These solutions are constructed along  granular computations guided by IS control. This process is facilitated by advanced reasoning techniques and/or expert collaboration. The IS control behavior is based on adaptive models of so-called complex games (see Section~\ref{subsec-Control-of-c-gran}), composed of rules that dictate necessary transformations within the current state of granular computation when relevant conditions are satisfied (to a satisfactory degree). The quality of approximate solutions is evaluated using appropriate quality measures. A primary objective is to generate, within the space of approximate solutions for a given problem, examples of approximate solutions constructed along granular computations definitively (with certainty) belonging to the concept of high-quality solutions, {\em i.e.}, the lower approximation of this concept. It is important to note that such examples are often unavailable beforehand.



\begin{figure}[htb]
\centering
\includegraphics[width=5.5in]{fig-new-rs.pdf}
\caption{From approximation of concepts to approximate solutions of problems.
}
\label{fig-new-rs}
\end{figure}

In Fig.~\ref{fig-perspectives-rs} is presented a context in which rough sets should be considered in IS.

\begin{figure}[ht!]
\centering
\includegraphics[width=4.5in]{fig-perspectives-rs.pdf}
\caption{Rough sets in IS (AM-attention module, IM-implementational module).}
\label{fig-perspectives-rs}
\end{figure}

%In this context (see Fig.~\ref{fig-perspectives-rs}), the rough set approach is not based on a single information or decision system (data table). On the basis of perceived data sets with help of experts different multi-relational approximation spaces $AS_1,$ $\ldots,$ $AS_k$ corresponding to different information (decision) systems are created. From these spaces using different  techniques families $\mathcal{F}_1, \ldots, \mathcal{F}_k$ of multi-relational approximation spaces are induced. They create the basis for optimization in searching for the high quality complex games. It is worthwhile to mention that in the optimization process aggregation and de-aggregation concerning granulation and de-granulation of information (decision) systems is performed in searching for the relevant  multi-relational approximation spaces. Also here, as it was mentioned before in the discussion on the proposal concerning information granulation by Zadeh, dialogue with expewrts is unavoidable. It should be noted that, in learning of the complex games a family of concepts labeled by specifications of transformations representing specifications of actions should be constructed rather then an approximation of a single concept. In testing is checked the quality (measured by the relevant for applications quality measures) of the discovered complex games and their adaptation is performed according the the results of testing. Moreover, the attention module (AM) supported by the relevant reasoning techniques and cooperation with experts allows the IcS control to acquire new data sets enabling the control to make the adaptation more efficient.


In the context of Fig.~\ref{fig-perspectives-rs}, the rough set approach departs from traditional methods that rely on a single information or decision system (data table). Instead, it utilizes perceived data sets with AM. These data sets are used to create multiple multi-relational approximation spaces, denoted as $AS_1,$ $\ldots,$ $AS_k,$ each corresponding to a distinct information or decision system. Various techniques are then applied to these spaces to generate a family of multi-relational approximation spaces, denoted as  $\mathcal{F}_{AS_1, \ldots, AS_k}.$ This family serves as the basis for optimization algorithms searching for high-quality complex games.

%%%%%%%%%%%%new
In Fig.~\ref{fig-perspectives-adaptive-rs} is presented a more general scheme illustrating the role of adaptive rough sets in the c-granule control.
\begin{figure}[ht!]
\centering
\includegraphics[width=4.5in]{fig-perspectives-adaptive-rs.pdf}
\caption{Adaptive rough sets in the c-granule control.}
\label{fig-perspectives-adaptive-rs}
\end{figure}


In the upper part of the figure, a learning scheme for a complex game is presented. It begins with generic (basic) granular spaces $GS_1(\Theta_1),\ldots, GS_k(\Theta_k)$, characterized by parameters $\Theta_1, \ldots, \Theta_k.$ These spaces consist of c-granules that correspond to sensory measurements. The informational layers of these c-granules store data represented by information granules ({\em e.g.}, information systems or decision systems) perceived in the physical space through sensory measurements. The relationships within these granular spaces are characterized by (partial) inclusion among the information granules. The parameters $\Theta_1, \ldots, \Theta_k$ correspond to factors such as the location of the measurement, the time at which it was taken, the reasoning behind the measurement, and the methods used.
Next, a network of granular spaces, denoted as $NGS_{GS_1(\Theta_1),\ldots, GS_k(\Theta_k)}$, is constructed from these granular spaces. This network consists of c-granules of various types ({\em e.g.}, related to different levels of hierarchical modeling), along with interfaces that specify how higher-level c-granules are constructed from simpler ones. From this network, a granular network $GN(\Lambda)$ is formed by selecting relevant c-granules and the relationships between them. The parameters in $\Lambda$ specify possible selections of these c-granules and their relationships within the network.





%%\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%dodane zdanie
%
%Now, let us turn back to our discussion on optimization.
%
%%%%%%%%%%%%%%%%

Following this, the optimization of the complex game focuses on selecting parameters from
$\Theta_1, \ldots, \Theta_k, \Theta, \Lambda.$ The constructed complex game is evaluated based on relevant fragments of computations generated from the granular network. Various quality measures, dependent on the target goals of the specific c-granule ({\em e.g.}, information system), are employed in the evaluation. If the quality is unsatisfactory, the complex game is modified using an adaptive strategy. The adaptive module (AdM) is responsible for managing the adaptation in the control of c-granules.

The figure also illustrates the IM module, previously explained, which is responsible
for implementing the specifications of transformations from the complex game in the physical world.
Additionally, it shows the reasoning module (RM), attention module (AM) and the dialogue module (DM),
which facilitates interaction between the c-granule and experts to support its behavior.

The c-granule with control aims to generate computations over granular networks that meet specific target goals. As previously mentioned, the quality of these computations may depend on either the final state of the computation or the entire computation process, depending on the task specifications realized by the c-granule. Degrees of computation quality correspond to different regions of approximation ({\em e.g.}, lower approximation region, upper approximation region, or boundary region) of potential solutions for the specified task realized by the c-granule. It is important to note that in some tasks executed by c-granules, positive examples of objects from certain approximation regions may not be available, unlike the scenarios typically encountered in machine learning (ML). In other words, the c-granule must discover examples of such objects by constructing them along granular computations.

%%%%%%%%%%%%%
In the considered context, the goal is to generate examples of computations that belong to the lower approximation of the concept being considered. These examples, referred to as positive examples, are not provided a priori. The objective is to discover complex games for controlling c-granules that enable the generation of such computations representing positive examples. Often, it is necessary to adapt the initial complex game due to changes in the physical world with which the c-granules interact as well as due to recognized mistakes performed in searching. Constructing positive examples requires conducting experiments that involve interactions with the physical world.

%%%%%%%%%%%%%%%%%%%%%%

 Approximate solutions are constructed along computations over granular networks. In the space of generated computations by control one can distinguish computations of the high quality, {\em i.e.}, computations generating approximate solutions of the high quality. Such computations belong to the lower approximation of the concept consisting computations of the high quality. Other computations may belong to computations of acceptable quality (creating  upper approximation of the concept consisting of the high quality computations) or to complement of such concept (creating complement of the upper approximation) (see Fig.~\ref{fig-approximation-regions}).

\begin{figure}[ht!]
\centering
\includegraphics[width=4.5in]{fig-approximation-regions.pdf}
\caption{Approximation regions in the space of computations of c-granule control.}
\label{fig-approximation-regions}
\end{figure}

%%%%%%%%%%%%%%%


It's important to note that the optimization process involves aggregation and de-aggregation, which correspond to granulation and de-granulation of the underlying information and decision systems, respectively. This process is crucial for identifying relevant multi-relational approximation spaces. As Zadeh previously emphasized in his proposal on information granulation \cite{ch12-Zadeh-RNC-book,ch10-zadeh01new}, in decomposition of vague concepts ongoing dialogue with experts remains essential.

Furthermore, unlike traditional approaches that approximate a single concept, our method aims to construct a family of concepts in the learning phase of complex games. These concepts are labeled by specifications of transformations, which represent the specifications of actions. During testing, the quality of the discovered complex games is evaluated using application-specific quality measures. Additionally, based on the testing results, the games are adapted. AM supported by relevant reasoning techniques and expert collaboration, empowers IS to acquire new data sets. This allows for more efficient adaptation.

%%%%%%%%%%%
In our discussion on GrC and IGrC, we highlighted the importance of developing relevant quality measures (metrics) for constructed granules. We have mentioned various proposals for metrics used in ML, particularly those related to the Minimum Description Length (MDL) principle and confusion matrices. For instance, we can consider quality measures for features treated as granules in machine learning, as well as metrics used in the optimization of parameterized granules, such as meta-parameters in learning algorithms.

In GrC, it is proposed to use the principle of justifiable granularity as a basis for developing quality measures for granules \cite{ch12-Pedrycz-principle-of-just-ration2013,ch12-Pedrycz-justifable2022}. In IGrC, we suggest constructing these measures based on the relevance of granules as computational building blocks necessary for understanding perceived situations, enabling better decision-making. Consequently, quality measures should be considered in the context of c-granule control. A significant challenge lies in developing measures that support the estimation of the quality of c-granules within this control framework. This will allow for the evaluation of the behaviors of such granules, particularly regarding the quality of their control. Another example of challenges concerns developing measures for estimation of the quality of performed reasoning by c-granule control aiming to support control in making decisions concerning its behavior.

It can be noted that our discussion can also be applied to fuzzy sets \cite{ch10-zadeh65fuzzy}. In this case, the modeling of fuzzy membership functions also requires interactions with the physical environment. The detailed discussion will be presented elsewhere.

\section{Conclusions}

In this paper, we discussed GrC and its extension to IGrC as the basis for the design and analysis of IS's dealing with complex phenomena. The following conclusions summarize our considerations:

\begin{itemize}

\item{} The presented approach is based on a substantial generalization of the existing rough set approaches, including generalization of approximation spaces to dynamic networks of granular spaces and generated  computations over granular networks. These entities are used by control of c-granule to discover the relevant computational building blocks for cognition in the form of c-granules, granular networks and computations over them. By using discovered complex games the c-granule control is aiming to understand the perceived situations for making the right decisions  in realization of the specified tasks along generated granular computations.

\item{} The control mechanism of IS based on the rough set theory framework, aims to adaptively learn complex games. This allows IS to generate  computations over granular networks leading to high-quality approximate solutions.

\item{} The control system, aided by AM, continuously searches for relevant data sets represented in multi-relational approximation spaces as $AS_1,$ $\ldots, AS_k,$ or in more general granular spaces. The AM is leveraging advanced reasoning techniques in collaboration with domain experts.

\item{} These approximation spaces can be extended, in cooperation with experts, into a large family of multi-relational approximation spaces $\mathcal{F}_{AS_1, \ldots, AS_k}$ or networks of granular spaces and extracted from them granular networks. The system then performs optimization to identify high-quality complex games.

\item{} These complex games are adaptively modified based on observed changes in their performance. As previously discussed, complex games consist of sets of rules. The predecessors  of these rules are classifiers for often complex, imprecise and vague concepts triggering specifications of transformations appearing on the right hand side of rules to be applied to granular networks when a rule is chosen for execution by the IS control. One should note that IM may require to make multi-level decomposition of the specification of transformation to be realized before it can be directly embedded into the physical world.

\item{} The universes of objects of information (decision) systems constructed by control of IS are formed by expressions satisfying their types expressing properties of fragments of granular computations realized by IS.

%the solutions to the problem IcS is trying to solve, and the concepts being approximated within this universe correspond to different quality layers of these solutions.

\item{} In many cases, IS needs to generate the granular computation leading to high-quality solutions even when no known examples of such solutions exist and only some negative examples are available. This is the case when, {\em e.g.}, IS is searching for  new chemical compounds with some specified properties. The success of this generation process heavily relies on the reasoning techniques supporting the IS control. The quality of the generated solution, determined by the quality of the used complex game, often depends on the behavior of this complex game over the entire generated granular computation, not  its final state only.

\end{itemize}

In summary, this approach highlights that approximation problems in IS are significantly more complex than those typically encountered in rough set applications, so far. The success of the control system heavily depends on the quality of both reasoning techniques and the dialogue with domain experts.

Our considerations are opening a new directions for the new fascinating research on the generalized rough set approach in IcS. This research based on IGrC, using also many existing partial results from different areas like  multi-agent systems, perception and action, machine learning, natural language processing etc., may lead to creation solid foundations for the IcS design and analysis. This research will be important in the realization of of a bit generalized goal formulated in \cite{ch12-Bower-book}:
\begin{quotation}
{\em Tomorrow, I believe,}[we will use INTELLIGENT SYSTEMS] {\em to support our decisions in defining our research strategy and specific aims, in managing our experiments,
in collecting our results, interpreting our data, in incorporating the findings of others,
in disseminating our observations, in extending (generalizing) our experimental observations
 through exploratory discovery and modeling -- in directions completely unanticipated.
}
\end{quotation}

In the future, we will continue the research on the foundations of IGrC aiming to extend the current results to societies of c-granules with control as well as dialogues with experts as well as chatbots, LLM and Agentic Systems.
Designing societies of c-granules with control providing the required behaviour of society in the environment of another given society of c-granules is a challenge ({\em e.g.}, discovery of learning algorithms and classifiers, new medicine or chemical compounds, control of autonomous vehicles or design of robots \cite{ch12-RussellNorvig-bookAI,ch12-friedman01elements,ch12-Drug-Disovery-book-2021,ch12-Robot-design-PNAS}).


We will continue our research on the foundations of IGrC, aiming to expand our current findings to include societies of c-granules. This will involve both further studies on control of c-granules and dialogues with experts, chatbots, and large language models (LLMs) as well as Agentic Systems.

One of the challenges is to develop  the granulated deep learning approach on the basis of the IGrC model \cite{ch10-SankarPal-2021-RS-DeepLearning2021}. Moreover, also a combination of the rough set approach based on IGrC with the Lifelong Learning (LL) \cite{ch12-LifelongML-2018} is a challenge. LL calls for techniques of learning in the dynamic and open world or environment in a self-supervised manner what is consistent with the aims of IGrC. Many recent applications ({\em e.g.}, chatbots, self-driving cars, or AI systems interacting with humans/physical environments) require to cope with their dynamic and open environments. Hence, they  should   continuously learn new things in order to function well. Techniques supporting realization of this goal should be based on the relevant computing model.

The proposed IGrC model may also be treated as a step toward realization of combination of the physical structure and thinking behavior of the brain \cite{ch12-AI-meets-physics-2024}:
\begin{quotation}
{\em
We should combine the physical structure and thinking behavior of the brain, add physical priors, break through the bottleneck of computing power, realize low-power, low-parameter, high-speed, high-precision, non-depth AI models, and develop more efficient artificial intelligence technology.
}
\end{quotation}
In addition, further research on IGrC may shed new light on the relationships between physical and informational quantities (see, {\em e.g.}, \cite{Vopson-info-mass-2019}).

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