Forgetting is an important concept in knowledge representation and automated reasoning with widespread applications across a number of disciplines. A standard forgetting operator, characterized in [26] in terms of model-theoretic semantics and primarily focusing on the propositional case, opened up a new research subarea. In this paper, a new operator called weak forgetting, dual to standard forgetting, is introduced and both together are shown to offer a new more uniform perspective on forgetting operators in general. Both the weak and standard forgetting operators are characterized in terms of entailment and inference, rather than a model theoretic semantics. This naturally leads to a useful algorithmic perspective based on quantifier elimination and the use of Ackermanns Lemma and its fixpoint generalization. The strong formal relationship between standard forgetting and strongest necessary conditions and weak forgetting and weakest sufficient conditions is also characterized quite naturally through the entailment based, inferential perspective used. The framework used to characterize the dual forgetting operators is also generalized to the first-order case and includes useful algorithms for computing first-order forgetting operators in special cases. Practical examples are also included to show the importance of both weak and standard forgetting in modeling and representation.

The Bdi model of rational agency has been studied for over three decades. Many robust multiagent systems have been developed, and a number of Bdi logics have been studied. Following this intensive development phase, the importance of integrating Bdi models with inconsistency handling and revision theory have been emphasized. There is also a demand for a tighter connection between Bdi-based implementations and Bdi logics. In this paper, we address these postulates by introducing a novel, paraconsistent logical Bdi model close to implementation, with building blocks that can be represented as Sql/rule-based databases. Importantly, tractability is achieved by reasoning as querying. This stands in a sharp contrast to the high complexity of known Bdi logics. We also extend belief shadowing, a shallow and lightweight alternative to deep and computationally demanding belief revision, to encompass agents motivational attitudes.

Reasoning about uncertainty is one of the main cornerstones of Knowledge Representation. More recently, combining logic with probability has been of major interest. Rough set methods have been proposed for modeling incompleteness and imprecision based on indiscernibility and its generalizations and there is a large body of work in this direction. More recently, the classical theory has been generalized to include probabilistic rough set methods of which there are also a great variety of proposals. Pragmatic, easily accessible, and easy to use tools for specification and reasoning with this wide variety of methods is lacking. It is the purpose of this paper to fill in that gap where the focus will be on probabilistic rough set methods. A landscape of (probabilistic) rough set reasoning methods and the variety of choices involved in specifying them is surveyed first. While doing this, an abstract generalization of all the considered approaches is derived which subsumes each of the methods. One then shows how, via this generalization, one can specify and reason about any of these methods using PROBLOG, a popular and widely used probabilistic logic programming language based on PROBLOG. The paper also considers new techniques in this context such as the use of probabilistic target sets when defining rough sets and the use of partially specified base relations that are also probabilistic. Additionally, probabilistic approaches using tolerance spaces are proposed. The paper includes a rich set of examples and provides a framework based on a library of generic PROBLOG relations that make specification of any of these methods, straightforward, efficient and compact. Complete, ready to run PROBLOG code is included in the Appendix for all examples considered.

Reasoning about uncertainty is one of the main cornerstones of Knowledge Representation. Formal representations of uncertainty are numerous and highly varied due to different types of uncertainty intended to be modeled such as vagueness, imprecision and incompleteness. There is a rich body of theoretical results that has been generated for many of these approaches. It is often the case though, that pragmatic tools for reasoning with uncertainty lag behind this rich body of theoretical results. Rough set theory is one such approach for modeling incompleteness and imprecision based on indiscernibility and its generalizations. In this paper, we provide a pragmatic tool for constructively reasoning with generalized rough set approximations that is based on the use of Answer Set Programming (Asp). We provide an interpretation of answer sets as (generalized) approximations of crisp sets (when possible) and show how to use Asp solvers as a tool for reasoning about (generalized) rough set approximations situated in realistic knowledge bases. The paper includes generic Asp templates for doing this and also provides a case study showing how these techniques can be used to generate reducts for incomplete information systems. Complete, ready to run clingo Asp code is provided in the Appendix, for all programs considered. These can be executed for validation purposes in the clingo Asp solver.

Ignorance, inconsistency, nonsense and similar phenomena are omnipresent in everyday reasoning. They have been intensively studied, especially in the area of multiple-valued logics. Therefore we develop a framework for belief bases, combining multiple-valued and probabilistic reasoning, with the main focus on the way belief bases are actually used and accessed through queries.As an implementation tool we use a probabilistic programming language PROBLOG. Though based on distribution semantics with the independence assumption, we show how its constructs can successfully be used in implementing the considered logics and belief bases. In particular, we develop a technique for shifting probabilistic dependencies to the level of symbolic parts of belief bases.We also discuss applications of the framework in reasoning with Likert-type scales, widely exploited in questionnaire-based experimental research in psychology, economics, sociology, politics, public opinion measurements, and related areas.

Answer Set Programming (ASP) is nowadays a dominant rule-based knowledge representation tool. Though existing ASP variants enjoy efficient implementations, generating an answer set remains intractable. The goal of this research is to define a new ASP-like rule language, 4SP, with tractable model generation. The language combines ideas of ASP and a paraconsistent rule language 4QL. Though 4SP shares the syntax of ASP and for each program all its answer sets are among 4SP models, the new language differs from ASP in its logical foundations, the intended methodology of its use and complexity of computing models. As we show in the paper, 4QL can be seen as a paraconsistent counterpart of ASP programs stratified with respect to default negation. Although model generation for 4QL programs is tractable, dropping stratification makes it intractable for both 4QL and ASP. To retain tractability while allowing non-stratified programs, in 4SP we introduce trial expressions interlacing programs with hypotheses as to the truth values of default negations. This allows us to develop a model generation algorithm with deterministic polynomial time complexity. We also show relationships among 4SP, ASP and 4QL.

Contemporary systems situated in real-world open environments frequently have to cope with incomplete and inconsistent information that typically increases complexity of reasoning and decision processes. Realistic modeling of such informationally complex environments calls for nuanced tools. In particular, incomplete and inconsistent information should neither trivialize nor stop both reasoning or planning. The paper introduces ACTLOG, a rule-based four-valued language designed to specify actions in a paraconsistent and paracomplete manner. ACTLOG is an extension of 4QL(Bel), a language for reasoning with paraconsistent belief bases. Each belief base stores multiple world representations. In this context, ACTLOGs action may be seen as a belief bases transformer. In contrast to other approaches, ACTLOG actions can be executed even when the underlying belief base contents is inconsistent and/or partial. ACTLOG provides a nuanced action specification tools, allowing for subtle interplay among various forms of nonmonotonic, paraconsistent, paracomplete and doxastic reasoning methods applicable in informationally complex environments. Despite its rich modeling possibilities, it remains tractable. ACTLOG permits for composite actions by using sequential and parallel compositions as well as conditional specifications. The framework is illustrated on a decontamination case study known from the literature.

We apply a non-classical four-valued logic in the process of reasoning regarding strategies for cops in a modified game of \"Cops and Robber\" played on a graph. We extend the game by introducing uncertainty in a form of random failures of detecting devices. This is realized by allowing that a robber can be detected in a node only with the given probability P-A. Additionally, with the probability P-F, cops can be given a false-positive, i.e., they are informed that the robber is located at some node, whereas it is located somewhere else. Consequently, non-zero P-F introduces a measurement noise into the system. All the cops have access to information provided by the detectors and can communicate with each other, so they can coordinate the search. By adjusting the number of detectors,P-A, and P-F we can achieve a smooth transition between the two well-known variants of the game: \"with fully visible robber\" and \"with invisible robber\". We compare a simple probabilistic strategy for cops with the non-parametric strategy based on reasoning with a four-valued paraconsistent logic. It is shown that this novel approach leads to a good performance, as measured by the required mean catch-time. We conclude that this type of reasoning can be applied in real-world applications where there is no knowledge about the underlying source of errors which is particularly useful in robotics.

Modern artificial systems, such as cooperative traffic systems or swarm robotics, are made of multiple autonomous agents, each handling uncertain, partial and potentially inconsistent information, used in their reasoning and decision making. Graded reasoning, being a suitable tool for addressing phenomena related to such circumstances, is investigated in the literature in many contexts – from graded modal logics to various forms of approximate reasoning. In this paper we first introduce a family of many-valued paraconsistent logics parametrised by a number of truth/falsity/inconsistency grades allowing one to handle multiple truth-values at the desired level of accuracy. Second, we define a corresponding family of rule-based languages with graded truth-values as first-class citizens, enjoying tractable query evaluation. In addition, we introduce introspection operators allowing one to resolve inconsistencies and/or lack of information in a non-monotonic manner. We illustrate and discuss the use of the framework in an autonomous robot scenario.

We introduce a Horn description logic called Horn-DL, which is strictly and essentially richer than Horn-Reg(1), Horn-SHTQ and Horn-SROIQ, while still has PTime data complexity. In comparison with Horn-SROIQ, Horn-DL additionally allows the universal role and assertions of the form irreflexive(s), -s(a, b), a b. More importantly, in contrast to all the well-known Horn fragments epsilon L, DL-Lite, DLP, Horn-SHIQ, and Horn-SROIQ of description logics, Horn-DL allows a form of the concept constructor \"universal restriction\" to appear at the left hand side of terminological inclusion axioms. Namely, a universal restriction can be used in such places in conjunction with the corresponding existential restriction. We develop the first algorithm with PTime data complexity for checking satisfiability of Horn-DL knowledge bases.

This paper discusses an implementation of four speech acts: assert, concede, request and challenge in a paraconsistent framework. A natural four-valued model of interaction yields multiple new cognitive situations. They are analyzed in the context of communicative relations, which partially replace the concept of trust. These assumptions naturally lead to six types of situations, which often require performing conflict resolution and belief revision. The particular choice of a rule-based, DATALOC. like query language 4QL as a four-valued implementation framework ensures that, in contrast to the standard two-valued approaches, tractability of the model is achieved.

We study a Horn fragment called Horn-RegI of the regular description logic with inverse RegI, which extends the description logic ALC with inverse roles and regular role inclusion axioms characterized by finite automata. In contrast to the well-known Horn fragments EL, DL-Lite, DLP, Horn-SHIQ and Horn-SROIQ of description logics, Horn-RegI allows a form of the concept constructor \"universal restriction\" to appear at the left hand side of terminological inclusion axioms, while still has PTIME data complexity. Namely, a universal restriction can be used in such places in conjunction with the corresponding existential restriction. We provide an algorithm with PTIME data complexity for checking satisfiability of Horn-RegI knowledge bases.

We develop a new Web ontology rule language, called WORL, which combines a variant of OWL 2 RL with eDatalog ¬ . We allow additional features like negation, the minimal number restriction and unary external checkable predicates to occur at the left-hand side of concept inclusion axioms. Some restrictions are adopted to guarantee a translation into eDatalog ¬ . We also develop the well-founded semantics and the stable model semantics for WORL as well as the standard semantics for stratified WORL (SWORL) via translation into eDatalog ¬ . Both WORL with respect to the well-founded semantics and SWORL with respect to the standard semantics have PTime data complexity. In contrast to the existing combined formalisms, in WORL and SWORL negation in concept inclusion axioms is interpreted using nonmonotonic semantics.

This paper focuses on approximate reasoning based on the use of approximation spaces. Approximation spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak. Approximation spaces are used to define neighborhoods around individuals and rough inclusion functions. These in turn are used to define approximate sets and relations. In any of the approaches, one would like to embed such relations in an appropriate logical theory which can be used as a reasoning engine for specific applications with specific constraints. We propose a framework which permits a formal study of the relationship between properties of approximations and properties of approximation spaces. Using ideas from correspondence theory, we develop an analogous framework for approximation spaces. We also show that this framework can be strongly supported by automated techniques for quantifier elimination.

The current article is devoted to extensions of the rule query language 4QL proposed by Małuszyński and Szałas. 4QL is a Datalog^¬¬-like language, allowing one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for lightweight versions of well-known forms of non-monotonic reasoning. In addition, 4QL is tractable w.r.t. data complexity and captures PTime queries. In the current article we relax most of restrictions of 4QL, obtaining a powerful but still tractable query language 4QL⁺. In its development we mainly focused on its pragmatic aspects: simplicity, tractability and generality. In the article we discuss our approach and choices made, define a new, more general semantics and investigate properties of 4QL⁺.

<em>In this paper we study automated reasoning in the modal logic CPDLreg which is a combination of CPDL (Propositional Dynamic Logic with Converse) and REGc (Regular Grammar Logic with Converse). The logic CPDL is widely used in many areas, including program verification, theory of action and change, and knowledge representation. On the other hand, the logic REGc is applicable in reasoning about epistemic states and ontologies (via Description Logics). The modal logic CPDLreg can serve as a technical foundation for reasoning about agency. Even very rich multi-agent logics can be embedded into CPDLreg via a suitable translation. Therefore, CPDLreg can serve as a test bed to implement and possibly verify new ideas without providing specific automated reasoning techniques for the logic in question. This process can to a large extent be automated. The idea of embedding various logics into CPDLreg is illustrated on a rather advanced logic TEAMLOGK designed to specify teamwork in multi-agent systems. Apart from defining informational and motivational attitudes of groups of agents, TEAMLOGK allows for grading agents' beliefs, goals and intentions. The current paper is a companion to our paper (Dunin-Kęplicz et al., 2010a). The main contribution are proofs of soundness and completeness of the tableau calculus for CPDLreg provided in (Dunin-Kęplicz et al., 2010a).</em>

Grammar logics were introduced by Fariñas del Cerro and Penttonen in 1988 and have been widely studied. In this paper we consider regular grammar logics with converse (<em>REG</em> ^<em>c</em>logics) and present sound and complete tableau calculi for the general satisfiability problem of <em>REG</em> ^<em>c</em>logics and the problem of checking consistency of an ABox w.r.t. a TBox in a <em>REG</em> ^<em>c</em>logic. Using our calculi we develop ExpTime (optimal) tableau decision procedures for the mentioned problems, to which various optimization techniques can be applied. We also prove a new result that the data complexity of the instance checking problem in <em>REG</em> ^<em>c</em>logics is coNP-complete.

The paper discusses properties of 4QL, a DATALOG¬¬-like query language, originally outlined by Małuszy´nski and Szałas (Małuszy´nski &amp; Szałas, 2011). 4QL allows one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for “lightweight” versions of known forms of nonmonotonic reasoning. Negated literals in heads of rules may naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some literals. Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in (Małuszy´nski et al., 2008; Vitória et al., 2009) with truth values: ‘true’, ‘false’, ‘inconsistent’ and ‘unknown’. In addition, 4QL is tractable w.r.t. data complexity and captures PTIME queries. Even though DATALOG¬¬ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties. In the current paper we:<ul><li>investigate properties of well-supported models of 4QL</li><li>prove the correctness of the algorithm for computing well-supported models</li><li>show that 4QL has PTIME data complexity and captures PTIME.</li></ul>

Description logics (DLs) are a family of knowledge representation languages which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. DLs can be used, for example, for conceptual modeling or as ontology languages. In fact, OWL (Web Ontology Language), recommended by W3C, is based on description logics. In the current paper we give the first direct ExpTime (optimal) tableau decision procedure, which is not based on transformation or on the pre-completion technique, for checking satisfiability of a knowledge base in the description logic <em>SH</em><img src=\"http://www.springerlink.com/jsMath/fonts/cmsy10/alpha/120/char53.png\" /><img src=\"http://www.springerlink.com/jsMath/fonts/cmsy10/alpha/120/char48.png\" />. Our procedure uses sound global caching and can be implemented as an extension of the highly optimized tableau prover TGC to obtain an efficient program for the mentioned satisfiability problem.

We reformulate Pratts tableau decision procedure of checking satisfiability of a set of formulas in PDL. Our formulation is simpler and its implementation is more direct. Extending the method we give the first Ex PT m E (optimal) tableau decision procedure not based on transformation for checking consistency of an ABox w.r.t. a TBox in PDL (here, PDL is treated as a description logic). We also prove a new result that the data complexity of the instance checking problem in PDL is coNP-complete.

In natural language we often use graded concepts, reflecting different intensity degrees of certain features. Whenever such concepts appear in a given real-life context, they need to be appropriately expressed in its models. In this paper, we provide a framework which allows for extending the BGI model of agency by grading beliefs, goals and intentions. We concentrate on TEAMLOG [6, 7, 8, 9, 12] and provide a complexity-optimal decision method for its graded version TEAMLOG(K) by translating it into CPDLreg (propositional dynamic logic with converse and \"inclusion axioms\" characterized by regular languages). We also develop a tableau calculus which leads to the first EXPTIME (optimal) tableau decision procedure for CPDLreg. As CPDLreg is suitable for expressing complex properties of graded operators, the procedure can also be used as a decision tool for other multiagent formalisms.

In this paper we present a framework for fusing approximate knowledge obtained from various distributed, heterogenous knowledge sources. This issue is substantial in modeling multi-agent systems, where a group of loosely coupled heterogeneous agents cooperate in achieving a common goal. In paper [5] we have focused on defining general mechanism for knowledge fusion. Next, the techniques ensuring tractability of fusing knowledge expressed as a Horn subset of propositional dynamic logic were developed in [13,16]. Propositional logics may seem too weak to be useful in real-world applications. On the other hand, propositional languages may be viewed as sublanguages of first-order logics which serve as a natural tool to define concepts in the spirit of description logics [2]. These notions may be further used to define various ontologies, like e. g. those applicable in the Semantic Web. Taking this step, we propose a framework, in which our Horn subset of dynamic logic is combined with deductive database technology. This synthesis is formally implemented in the framework of HSPDL architecture. The resulting knowledge fusion rules are naturally applicable to real-world data.

In this paper we investigate a technique for fusing approximate knowledge obtained from distributed, heterogeneous information sources. This issue is substantial, e.g., in modeling multiagent systems, where a group of loosely coupled heterogeneous agents cooperate in achieving a common goal. Information exchange, leading ultimately to knowledge fusion, is a natural and vital ingredient of this process. We use a generalization of rough sets and relations [30], which depends on allowing arbitrary similarity relations. The starting point of this research is [6], where a framework for knowledge fusion in multiagent systems is introduced. Agents individual perceptual capabilities are represented by similarity relations, further aggregated to express joint capabilities of teams, This aggregation, expressing a shift from individual to social level of agents activity, has been formalized by means of dynamic logic. The approach of Doherty et al. (2007) [6] uses the full propositional dynamic logic, which does not guarantee tractability of reasoning. Our idea is to adapt the techniques of Nguyen [26-28] to provide an engine for tractable approximate database querying restricted to a Horn fragment of serial dynamic logic. We also show that the obtained formalism is quite powerful in applications.

In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph’s nodes. Such theories, which we call <em>annotation theories</em>, can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21].

Mathematical theory of voting and social choice has attracted much attention. In the general setting one can view social choice as a method of aggregating individual, often conflicting preferences and making a choice that is the best compromise. How preferences are expressed and what is the “best compromise” varies and heavily depends on a particular situation. The method we propose in this paper depends on expressing individual preferences of voters and specifying properties of the resulting ranking by means of first-order formulas. Then, as a technical tool, we use methods of second-order quantifier elimination to analyze and compute results of voting. We show how to specify voting, how to compute resulting rankings and how to verify voting protocols.

In the paper we present a technique for eliminating quantifiers of arbitrary order, in particular of first-order. Such a uniform treatment of the elimination problem has been problematic up to now, since techniques for eliminating first-order quantifiers do not scale up to higher-order contexts and those for eliminating higher-order quantifiers are usually based on a form of monotonicity w.r.t implication (set inclusion) and are not applicable to the first-order case. We make a shift to arbitrary relations \"ordering\" the underlying universe. This allows us to incorporate background theories into higher-order quantifier elimination methods which, up to now, has not been achieved. The technique we propose subsumes many other results, including the Ackermann's lemma and various forms of fixpoint approaches when the \"ordering\" relations are interpreted as implication and reveals the common principle behind these approaches.

In real world applications robots and software agents often have to be equipped with higher level cognitive functions that enable them to reason, act and perceive in changing, incompletely known and unpredictable environments. One of the major tasks in such circumstances is to fuse information from various data sources. There are many levels of information fusion, ranging from the fusing of low level sensor signals to the fusing of high level, complex knowledge structures. In a dynamically changing environment even a single agent may have varying abilities to perceive its environment which are dependent on particular conditions. The situation becomes even more complex when different agents have different perceptual capabilities and need to communicate with each other. In this paper, we propose a framework that provides agents with the ability to fuse both low and high level approximate knowledge in the context of dynamically changing environments while taking account of heterogeneous and contextually limited perceptual capabilities. To model limitations on an agent's perceptual capabilities we introduce the idea of partial tolerance spaces. We assume that each agent has one or more approximate databases where approximate relations are represented using lower and upper approximations on sets. Approximate relations are generalizations of rough sets. It is shown how sensory and other limitations can be taken into account when constructing and querying approximate databases for each respective agent. Complex relations inherit the approximativeness of primitive relations used in their definitions. Agents then query these databases and receive answers through the filters of their perceptual limitations as represented by (partial) tolerance spaces and approximate queries. The techniques used are all tractable. © 2005 Elsevier B.V. All rights reserved.

Second-order quantifier elimination in the context of classical logic emerged as a powerful technique in many applications, including the correspondence theory, relational databases, deductive and knowledge databases, knowledge representation, commonsense reasoning and approximate reasoning. In the current paper we first generalize the result of Nonnengart and Szalas [17] by allowing second-order variables to appear within higher-order contexts. Then we focus on a semantical analysis of conditionals, using the introduced technique and Gabbay's semantics provided in [10] and substantially using a third-order accessibility relation. The analysis is done via finding correspondences between axioms involving conditionals and properties of the underlying third-order relation. © 2007 Springer Science+Business Media B.V.

This paper focuses on approximate reasoning based on the use of similarity spaces. Similarity spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak [17, 18]. Similarity spaces are used to define neighborhoods around individuals and these in turn are used to define approximate sets and relations. In any of the approaches, one would like to embed such relations in an appropriate logic which can be used as a reasoning engine for specific applications with specific constraints. We propose a framework which permits a formal study of the relationship between approximate relations, similarity spaces and three-valued logics. Using ideas from correspondence theory for modal logics and constraints on an accessibility relation, we develop an analogous framework for three-valued logics and constraints on similarity relations. In this manner, we can provide a tool which helps in determining the proper three-valued logical reasoning engine to use for different classes of approximate relations generated via specific types of similarity spaces. Additionally, by choosing a three-valued logic first, the framework determines what constraints would be required on a similarity relation and the approximate relations induced by it. Such information would guide the generation of approximate relations for specific applications.

Description logics refer to a family of formalisms concentrated around concepts, roles and individuals. They belong to the most frequently used knowledge representation formalisms and provide a logical basis to a variety of well known paradigms. The main reasoning tasks considered in the area of description logics are those reducible to subsumption. On the other hand, any knowledge representation system should be equipped with a more advanced reasoning machinery. Therefore in the current paper we make a step towards integrating description logics with second-order reasoning. One of the important motivations behind introducing second-order formalism follows from the fact that many forms of commonsense and nonmonotonic reasoning used in AI can be modelled within the second-order logic. To achieve our goal we first extend description logics with a possibility to quantify over concepts. Since one of the main criticisms against the use of second-order formalisms is their complexity, we next propose second-order quantifier elimination techniques applicable to a large class of description logic formulas. Finally we show applications of the techniques, in particular in reasoning with circumscribed concepts and approximated terminological formulas.

This paper describes an experimental platform for approximate knowledge databases called the Approximate Knowledge Database (AKDB), based on a semantics inspired by rough sets. The implementation is based upon the use of a standard SQL database to store logical facts, augmented with several query interface layers implemented in JAVA through which extensional, intensional and local closed world nonmonotonic queries in the form of crisp or approximate logical formulas can be evaluated tractably. A graphical database design user interface is also provided which simplifies the design of databases, the entering of data and the construction of queries. The theory and semantics for AKDBs is presented in addition to application examples and details concerning the database implementation.

Currently, there is a great deal of interest in developing tools for the generation and use of ontologies on the WWW. These knowledge structures are considered essential to the success of the semantic web, the next phase in the evolution of the WWW. Much recent work with ontologies assumes that the concepts used as building blocks are crisp as opposed to approximate. It is a premise of this paper that approximate concepts and ontologies will become increasingly more important as the semantic web becomes a reality. We propose a framework for specifying, generating and using approximate ontologies. More specifically, (1) a formal framework for defining approximate concepts, ontologies and operations on approximate concepts and ontologies is presented. The framework is based on intuitions from rough set theory, (2) algorithms for automatically generating approximate ontologies from traditional crisp ontologies or from large data sets together with additional knowledge are presented. The knowledge will generally be related to similarity measurements between individual objects in the data sets, or constraints of a logical nature which rule out particular constellations of concepts and dependencies in generated ontologies. The techniques for generating approximate ontologies are parameterizable. The paper provides specific instantiations and examples.

We introduce the notion of a meta-query on relational databases and a technique which can be used to represent and solve a number of interesting problems from the area of knowledge representation using logic. The technique is based on the use of quantifier elimination and may also be used to query relational databases using a declarative query language called SHQL (Semi-Horn Query Language), introduced in [6]. SHQL is a fragment of classical first-order predicate logic and allows us to define a query without supplying its explicit definition. All SHQL queries to the database can be processed in polynomial time (both on the size of the input query and the size of the database). We demonstrate the use of the technique in problem solving by structuring logical puzzles from the Knights and Knaves domain as SHQL meta-queries on relational databases. We also provide additional examples demonstrating the flexibility of the technique. We conclude with a description of a newly developed software tool, The Logic Engineer, which aids in the description of algorithms using transformation and reduction techniques such as those applied in the meta-querying approach.

In this paper, we consider the problem of expressing and computing queries on relational deductive databases in a purely declarative query language, called SHQL (Semi-Horn Query Language). Assuming the relational databases in question are ordered, we show that all SHQL queries are computable in PTIME (polynomial time) and the whole class of PTIME queries is expressible in SHQL. Although similar results have been proven for fixpoint languages and extensions to datalog, the claim is that SHQL has the advantage of being purely declarative, where the negation operator is interpreted as classical negation, mixed quantifiers may be used and a query is simply a restricted first-order theory not limited by the rule-based syntactic restrictions associated with logic programs in general. We describe the PTIME algorithm used to compute queries in SHQL which is based in part on quantifier elimination techniques and also consider extending the method to incomplete relational databases using intuitions related to circumscription techniques.

We first define general domain circumscription (GDC) and provide it with a semantics. GDC subsumes existing domain circumscription proposals in that it allows varying of arbitrary predicates, functions, or constants, to maximize the minimization of the domain of a theory. We then show that for the class of semi-universal theories without function symbols, that the domain circumscription of such theories can be constructively reduced to logically equivalent first-order theories by using an extension of the DLS algorithm, previously proposed by the authors for reducing second-order formulas. We also show that for a certain class of domain circumscribed theories, that any arbitrary second-order circumscription policy applied to these theories is guaranteed to be reducible to a logically equivalent first-order theory. In the case of semi-universal theories with functions and arbitrary theories which are not separated, we provide additional results, which although not guaranteed to provide reductions in all cases, do provide reductions in some cases. These results are based on the use of fixpoint reductions.

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.

In the current paper we present a powerful technique of obtaining natural deduction proof systems for first-order fixpoint logics. The term fixpoint logics refers collectively to a class of logics consisting of modal logics with modalities definable at meta-level by fixpoint equations on formulas. The class was found very interesting as it contains most logics of programs with e.g. dynamic logic, temporal logic and the ¯-calculus among them. In this paper we present a technique that allows us to derive automatically natural deduction systems for modal logics from fixpoint equations defining the modalities

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 <em>semi-Horn formulas</em>. 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.

In this paper we study correspondences between modal proof rules and the classical logic. The method we apply is based on an Ackermann's technique of eliminating second-order quantifiers from formulas. We show that the process of finding suitable correspondences can be reduced to a few simple steps. Moreover, the whole technique can be fully mechanized. We thus provide the reader with a powerful tool, useful in automated translations between modal logics and the classical logic.

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 propositional modal schemas. Given a modal schema and a semantics base method of translating propositional modal formulae into classical first-order ones, we try to derive automatically classica first-order formulae characterizing precisely the class of frames validating the schema. The technique we consider can, in many cases, be easily applied even without computer support. Although we mainly concentrate on Kripke semantics, the technique we apply is much more general, as it is based on elimination of second-order quantifiers from formulae. We show many examples of application of the method. These can also serve as new, automated proofs of considered correspondences.

We introduce and discuss a notion of strictly arithmetical completeness related to relative completeness of Cook (1978) and arithmetical completeness of Harel (1978). We present a powerful technique of obtaining strictly arithmetical axiomatizations of logics of programs. Given a model-theoretic semantics of a logic and a set of formulae defining (in a metalanguage) its nonclassical connectives, we automatically derive strictly arithmetically complete and sound proof systems for this logic. As examples of application of the technique we obtain new axiomatizations of algorithmic logic, (concurrent) dynamic logic and temporal logic.

An alternative method of describing semantics of cause-effect structures is presented. It is based on a model of discrete dynamic systems. The model is similar to a condition-event Petri net, differing in the way restrictions on the alterability of actions are imposed.

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.

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.

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 <em>Σ</em>⁰_<em>n</em> nor <em>Π</em>⁰_<em>n</em> for any natural number <em>n</em>). Thus we show that there is no finitistic and complete axiomatization of the considered logic.