Andrzej Szałas

Computing Circumscription Revisited: A Reduction Algorithm

In recent years, a great deal of attention has been devoted to logics of common-sense reasoning. Among the candidates proposed, circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic reasoning, but difficult to apply in practice. The major reason for this is the second-order nature of circumscription axioms and the difficulty in finding proper substitutions of predicate expressions for predicate variables. One solution to this problem is to compile, where possible, second-order formulas into equivalent first-order formulas. Although some progress has been made using this approach, the results are not as strong as one might desire and they are isolated in nature. In this article, we provide a general method that can be used in an algorithmic manner to reduce certain circumscription axioms to first-order formulas. The algorithm takes as input an arbitrary second-order formula and either returns as output an equivalent first-order formula, or terminates with failure. The class of second-order formulas, and analogously the class of circumscriptive theories that can be reduced, provably subsumes those covered by existing results. We demonstrate the generality of the algorithm using circumscriptive theories with mixed quantifiers (some involving Skolemization), variable constants, nonseparated formulas, and formulas with n-ary predicate variables. In addition, we analyze the strength of the algorithm, compare it with existing approaches, and provide formal subsumption results.

On the Correspondence Between Modal and Classical Logic: an Automated Approach

The current paper is devoted to automated techniques in the correspondence theory. The theory we deal with concerns the problem of finding classical first-order axioms corresponding to propositiona ...

Concerning the semantic consequence relation in first-order temporal logic

In this paper we consider the first-order temporal logic with linear and discrete time. We prove that the set of tautologies of this logic is not arithmetical (i.e., it is neither Σ0n nor Π0n for any natural number n). Thus we show that there is no finitistic and complete axiomatization of the considered logic.

Knowledge Representation Techniques. : a rough set approach

The basis for the material in this book centers around a long term research project with autonomous unmanned aerial vehicle systems. One of the main research topics in the project is knowledge repr ...

Incompleteness of first-order temporal logic with until

The results presented in this paper concern the axiomatizability problem of first-order temporal logic with linear and discrete time. We show that the logic is incomplete, i.e., it cannot be provided with a finitistic and complete proof system. We show two incompleteness theorems. Although the first one is weaker (it assumes some first-order signature), we decided to present it, for its proof is much simpler and contains an interesting fact that finite sets are characterizable by means of temporal formulas. The second theorem shows that the logic is incomplete independently of any particular signature.

Tolerance Spaces and Approximative Representational Structures

In traditional approaches to knowledge representation, notions such as tolerance measures on data, distance between objects or individuals, and similarity measures between primitive and complex data structures are rarely considered. There is often a need to use tolerance and similarity measures in processes of data and knowledge abstraction because many complex systems which have knowledge representation components such as robots or software agents receive and process data which is incomplete, noisy, approximative and uncertain. This paper presents a framework for recursively constructing arbitrarily complex knowledge structures which may be compared for similarity, distance and approximativeness. It integrates nicely with more traditional knowledge representation techniques and attempts to bridge a gap between approximate and crisp knowledge representation. It can be viewed in part as a generalization of approximate reasoning techniques used in rough set theory. The strategy that will be used is to define tolerance and distance measures on the value sets associated with attributes or primitive data domains associated with particular applications. These tolerance and distance measures will be induced through the different levels of data and knowledge abstraction in complex representational structures. Once the tolerance and similarity measures are in place, an important structuring generalization can be made where the idea of a tolerance space is introduced. Use of these ideas is exemplified using two application domains related to sensor modeling and communication between agents.

Using Contextually Closed Queries for Local Closed-World Reasoning in Rough Knowledge Databases

Soft computing comprises various paradigms dedicated to approximately solving real-world problems, e.g., in decision making, classification or learning; among these paradigms are fuzzy sets, rough ...

On the Correspondence between Approximations and Similarity

This paper focuses on the use and interpretation of approximate databases where both rough sets and indiscernibility partitions are generalized and replaced by approximate relations and similarity spaces. Similarity spaces are used to define neighborhoods around individuals and these in turn are used to define approximate sets and relations. There is a wide spectrum of choice as to what properties the similarity relation should have and how this affects the properties of approximate relations in the database. In order to make this interaction precise, we propose a technique which permits specification of both approximation and similarity constraints on approximate databases and automatic translation between them. This technique provides great insight into the relation between similarity and approximation and is similar to that used in modal correspondence theory. In order to automate the translations, quantifier elimination techniques are used.

A complete axiomatic characterization of first-order temporal logic of linear time

As shown in (Szalas, 1986, 1986, 1987) there is no finitistic and complete axiomatization of First-Order Temporal Logic of linear and discrete time. In this paper we give an infinitary proof system for the logic. We prove that the proof system is sound and complete. We also show that any syntactically consistent temporal theory has a model. As a corollary we obtain that the Downward Theorem of Skolem, Lowenheim and Tarski holds in the case of considered logic.

Elimination of Predicate Quantifiers

Formulae of higher-order predicate logic are difficult to handle with automated inference systems. Some of these formulae, however, are equivalent to formulae of first-order predicate logic or even propositional logic. For example the formula of second-order predicate logic P (P P) is trivially equivalent to the propositional constant false. In applications where formulae of higher-order predicate logic occur naturally it is very useful to determine whether the given formula is in fact equivalent to a simpler formula of first-order or propositional logic. Typical applications where this occurs are predicate minimization by circumscription, correspondence theory in non-classical logic, and simple versions of set theory. In these areas we are faced with formulae of second-order predicate logic with existentially or universally quantified predicate variables and we want to simplify them by computing equivalent first-order formulae.