## Abstracts of Fundamenta Informaticae Volume 28.

Number 1-2

J. J. Alferes, L. M. Pereira, T. C. Przymusinski
Belief Revision in Non-Monotonic Reasoning and Logic Programming.
pages 1-22
In order to be able to explicitly reason about beliefs, we've introduced a non-monotonic formalism, called the Autoepistemic Logic of Beliefs, AEB$, obtained by augmenting classical propositional logic with a belief operator, B. For this language we've defined the static autoepistemic expansions semantics. The resulting non-monotonic knowledge representation framework turned out to be rather simple and yet quite powerful. Moreover, it has some very natural properties which sharply contrast with those of Moore's AEL. While static expansions seem to provide a natural and intuitive semantics for many belief theories, and, in particular, for all affirmative belief theories (which include the class of all normal and disjunctive logic programs), they often can lead to inconsistent expansions for theories in which (subjective) beliefs clash with the known (objective) information or with some other beliefs. In particular, this applies to belief theories (and to logic programs) with strong or explicit negation. In this paper we generalize AEB to avoid the acceptance of inconsistency provoking beliefs. We show how such AEB theories can be revised to prevent belief originated inconsistencies, and also to introduce declarative language level control over the revision level of beliefs, and apply it to the domains of diagnosis and declarative debugging. The generality of our$AEB\$ framework can capture and justify the methods that have been deployed to solve similar revision problems within the logic programming paradigm.
K. R. Apt, R. Ben-Eliyahu
Meta-variables in Logic Programming, or in Praise of Ambivalent Syntax.
pages 23-36
We show here that meta-variables of Prolog admit a simple declarative interpretation. This allows us to extend the usual theory of SLD-resolution to the case of logic programs with meta-variables, and to establish soundness and strong completeness of the corresponding extension of the SLD-resolution. The key idea is the use of ambivalent syntax which allows us to use the same symbols as function and relation symbols.
We also study the problem of absence of run-time errors in presence of meta-variables. We prove that this problem is undecidable. However, we also provide some sufficient and polynomial-time-decidable conditions which imply absence of run-time errors.
H.A.Blair
Designing Dependencies.
pages 37-54
Given a binary recursively enumerable relation R, one or more logic programs over a language L can be constructed and interconnected to produce a dependency relation D on selected predicates within the Herbrand base BL isomorphic to R. D can be, optionally, a positive, negative or mixed dependency relation. The construction is applied to representing any effective game of the type introduced by Gurevich and Harrington, which they used to prove Rabin's decision method for S2S, as the dependency relation of a logic program. We allow games over an infinite alphabet of possible moves. We use this representation to reveal a common underlying reason, having to do with the shape of a program's dependency relation, for the complexity of several logic program properties.
A. Blikle
Why Denotational? Remarks on Applied Denotational Semantics.
pages 55-85
This is an essay where the author expresses his views on applied denotational semantics. In the author's opinion, whether a software system has or does not have a sufficiently abstract denotational semantics should be regarded as a pragmatic attribute of the system rather than merely as a mathematical attribute of its description. In a software system with denotational semantics structured programming is feasible and for such systems there is a routine method of developing program-correctness logic. All that may not be the case if denotationality is not ensured. On the other hand, a non-denotational semantics can be always artificially made denotational on the expense of lowering its level of abstraction. This leads to an important pragmatic question: to what extent and in which situations can we sacrifice denotationality and/or abstraction of a semantics? All discussions are carried on an algebraic ground but the paper is not very technical and contains a short introduction to the algebraic theory of denotational semantics.
J. Dix, G. Gottlob, W. Marek
Reducing Disjunctive to Non-Disjunctive Semantics by Shift-Operations.
pages 87-100
It is wellknown that Minker's semantics GCWA for positive disjunctive programs P, i.e.~to decide if a literal is true in all minimal models of P is Pi^P_2-complete. This is in contrast to the same entailment problem for semantics of non-disjunctive programs such as STABLE and SUPPORTED (both are co-NP-complete) as well as M^{supp}_P and WFS (that are even polynomial).
Recently, the idea of reducing disjunctive to non-disjunctive programs by using so called shift-operations was introduced independently by Bonatti, Dix/Gottlob/Marek, and Schaerf. In fact, Schaerf associated to each semantics SEM for normal programs a corresponding semantics Weak-SEM for disjunctive programs and asked for the properties of these weak semantics, in particular for the complexity of their entailment relations. While Schaerf concentrated on Weak-STABLE and Weak-SUPPORTED, we investigate the weak versions of Apt/Blair/Walker's stratified semantics M^{supp}_P and of Van Gelder/Ross/Schlipf's well-founded semantics WFS.
We show that credulous entailment for both semantics is NP-complete (consequently, sceptical entailment is co-NP--complete). Thus, unlike GCWA, the complexity of these semantics belongs to the first level of the polynomial hierarchy. Note that, unlike Weak-WFS, the semantics Weak-M^{supp}_P is not always defined: testing consistency of Weak-M^{supp}_P is also NP--complete.
We also show that Weak-WFS and Weak-M^{supp}_P are cumulative and rational and that, in addition, Weak-WFS satisfies some of the well-behaved principles introduced by Dix.
M.Fitting
A Modal Herbrand Theorem.
pages 101-122
We state and prove a modal Herbrand theorem that is, we believe, a more natural analog of the classical version than has appeared before. The statement itself requires the enlargement of the usual machinery of first-order modal logic---we use the device of predicate abstraction, something that has been considered elsewhere as well. This expands the expressive power of modal logic in a natural way. Our proof of the modal version of Herbrand's theorem uses a tableau system that takes predicate abstraction into account. It is somewhat simpler than other systems for the same purpose that have previously appeared.
G. Gottlob, M. Truszczynski
Approximating the Stable Model Semantics is Hard.
pages 123-128
In this paper we investigate the complexity of problems concerned with approximating the stable model semantics. We show that under rather weak assumptions it is NP-hard to decide whether the size of a polynomially computable approximation is within a constant factor from the size of the intersection (union) of stable models of a program. We also show that unless P=NP, no approximation exists that uniformly bounds the intersection (union) of stable models.
J. Hsiang, A. Wasilewska
Automating Algebraic Proofs in Algebraic Logic.
pages 129-140
We present here an effective proof theory that allows one to reason within algebras of algebraic logic in a purely syntactic, algebraic fashion. We demonstrate the effectiveness of the method by discussing our automated proofs of problems and theorems taken from Professor Helena Rasiowa's book An Algebraic Approach to Non-Classical Logics, Studies in Logic and Fundation of Mathematics, Volume 78, North Holland Publishing Company, Amsterdam, London - PWN, Warsaw (1974). We include a detailed proof of a problem presented to us by Professor Helena Rasiowa in June 1993. Most of these proofs are the first direct proofs ever discovered, and they are produced by the computer without human assistance.
G. Mirkowska, A. Salwicki
The Algebraic Specifications do not have the Tennenbaum Property.
pages 141-152
It is commonly believed that a programmable model satisfying the axioms of a given algebraic specification guarantees good properties and is a correct implementation of the specification. This convinction might be related to the Tennenbaum's property[?] of the arithmetic: every computable model of the Peano arithmetic of natural numbers is isomorphic to the standard model.
Here, on the example of stacks, we show a model satisfying all axioms of the algebraic specification of stacks which can not be accepted as a good model in spite of the fact that it is defined by a program. For it enables to ''pop'' a stack infinitely many times.
J. A. Plaza
Logic Programming from the Perspective of Algebraic Semantics.
pages 153-164
We present an approach to foundations of logic programming in which the connection with algebraic semantics becomes apparent. The approach is based on omega-Herbrand models instead of conventional Herbrand models. We give a proof of Clark's theorem on completeness of SLD-resolution by methods of the algebraic semantics. We prove the existence property for definite programs.
V. Pratt, J. Tiuryn
Satisfiability of Inequalities in a Poset.
pages 165-182
We consider tractable and intractable cases of the satisfiability problem for conjunctions of inequalities between variables and constants in a fixed finite poset. We show that crowns are intractable. We study members and closure properties of the class of tractable posets. We define a feasible poset to be one whose potential obstacles to satisfiability are representable by a certain formula of the first-order extended by the least fixed operator. For bipartite posets we give a complete classification of hard posets and feasible ones.
B. A. Trakhtenbrot
On the Power of Compositional Proofs for Nets: Relationships Between Completenesess and Modularity.
pages 183-195
The intention of the present paper is to elucidate at a more abstract (and hopefully more persuasive) level the relationship between modularity of net semantics and completeness of compositional proofs for nets. This is to be achieved by a separation of concerns which is not reflected in common terminology.
P. Urzyczyn
Positive Recursive Type Assignment.
pages 197-209
We consider several different definitions of type assignment with positive recursive types, from the point of view of their typing ability. We discuss the relationships between these systems. In particular, we show that the class of typable pure lambda terms remains the same for different type disciplines involving positive type fixpoints, and that type reconstruction is decidable.
Number 3-4
M. Banerjee, M. K. Chakraborty
Rough Sets Through Algebraic Logic
While studying rough equality within the framework of the modal system S5, an algebraic structure called rough algebra [1], came up. Its features were abstracted to yield a topological quasi-Boolean algebra (tqBa). In this paper, it is observed that rough algebra is more structured than a tqBa. Thus, enriching the tqBa with additional axioms, two more structures, viz. {\em pre-rough algebra and rough algebra, are defined. Representation theorems of these algebras are also obtained. Further, the corresponding logical systems L1, L2 are proposed and eventually, is L2 proved to be sound and complete with respect to a rough set semantics.
Alexander Bochman
On a Logical Basis of Normal Logic Programs
pages 223--245
In this paper we develop a logical framework for normal logic programs, called default consequence relations. We give a representation of major semantics for logic programs in this formalism and study the question what logics are adequate for the latter. It is shown that, in general, default consequence relations based on three-valued inference are appropriate for these semantics, though different semantics admit different kinds of inference.
Nguyen Cat Ho
A Method in Linguistic Reasoning on a Knowledge Base Representing by Sentences with Linguistic Belief
pages 247--259
In this paper we shall introduce a notion of formal knowledge base consisting of statements associated with linguistic truth degrees and a set of inference rules handling this kind of statements. A deductive reasoning method based on these inference rules will be examined. The logical foundation of this method is linguistic valued logic based on extended hedge algebras and, based on this basis, the consistency of the knowledge base will be considered.
P. Doherty, W. Lukaszewicz, A. Szalas
A Reduction Result for Circumscribed Semi-Horn Formulas
pages 261--271
Circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic and commonsense reasoning, but difficult to apply in practice due to the use of second-order formulas. One proposal for dealing with the computational problems is to identify classes of first-order formulas whose circumscription can be shown to be equivalent to a first-order formula. In previous work, we presented an algorithm which reduces certain classes of second-order circumscription axioms to logically equivalent first-order formulas. The basis for the algorithm is an elimination lemma due to Ackermann. In this paper, we capitalize on the use of a generalization of Ackermann's Lemma in order to deal with a subclass of universal formulas called semi-Horn formulas. Our results subsume previous results by Kolaitis and Papadimitriou regarding a characterization of circumscribed definite logic programs which are first-order expressible. The method for distinguishing which formulas are reducible is based on a boundedness criterion. The approach we use is to first reduce a circumscribed semi-Horn formula to a fixpoint formula which is reducible if the formula is bounded, otherwise not. In addition to a number of other extensions, we also present a fixpoint calculus which is shown to be sound and complete for bounded fixpoint formulas.
J. Green, N. Horne, E. Orlowska, P. Siemens
A Rough Set Model of Information Retrieval
pages 273--296
The purpose of this paper is to define and investigate document retrieval strategies based on rough sets.
A subject classification index is modelled by means of a family of set-theoretic information systems. The family induces a power set hierarchy built from the set of keywords.
In the paper, the task of information retrieval in response to a query that refers to several levels of that hierarchy is defined. Retrieval strategies are proposed and discussed. The strategies are, roughly speaking, based on various types of similarity between sets of keywords included in a query and in a document. It is not necessarily required that the document matches the query; we allow for approximate matching. Measures of relevance of documents to a query are proposed for the given strategies.
D. Grigoriev, M. Karpinski, A. M. Odlyzkol
Short Proofs for Nondivisibility of Sparse Polynomials under the Extended Riemann Hypothesis
pages 297--301
We prove for the first time an existence of the short (polynomial size) proofs for nondivisibility of two sparse polynomials (putting thus this = problem is the class NP) under the Extended Riemann Hypothesis. The divisibility problem is closely related to the problem of rational interpolation. Its computational complexity was studied in [5], [4], and [6].
We prove also, somewhat surprisingly, the problem of deciding whether a rational function given by a black box equals to a polynomial belong to the parallel class NC (see, e.g., [KR 90]), provided we know the degree of its sparse representation.
A.Gomolinska
On Semantics for the Nonmonotonic Modal Formalization of the Logic of Acceptance and Rejection
pages 303--313
The logic of acceptance and rejection (AEL2) is a nonmonotonic formalism in the spirit of Moore's autoepistemic logic (AEL). AEL2 is to represent knowledge of an ideal rational agent who makes decisions about pieces of information. The case of uncertainty of the agent is considered as well.
Therefore the agent's choices are: acceptance, rejection, and the lack of decision. AEL is treated as the nonmonotonic KD45 modal logic. AEL2 may be formalized in a similar way. In the paper the problem of an appropriate semantics for the modal formalization of AEL2 is discussed.
G. Paun, L. Polkowski, A. Skowron
Parallel Communicating Grammar Systems with Negotiation
pages 315--330
In a parallel communicating grammar system, several grammars work together, synchronously, on their own sentential forms, and communicate on request. No restriction is imposed usually about the communicated strings. We consider here two types of restrictions, as models of the negotiation process in multi-agent systems: (1) conditions formulated on the transmitted strings, and (2) conditions formulated on the resulting string in the receiving components. These conditions are expressed in terms of regular languages, whose elements the mentioned strings should be. Moreover, we allow that the communicated string is a part (any one, a prefix, a maximal, a minimal one, etc.) of the string of the communicating component. We investigate these variants from the point of view of the generative capacity of the corresponding parallel communicating grammar systems.
Argumentation Systems
pages 331--346
Argumentation systems are useful instruments for decision support when the arguments are changed dynamically in dependence of the users and the situation. In argumentation systems different strategies are used when the system prepares the argumentation for a given situation and/or user. These strategies correspond to some many valued logics and some of these strategies have fuzzy and rough interpretations. Many information systems have interpretation in different argumentation systems. For the lower and upper approximations there are equivalent argumentation strategies that correspond to the well known logical systems. Moreover when a practical system chooses the arguments for the user it needs more compact argumentation then the list of all procedures it uses. Then the argumentation system finds arguments with the same persuade power as its argumentation.
W. Skarbek
Dynamic Behaviour of Spatial Signal OR-Graphs
pages 347--352
Spatial Signal OR Graphs (SSOG) are defined and specialized to fractal pattern generation (FSSOG) and cellular array pattern generation(CSSOG). It is proved that fractal patterns always stabilize after predictable transient time, while cellular pattern graphs enter in parallel mode into a stable state if the initial state is even, otherwise they enter into a cycle of length 2. If nondeterministic mode is considered then any G in SSOG enters a stable state with probability one.
Z. Suraj
Discovery of Concurrent Data Models from Experimental Tables: A Rough Set Approach
pages 353--376
The main objective of machine discovery is the determination of relations between data and of data models. In the paper we describe a method for discovery of data models represented by concurrent systems from experimental tables. The basic step consists in a determination of rules which yield a decomposition of experimental data tables; the components are then used to define fragments of the global system corresponding to a table. The method has been applied to automatic data models discovery = from experimental tables with Petri nets as models for concurrency.
K. Szlechta
A Two-Stage Approach to First Order Default Reasoning
pages 377--402
Our subject is the representation and analysis of simple first-order default statements of ordinary language, such as normally, birds fly''. There are, among other approaches, two kinds of analysis, both semantic in style. One interprets normally, birds fly'' along the lines of for every item x in the domain of discourse, the most normal models of x is a bird'' are models of x flies''''. This is the preferential models approach, first outlined by Bossu/Siegel and Shoham, and studied by Kraus, Lehmann, Magidor and others. The other interprets normally, birds fly'' along the lines of there is an important subset of the birds, all of whose elements fly''. This is the generalized quantifier approach, formulated and developed by the author. The purpose of the present paper is to show how the two approaches may usefully be combined into a single two-stage approach, and how such a combination provides an elegant account of certain problematic examples.
J. Winkowski
Concatenable Weighted Pomsets and Their Applications to Modelling Processes of Petri Nets
pages 403--421
Structures called concatenable weighted pomsets are introduced which can serve as models of processes of Petri nets, including nets with time features. Operations on such structures are defined which allow to combine them sequentially and in parallel. These operations correspond to natural operations on processes. They make the universe of concatenable weighted pomsets a partial algebra which appears to be a symmetric strict monoidal category. Sets of processes of timed and time Petri nets are characterized as subsets of this algebra.
J. Wroblewski
Theoretical Foundations of Order-Based Genetic Algorithms
pages 423--430
A lot of research on genetic algorithms theory is concentrated on classical, binary case. However, there are many other types of useful genetic algorithms (GA), e.g. tree-based (genetic programming), or order-based ones. This paper shows, that many of classical results can be transferred into the order-based GAs. The analysis includes the Schema Theorem and Markov chain modelling of order-based .