A Paraconsistent ASP-like Language with Tractable Model Generation

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 of well-supported models for 4QL programs is tractable, dropping stratification makes both 4QL and ASP intractable. 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

Semantics of logic programs with explicit negation

After a historical introduction, the bulk of the thesis concerns the study of a declarative semantics for logic programs. The main original contributions are: ² WFSX (Well–Founded Semantics with eXplicit negation), a new semantics for logic programs with explicit negation (i.e. extended logic programs), which compares favourably in its properties with other extant semantics. ² A generic characterization schema that facilitates comparisons among a diversity of semantics of extended logic programs, including WFSX. ² An autoepistemic and a default logic corresponding to WFSX, which solve existing problems of the classical approaches to autoepistemic and default logics, and clarify the meaning of explicit negation in logic programs. ² A framework for defining a spectrum of semantics of extended logic programs based on the abduction of negative hypotheses. This framework allows for the characterization of different levels of scepticism/credulity, consensuality, and argumentation. One of the semantics of abduction coincides with WFSX. ² O–semantics, a semantics that uniquely adds more CWA hypotheses to WFSX. The techniques used for doing so are applicable as well to the well–founded semantics of normal logic programs. ² By introducing explicit negation into logic programs contradiction may appear. I present two approaches for dealing with contradiction, and show their equivalence. One of the approaches consists in avoiding contradiction, and is based on restrictions in the adoption of abductive hypotheses. The other approach consists in removing contradiction, and is based in a transformation of contradictory programs into noncontradictory ones, guided by the reasons for contradiction

Properties and Applications of Programs with Monotone and Convex Constraints

We study properties of programs with monotone and convex constraints. We extend to these formalisms concepts and results from normal logic programming. They include the notions of strong and uniform equivalence with their characterizations, tight programs and Fages Lemma, program completion and loop formulas. Our results provide an abstract account of properties of some recent extensions of logic programming with aggregates, especially the formalism of lparse programs. They imply a method to compute stable models of lparse programs by means of off-the-shelf solvers of pseudo-boolean constraints, which is often much faster than the smodels system

Every normal logic program has a 2-valued semantics: theory, extensions, applications, implementations

Trabalho apresentado no âmbito do Doutoramento em Informática, como requisito parcial para obtenção do grau de Doutor em InformáticaAfter a very brief introduction to the general subject of Knowledge Representation and Reasoning with Logic Programs we analyse the syntactic structure of a logic program and how it can influence the semantics. We outline the important properties of a 2-valued semantics for Normal Logic Programs, proceed to define the new Minimal Hypotheses semantics with those properties and explore how it can be used to benefit some knowledge representation and reasoning mechanisms. The main original contributions of this work, whose connections will be detailed in the sequel, are: • The Layering for generic graphs which we then apply to NLPs yielding the Rule Layering and Atom Layering — a generalization of the stratification notion; • The Full shifting transformation of Disjunctive Logic Programs into (highly nonstratified)NLPs; • The Layer Support — a generalization of the classical notion of support; • The Brave Relevance and Brave Cautious Monotony properties of a 2-valued semantics; • The notions of Relevant Partial Knowledge Answer to a Query and Locally Consistent Relevant Partial Knowledge Answer to a Query; • The Layer-Decomposable Semantics family — the family of semantics that reflect the above mentioned Layerings; • The Approved Models argumentation approach to semantics; • The Minimal Hypotheses 2-valued semantics for NLP — a member of the Layer-Decomposable Semantics family rooted on a minimization of positive hypotheses assumption approach; • The definition and implementation of the Answer Completion mechanism in XSB Prolog — an essential component to ensure XSB’s WAM full compliance with the Well-Founded Semantics; • The definition of the Inspection Points mechanism for Abductive Logic Programs;• An implementation of the Inspection Points workings within the Abdual system [21] We recommend reading the chapters in this thesis in the sequence they appear. However, if the reader is not interested in all the subjects, or is more keen on some topics rather than others, we provide alternative reading paths as shown below. 1-2-3-4-5-6-7-8-9-12 Definition of the Layer-Decomposable Semantics family and the Minimal Hypotheses semantics (1 and 2 are optional) 3-6-7-8-10-11-12 All main contributions – assumes the reader is familiarized with logic programming topics 3-4-5-10-11-12 Focus on abductive reasoning and applications.FCT-MCTES (Fundação para a Ciência e Tecnologia do Ministério da Ciência,Tecnologia e Ensino Superior)- (no. SFRH/BD/28761/2006

Arithmetic and Modularity in Declarative Languages for Knowledge Representation

The past decade has witnessed the development of many important declarative languages for knowledge representation and reasoning such as answer set programming (ASP) languages and languages that extend first-order logic. Also, since these languages depend on background solvers, the recent advancements in the efficiency of solvers has positively affected the usability of such languages. This thesis studies extensions of knowledge representation (KR) languages with arithmetical operators and methods to combine different KR languages. With respect to arithmetic in declarative KR languages, we show that existing KR languages suffer from a huge disparity between their expressiveness and their computational power. Therefore, we develop an ideal KR language that captures the complexity class NP for arithmetical search problems and guarantees universality and efficiency for solving such problems. Moreover, we introduce a framework to language-independently combine modules from different KR languages. We study complexity and expressiveness of our framework and develop algorithms to solve modular systems. We define two semantics for modular systems based on (1) a model-theoretical view and (2) an operational view on modular systems. We prove that our two semantics coincide and also develop mechanisms to approximate answers to modular systems using the operational view. We augment our algorithm these approximation mechanisms to speed up the process of solving modular system. We further generalize our modular framework with supported model semantics that disallows self-justifying models. We show that supported model semantics generalizes our two previous model-theoretical and operational semantics. We compare and contrast the expressiveness of our framework under supported model semantics with another framework for interlinking knowledge bases, i.e., multi-context systems, and prove that supported model semantics generalizes and unifies different semantics of multi-context systems. Motivated by the wide expressiveness of supported models, we also define a new supported equilibrium semantics for multi-context systems and show that supported equilibrium semantics generalizes previous semantics for multi-context systems. Furthermore, we also define supported semantics for propositional programs and show that supported model semnatics generalizes the acclaimed stable model semantics and extends the two celebrated properties of rationality and minimality of intended models beyond the scope of logic programs

Stability, Supportedness, Minimality and Kleene Answer Set Programs

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.CADICSELLIITCUASSHERPANFFP