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.

Inheritance has intensively been studied in both object-oriented programming (Oop) and knowledge representation and reasoning (KRR). On the other hand, the approaches to multiple inheritance and related method resolution, developed in both domains, remain separated. The primary goal of this paper is to demonstrate how these approaches may be integrated using inheritance expressions. In particular, we examine inheritance as a belief bases management machinery designed to operate in dynamically changing environments where objects are embedded and act. We focus on objects that are belief bases containers, potentially participating in complex distributed reasoning scenarios. We show that inheritance expressions, inspired both by Oop and KRR, provide a simple yet flexible and powerful means for expressing inheritance and related belief/knowledge fusion.

For over three decades researchers have been studying the BDI modelof agency. Many robust multiagent systems have been developed, and a numberof BDI logics have been studied. Following this intensive development phase, theimportance of integrating BDI models with inconsistency handling and revisiontheory have been emphasized. There is also a demand for a tighter connectionbetween BDI-based implementations and BDI logics. In this paper, we addressthese postulates by introducing a novel, paraconsistent logical BDI model close toimplementation, with building blocks that can be represented as SQL/rule-baseddatabases. Importantly, tractability is achieved by reasoning as querying. Thisstands in a sharp contrast to the high complexity of BDI logics. We also extendbelief shadowing, a shallow and lightweight alternative to deep and computation-ally demanding belief revision, to encompass agents’ motivational attitudes.

Inheritance has intensively been investigated during the past decades in object-oriented programming and knowledge representation and reasoning areas. In the paper we focus on recently introduced inheritance expressions that allow one to represent dynamic concept hierarchies as well as fuse and disambiguate beliefs acquired by the objects involved. We focus on querying and reasoning about inheritance expressions using a four-valued paraconsistent formalism that has been developed over the last ten years. In particular, we show that querying inheritance expressions and formulas can be efficiently implemented. In addition, we provide tableaux for general reasoning purposes. Complexity of the investigated tools is also analyzed.

The majority of contemporary software systems are developed using object-oriented tools and methodologies, where constructs like classes, inheritance and objects are first-class citizens. In the current paper we provide a novel formal framework for many-valued object-oriented inheritance in rule-based query languages. We also relate the framework to rough set-like approximate reasoning. Rough sets and their generalizations have intensively been studied and applied. However, the mainstream of the area mainly focuses on the context of information and decision tables. Therefore, approximations defined in the much richer object-oriented contexts generalize known approaches. 

As a side-effect of the Covid-19 pandemic, significant decreases in medical procedures for noncommunicable diseases have been observed. This calls for a decision support assisting in the analysis of opportunities to relocate procedures among hospitals in an efficient or, preferably, optimal manner. In the current paper we formulate corresponding decision problems and develop linear (mixed integer) programming models for them. Since solving mixed integer programming problems is NP-complete, we verify experimentally their usefulness using real-world data about urological procedures. We show that even for large models, with millions of variables, the problems’ instances are solved in perfectly acceptable time.

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.

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.

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.

Recent work in the area of Knowledge Representation and Reasoning has focused on modification and optimization of knowledge bases (KB) through the use of forgetting operators of the form forget(KB, (R) over bar), where (R) over bar is a set of relations in the language signature used to specify the KB. The result of this operation is a new KB where the relations in (R) over bar are removed from the KB in a principled manner resulting in a more efficient representation of the KB for different purposes. The forgetting operator is also reflected semantically in terms of the relation between the original models of the KB and the models for the revised KB after forgetting. In this paper, we first develop a rough reasoning framework where our KBs consist of rough formulas with a semantics based on a generalization of Kleene algebras. Using intuitions from the classical case, we then define a forgetting operator that can be applied to rough KBs removing rough relations. A constructive basis for generating a new KB as the result of applying the forgetting operator to a rough KB is specified using second-order quantifier elimination techniques. We show the application of this technique with some practical examples.

Belief shadowing is a relatively recent approach to belief change. In essence, shadowing depends on accepting beliefs of others at the expense of dismissing, perhaps temporarily, some of the own ones. As a transient belief change, it is useful when an agent, acting as a group member or playing a particular role, has to adopt adequate "external" beliefs. So far two forms of shadowing, single and multiple, have been considered. While the former specifies shadowing when an agent belongs to a single group or plays a single role, multiple shadowing relaxes this restriction.In the paper we generalize shadowing to arbitrary finitely many-valued logics and consider more complex semantical structures allowing arbitrarily nested sets of worlds. We show that in this general setting multiple shadowing can be represented by single shadowing. The complexity of queries involving such generic shadowing operators is also analyzed.

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.

Dual tableaux were introduced by Rasiowa and Sikorski (1960) as a cut free deduction system for classical first-order logic. In the current paper, a sound and complete proof procedure based on dual tableaux is proposed for

R₃

which is the standard Kleene logic augmented with a weak negation connective and an implication connective proposed, in another context, by Shepherdson (1989).

R₃

is used as a basis for defining Kleene Answer Set Programs (

ASP^K

programs). The semantics for

ASP^K

programs is based on strongly supported models. Both entailment procedures and model generation procedures for normal and non-normal

ASP^K

programs are proposed based on the use of dual tableaux and a model filtering technique. The dual tableau proof procedure extended with a model filtering technique is shown to be sound and complete for

ASP^K

programs, both normal and non-normal. Since there is a direct relationship between answer sets for classical ASP programs and

R₃

models for

ASP^K

programs, it can be shown that the sound and complete dual tableaux proof procedure with filtering for ASPK" role="presentation">ASPKprograms is also sound and complete for classical normal ASP programs. For classical non-normal ASP programs, the proof procedure is only sound, since an alternative semantics for disjunction is used in

ASP^K

The paper introduces ActLog, a rule-based language capable of specifying actions paraconsistently. ActLog is an extension of 4QL Bel " role="presentation"> Bel , a rule-based language for reasoning with paraconsistent and paracomplete belief bases and belief structures. Actions considered in the paper act on belief bases rather than states represented as sets of ground literals. Each belief base stores multiple world representations which can be though of as a representation of possible states. In this context ActLog’s action may be then seen as a method of transforming one belief base into another. In contrast to other approaches, ActLog permits to execute actions even if the underlying belief base state is partial or inconsistent. Finally, the framework introduced in this paper is tractable.

Stanisław Jaśkowski, in his 1948–1949 papers on propositional calculus for contradictory deductive systems, proposed discursive logic D₂. The main motivation behind D₂ is the need to properly deal with contradictions that naturally appear in many areas of philosophy and discourse. The intuitive justification of this logic reflects knowledge fusion occurring when “the theses advanced by several participants in a discourse are combined into a single system.” This point of view was seminal in the mid twentieth century and remains visionary nowadays.

In contemporary autonomous systems operating in dynamic, unpredictable information-rich environments, distributed reasoning routinely takes place. This explains the key role of knowledge fusion, among others, in Distributed Artificial Intelligence. Therefore, different types of modern knowledge and belief bases become primarily concerned with inconsistent or lacking information. This requirement leads to recent approaches to paraconsistent and paracomplete reasoning, where nonmonotonic techniques for disambiguating inconsistencies and completing missing knowledge can be applied.

In this chapter we remind Jaśkowski’s seminal, pioneering work on paraconsistent reasoning and indicate some of its relations to contemporary research on reasoning in Distributed AI.

This paper is devoted to paraconsistent approximate reasoning with graded truth-values. In the previous research we introduced a family of many-valued logics parameterized by a variable number of truth/falsity grades together with a corresponding family of rule languages with tractable query evaluation. Such grades are shown here to be a natural qualitative counterpart of quantitative measures used in various forms of approximate reasoning. The developed methodology allows one to obtain a framework unifying heterogeneous reasoning techniques, providing also the logical machinery to resolve partial and incoherent information that may arise after unification. Finally, we show the introduced framework in action, emphasizing its expressiveness in handling heterogeneous approximate reasoning in realistic scenarios.

This paper introduces 4QL(Bel), a four-valued rule language designed for reasoning with paraconsistent and paracomplete belief bases as well as belief structures. Belief bases consist of finite sets of ground literals providing (partial and possibly inconsistent) complementary or alternative views of the world. As introduced earlier, belief structures consist of constituents, epistemic profiles and consequents. Constituents and consequents are belief bases playing different roles. Agents perceive the world forming their constituents, which are further transformed into consequents via the agents or groups epistemic profile. In order to construct 4QL(Bel), we extend 4QL, a four-valued rule language permitting for many forms of reasoning, including doxastic reasoning. Despite the expressiveness of 4QL(Bel), we show that its tractability is retained.

Classical Answer Set Programming is a widely known knowledge representation framework based on the logic programming paradigm that has been extensively studied in the past decades. Semantic theories for classical answer sets are implicitly three-valued in nature, yet with few exceptions, computing classical answer sets is based on translations into classical logic and the use of SAT solving techniques. In this paper, we introduce a variation of Kleene three-valued logic with strong connectives, R3" role="presentation">R3, and then provide a sound and complete proof procedure for R3" role="presentation">R3 based on the use of signed tableaux. We then define a restriction on the syntax of R3" role="presentation">R3 to characterize Kleene ASPs. Strongly-supported models, which are a subset of R3" role="presentation">R3 models are then defined to characterize the semantics of Kleene ASPs. A filtering technique on tableaux for R3" role="presentation">R3 is then introduced which provides a sound and complete tableau-based proof technique for Kleene ASPs. We then show a translation and semantic correspondence between Classical ASPs and Kleene ASPs, where answer sets for normal classical ASPs are equivalent to strongly-supported models. This implies that the proof technique introduced can be used for classical normal ASPs as well as Kleene ASPs. The relation between non-normal classical and Kleene ASPs is also considered.

In this extended abstract, we discuss the use of iteratively-supported formulas (ISFs) as a basis for computing strongly-supported models for Kleene Answer Set Programs (ASPK). ASPK programs have a syntax identical to classical ASP programs. The semantics of ASPK programs is based on the use of Kleene three-valued logic and strongly-supported models. For normal ASPK programs, their strongly supported models are identical to classical answer sets using stable model semantics.  For disjunctive ASPK programs, the semantics weakens the minimality assumption resulting in a classical interpretation for disjunction. We use ISFs to characterize strongly-supported models and show that they are polynomially bounded.

This book is in celebration of Rineke Verbrugge's 50th birthday. It is a product of an incredible effort on the part of Rineke's teachers, colleagues, students and friends who have all been won over by her ever-encouraging and positive presence in academia and also in daily life. Pertaining to Rineke's research interests, the book features eight articles on a wide range of topics - from theories of arithmetic to a study on autism. The papers on hybrid logic, formal theories of belief, probability, goals, social networks, and bisimulations enrich the logic section of the book while papers on cognitive strategizing and social cognition bring up the cognitive perspective. The themes themselves provide a compelling perception of the vast expanse of Rineke's academic interests and endeavours. A series of personal comments, stories, anecdotes, and pictures constitute the latter part of the book, adding a distinct personal touch to this volume.

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.

Answer Set Programming is a widely known knowledge representation framework based on the logic programming paradigm that has been extensively studied in the past decades. The semantic framework for Answer Set Programs is based on the use of stable model semantics. There are two characteristics intrinsically associated with the construction of stable models for answer set programs. Any member of an answer set is supported through facts and chains of rules and those members are in the answer set only if generated minimally in such a manner. These two characteristics, supportedness and minimality, provide the essence of stable models. Additionally, answer sets are implicitly partial and that partiality provides epistemic overtones to the interpretation of disjunctiver ules and default negation. This paper is intended to shed light on these characteristics by defining a semantic framework for answer set programming based on an extended first-order Kleene logic with weak and strong negation. Additionally, a definition of strongly supported models is introduced, separate from the minimality assumption explicit in stable models. This is used to both clarify and generate alternative semantic interpretations for answer set programs with disjunctive rules in addition to answer set programs with constraint rules. An algorithm is provided for computing supported models and comparative complexity results between strongly supported and stable model generation are provided.

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.

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.

The current paper falls into a bigger research programme concerning construction of modern belief structures applicable in multiagent systems. In previous papers we approached individual and group beliefs via querying paraconsistent belief bases. This framework, covering deterministic belief structures, turned out to be tractable under some natural restrictions on implementation. Moreover, we have indicated a four-valued query language 4QL as an implementation tool guaranteeing tractability and capturing all PTime -constructible belief structures.

In this paper we generalize our approach to the nondeterministic case. This is achieved by adjusting the key abstractions of epistemic profiles and belief structures to this new situation. Importantly, tractability of the approach is still maintained.

Dealing with heterogeneous information sources and reasoning techniques allowing for incomplete and inconsistent information is one of current challenges in the area of knowledge representation and reasoning. We advocate for 4QL, a rule-based query language, as a proper tool allowing one to address these challenges. To justify this point of view we discuss a rescue robotics scenario for which a simulator has been developed and tested. In particular, we present a planner using 4QL and, therefore, capable to deal with lack of knowledge and inconsistencies. Through the case study we show that our approach allows one to use lightweight knowledge representation tools: due to the use of 4QL tractability of modeling and reasoning is guaranteed and high usability 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 fragmentsEL, DL-Lite, DLP, Horn-SH IQ and Horn-SROIQof 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.

Various generalizations of fuzzy reasoning are frequently used in decision making. While in many application areas it is natural to assume that truth degrees of a property and its complement sum up to 1, such an assumption appears problematic, e.g., in modeling ignorance. Therefore, in some generalizations of fuzzy sets, degrees of membership in a set and in its complement are separated and are no longer required to sum up to 1. In frequent cases, this separation of positive and negative evidences for concept membership is more natural. As we discuss in the current paper, symbolic explanations of results of such forms of reasoning provide additional important information. In the present paper we address two related questions: (i) given generalized fuzzy connectives and a finite set of truth values T, find a finitely-valued logic over T, explaining fuzzy reasoning, and (ii) given a finitely-valued logic, find a fuzzy semantics, explained by the given logic. We also show examples illustrating usefulness of the approach.

It is known that the OWL 2RL Web Ontology Language Profile has PTime data complexity and can be translated into Datalog. However, the result of translation may consist of a Datalog program and a set of constraints in the form of negative clauses. Therefore, a knowledge base in OWL 2RL may be unsatisfiable. In the current paper we first identify a maximal fragment of OWL 2RL, called OWL 2RL+, with the property that every knowledge base expressed in OWL2RL+ can be translated to a Datalog program and hence is satisfiable. We then propose some extensions of OWL 2RL and OWL 2RL + that still have PTime data complexity. © 2014 Springer-Verlag Berlin Heidelberg.

In contemporary autonomous systems, like robotics, the need to apply group knowledge has been growing consistently with the increasing complexity of applications, especially those involving teamwork. However, classical notions of common knowledge and common belief, as well as their weaker versions, are too complex. Also, when modeling real-world situations, lack of knowledge and inconsistency of information naturally appear. Therefore, we propose a shift in perspective from reasoning in multi-modal logics to querying paraconsistent knowledge bases. This opens the possibility for exploring a new approach to group beliefs. To demonstrate expressiveness of our approach, examples of social procedures leading to complex belief structures are constructed via the use of epistemic profiles. To achieve tractability without compromising the expressiveness, as an implementation tool we choose 4QL, a four-valued rule-based query language. This permits both to tame inconsistency in individual and group beliefs and to execute the social procedures in polynomial time. Therefore, a marked improvement in efficiency has been achieved over systems such as (dynamic) epistemic logics with common knowledge and ATL, for which problems like model checking and satisfiability are PSPACE- or even EXPTIME-hard.

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 paper is devoted to belief fusion when information sources may deliver incomplete and inconsistent information. In such cases paraconsistent and commonsense reasoning techniques can be used to complete missing knowledge and disambiguate inconsistencies. We propose a novel, realistic model of distributed belief fusion and an implementation framework guaranteeing its tractability.

We introduce a Horn description logic called Horn-DL, which is strictly and essentially richer than Horn- SROIQ , while still has PTime data complexity. In comparison with Horn- SROIQ , HornDL 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 EL , DL-Lite, DLP, Horn- SHIQ , Horn- SROIQ of description logics, HornDL 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. In the long version of this paper, we present the first algorithm with PTime data complexity for checking satisfiability of HornDL knowledge bases.

The logic TeamLog proposed by Dunin-Kęplicz and Verbrugge is used to express properties of agents’ cooperation in terms of individual, bilateral and collective informational and motivational attitudes like beliefs, goals and intentions. In this paper we isolate a Horn fragment of TeamLog, called Horn-TeamLog, and we show that it has PTime data complexity.

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⁺.

This book is dedicated to the memory of Professor Zdzis{\l}aw Pawlak who passed away almost six year ago. He is the founder of the Polish school of Artificial Intelligence and one of the pioneers in Computer Engineering and Computer Science with worldwide influence. He was a truly great scientist, researcher, teacher and a human being.This book prepared in two volumes contains more than 50 chapters. This demonstrates that the scientific approaches  discovered by of Professor Zdzis{\l}aw Pawlak, especially the rough set approach as a tool for dealing with imperfect knowledge, are vivid and intensively explored by many researchers in many places throughout the world. The submitted papers prove that interest in rough set research is growing and is possible to see many new excellent results both on theoretical foundations and applications of rough sets alone or in combination with other approaches.We are proud to offer the readers this book. 

We study Horn fragments of serial multimodal logics which are characterized by regular grammars with converse. Such logics are useful for reasoning about epistemic states of multiagent systems as well as similarity-based approximate reasoning. We provide the first algorithm with PTIME data complexity for checking satisfiability of a Horn knowledge base in a serial regular grammar logic with converse.

Agents' beliefs can be incomplete and partially inconsistent. The process of agents' belief formation in such contexts has to be supported by suitable tools allowing one to express a variety of inconsistency resolving and nonmonotonic reasoning techniques.

In this paper we discuss 4QL*, a general purpose rule-based query language allowing one to use rules with negation in the premises and in the conclusions of rules. It is based on a simple and intuitive semantics and provides uniform tools for lightweight versions of well-known forms of nonmonotonic reasoning. In addition, it is tractable w.r.t. data complexity and captures PTIME queries, so can be used in real-world applications.

Reasoning in 4QL* is based on well-supported models. We simplify and at the same time generalize previous definitions of well-supported models and develop a new algorithm for computing such models.

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, DATALOG$^{\neg \neg}$-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.

A novel formalization of beliefs in multiagent systems has recently been proposed by Dunin-Kęplicz and Szałas. The aim has been to bridge the gap between idealized logical approaches to modeling beliefs and their actual implementations. Therefore the stages of belief acquisition, intermediate reasoning and final belief formation have been isolated and analyzed. In conclusion, a novel semantics reflecting those stages has been provided. This semantics is based on the new concept of epistemic profile, reflecting agent’s reasoning capabilities in a dynamic and unpredictable environment. The presented approach appears suitable for building complex belief structures in the context of incomplete and/or inconsistent information. One of original ideas is that of epistemic profiles serving as a tool for transforming preliminary beliefs into final ones. As epistemic profile can be devised both on an  individual and a  group level in analogical manner, a uniform treatment of single agent and group beliefs has been achieved.

In the current paper these concepts are further elaborated. Importantly, we indicate an implementation framework ensuring tractability of reasoning about beliefs, propose the underlying methodology and illustrate it on an example.