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\chapter{Adaptive Rough Sets and Adaptive Reasoning}\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=0.80\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}).

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