Knowledge Representation for the Semantic Web [2Ex] Lecture 6

Total Page:16

File Type:pdf, Size:1020Kb

Knowledge Representation for the Semantic Web [2Ex] Lecture 6 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Knowledge Representation for the Semantic Web Lecture 6: Answer Set Programming I Daria Stepanova partially based on slides by Thomas Eiter D5: Databases and Information Systems Max Planck Institute for Informatics WS 2017/18 1 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Unit Outline Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics 2 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics French Phrases, Italian Soda Six people sit at a round table. Each drinks a different kind of soda. Each plans to visit a different French-speaking country. The person who is planning a trip to Quebec, who drank either blueberry or lemon soda, didn't sit in seat number one. Jeanne didn't sit next to the person who enjoyed the kiwi soda. The person who has a plane ticket to Belgium, who sat in seat four or seat five, didn't order the cherry soda. ::: Question: What is each of them drinking, and where is each of them going? 3 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Sudoku 6 1 4 5 8 3 5 6 2 1 8 4 7 6 6 3 7 9 1 4 5 2 7 2 6 9 4 5 8 7 Task: Fill in the grid so that every row, every column, and every 3x3 box contains the digits 1 through 9. 4 / 45 + – a rule r, if Head(r) S = whenever Body (r) S and Body−(r) S = ; ∩ 6 ∅ ⊆ ∩ ∅ – a logic program (I is a model of ), if it satisfies all rules r of . P P P Definition 8. A model I of a logic program is called minimal, if there is no model I0 of , such that I0 I. P P ⊂ The semantics of ASP programs is given in terms of answer sets (or stable models). Definition 9. Given a positive logic program , an interpretation I is an answer set (stable model) of if I = and P P | P there does not exist I0 I, such that I0 = . AS( ) denotes the set of all answer sets. ⊂ | P P The notion of answer sets for logic programs with negation is defined using the Gelfond-Lifshitz reduct. Definition 10. Given a logic program and an interpretation I of the Gelfond-Lifshitz (GL-)reduct denoted by I is P P PGL constructed from ground( ) by P – deleting all rules r from ground( ) s.t. Body−(r) I = , and P ∩ 6 ∅ – deleting the negative body for all of the remaining rules. Note that for any program and interpretation the GL-reduct is positive. The definition of answer sets for arbitrary logic programs is then as follows: Definition 11. An answer set of a program is an interpretation I HB , such that I is an answer set of the positive P ⊆ P program I . PGL The availability of advanced reasoners makes the ASP formalism applicable to solving practical tasks. The standard reasoner comprises two components: a grounder and a solver. The list of the most well-known grounders include: dlv4, gringo5, lparse6, etc. Among the solvers one can mention dlv, clasp7, claspd8, gnt9. Potassco10 represents a collection of ASP reasoning tools, combining clasp and gringo into a system architecture. We now present an exampleIntroduction that illustratesHorn Logic Programming the basic notionsNegation of in the Logic Answer Programs Set ProgrammingAnswer Set Semantics that we have inroduced. Example 10. Consider a graph 3-colourability problem,Graph in 3-colouring which we are given a graph and three colours, e.g. red, blue and green, and we aim at finding the assignment of colours to the nodes of a graph in such a way that no adjacent nodes share the same color. The encoding of this problem is as follows. 2 3 1 4 6 5 (1) node(1 ... 6 ); (2) edge(1 , 2 ); ... (8) edge(5 , 6 ); Task: (9) coloured(V , red) not coloured(V , blue), not coloured(V , green), node(V ); Colour← the nodes of the graph in three colors such that none of the two (10) coloured(V , green) not coloured(V , blue), not coloured(V , red), node(V ); I = adjacent← nodes share the same colour. P (11) coloured(V , blue) not coloured(V , green), not coloured(V , red), node(V ); ← (12) coloured(V , C), coloured(V , C 0),C = C0; ⊥ ← 6 (13) coloured(V , C), coloured(V 0, C), edge(V , V 0) 5 / 45 ⊥ ← The facts of the programs describe the nodes and edges of the graph from above. The rules (9)-(11) state that each P node has to be colored in at least one of the three colours. The constraint (12) forbids the nodes to be colored in more then one color, while the constraint (13) says that two nodes connected via an edge must have different colours. 4 http://www.dlvsystem.com/ 5 http://potassco.sourceforge.net/ 6 http://www.tcs.hut.fi/Software/smodels/ 7 http://potassco.sourceforge.net/ 8 http://www.cs.uni-potsdam.de/claspD/ 9 http://www.tcs.hut.fi/Software/gnt/ 10 http://potassco.sourceforge.net/ Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Wanted! A general-purpose approach for modeling and solving these and many other problems. Issues: Diverse domains Spatial and temporal reasoning Constraints Incomplete information Frame problem Proposal: Answer-set programming (ASP) paradigm! 6 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Answer Set Programming Answer Set Programming (ASP) is a recent problem solving approach, based on declarative programming. The term was coined by Vladimir Lifschitz [1999,2002]. Proposed by other people at about the same time, e.g., by Marek and Truszczy´nski[1999] and Niemel¨a[1999]. It has roots in knowledge representation, logic programming, and nonmonotonic reasoning. At an abstract level, ASP relates to SAT solving and constraint satisfaction problems (CSPs). 7 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Answer Set Programming (cont'd) Important logic programming method Developed in the early 1990s by Gelfond and Lifschitz. Left: Michael Gelfond (Texas Tech Univ., Lubbock) Right: Vladimir Lifschitz (Univ. of Texas, Austin) Both are graduates from the Steklov Mathematical Institute, St.Petersburg (then: Leningrad). 8 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Answer Set Programming (cont'd) ASP is an approach to declarative programming, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in deductive databases logic programming with negation knowledge representation and nonmonotonic reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NPNP) in a uniform way 9 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Answer Set Programming (cont'd) ASP is an approach to declarative programming, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in deductive databases logic programming with negation knowledge representation and nonmonotonic reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NPNP) in a uniform way 9 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Declarative Programming 10 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Answer Set Programming (cont'd) ASP is an approach to declarative programming, combining a rich yet simple modeling language with high-performance solving capacities ASP has its roots in deductive databases logic programming with negation knowledge representation and nonmonotonic reasoning constraint solving (in particular, SATisfiability testing) ASP allows for solving all search problems in NP (and NPNP) in a uniform way 11 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Nonmonotonic Reasoning Nonmonotonicity means that conclusions may be invalidated in the light of new information. More specifically, an inference relation j= is nonmonotonic if it violates the monotonicity principle: if T j= φ and T ⊆ T 0, then T 0 j= φ. Note: inference in description logics is monotonic. Example: Monotonicity of description logics T = fBird v Flier; Bird(tweety)g T j= F lier(tweety) T 0 = T [ f:Flier(tweety)g T 0 j= F lier(tweety) (actually T 0 is inconsistent) 12 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics Nonmonotonic Reasoning Nonmonotonicity means that conclusions may be invalidated in the light of new information. More specifically, an inference relation j= is nonmonotonic if it violates the monotonicity principle: if T j= φ and T ⊆ T 0, then T 0 j= φ. Note: inference in description logics is monotonic. Example: Nonmonotonic inference If bird(x) holds and there is no evidence for :flies(x), then infer flies(x). I.e., if bird(x), assume flies(x) by default. 12 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics ASP Systems ASP gains increasing importance for knowledge representation High expressiveness Efficient solvers available: DLV, clasp,... Source: Wikipedia (Dec 6, 2017) 13 / 45 Fundamental characteristics: models, not proofs, represent solutions; requires techniques to compute models (rather than techniques to compute proofs) Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics ASP: General Idea ASP are logic programs; Their semantics adheres to the multiple preferred models approach: given as a selection of the collection of all classical models; selected (intended) models are called stable models or answer sets. 14 / 45 Introduction Horn Logic Programming Negation in Logic Programs Answer Set Semantics ASP: General Idea ASP are logic programs; Their semantics adheres to the multiple preferred models approach: given as a selection of the collection of all classical models; selected (intended) models are called stable models or answer sets.
Recommended publications
  • Stable Models and Circumscription
    Stable Models and Circumscription Paolo Ferraris, Joohyung Lee, and Vladimir Lifschitz Abstract The concept of a stable model provided a declarative semantics for Prolog programs with negation as failure and became a starting point for the development of answer set programming. In this paper we propose a new definition of that concept, which covers many constructs used in answer set programming and, unlike the original definition, refers neither to grounding nor to fixpoints. It is based on a syntactic transformation similar to parallel circumscription. 1 Introduction Answer set programming (ASP) is a form of declarative logic programming oriented towards knowledge-intensive search problems, such as product con- figuration and planning. It was identified as a new programming paradigm ten years ago [Marek and Truszczy´nski,1999, Niemel¨a,1999], and it has found by now a number of serious applications. An ASP program consists of rules that are syntactically similar to Prolog rules, but the computational mechanisms used in ASP are different: they use the ideas that have led to the creation of fast satisfiability solvers for propositional logic [Gomes et al., 2008]. ASP is based on the concept of a stable model [Gelfond and Lifschitz, 1988]. According to the definition, to decide which sets of ground atoms are \stable models" of a given set of rules we first replace each of the given rules by all its ground instances. Then we verify a fixpoint condition that is similar to the conditions employed in the semantics of default logic [Reiter, 1980] and autoepistemic logic [Moore, 1985] (see [Lifschitz, 2008, Sections 4, 5] for details).
    [Show full text]
  • Answer Set Programming: a Primer?
    Answer Set Programming: A Primer? Thomas Eiter1, Giovambattista Ianni2, and Thomas Krennwallner1 1 Institut fur¨ Informationssysteme, Technische Universitat¨ Wien Favoritenstraße 9-11, A-1040 Vienna, Austria feiter,[email protected] 2 Dipartimento di Matematica, Universita´ della Calabria, I-87036 Rende (CS), Italy [email protected] Abstract. Answer Set Programming (ASP) is a declarative problem solving para- digm, rooted in Logic Programming and Nonmonotonic Reasoning, which has been gaining increasing attention during the last years. This article is a gentle introduction to the subject; it starts with motivation and follows the historical development of the challenge of defining a semantics for logic programs with negation. It looks into positive programs over stratified programs to arbitrary programs, and then proceeds to extensions with two kinds of negation (named weak and strong negation), and disjunction in rule heads. The second part then considers the ASP paradigm itself, and describes the basic idea. It shows some programming techniques and briefly overviews Answer Set solvers. The third part is devoted to ASP in the context of the Semantic Web, presenting some formalisms and mentioning some applications in this area. The article concludes with issues of current and future ASP research. 1 Introduction Over the the last years, Answer Set Programming (ASP) [1–5] has emerged as a declar- ative problem solving paradigm that has its roots in Logic Programming and Non- monotonic Reasoning. This particular way of programming, in a language which is sometimes called AnsProlog (or simply A-Prolog) [6, 7], is well-suited for modeling and (automatically) solving problems which involve common sense reasoning: it has been fruitfully applied to a range of applications (for more details, see Section 6).
    [Show full text]
  • Reducts of Propositional Theories, Satisfiability Relations, And
    Reducts of propositional theories, satisfiability relations, and generalizations of semantics of logic programs Mirosław Truszczynski´ Department of Computer Science, University of Kentucky, Lexington, KY 40506-0046, USA [email protected] Abstract. Over the years, the stable-model semantics has gained a position of the correct (two-valued) interpretation of default negation in programs. How- ever, for programs with aggregates (constraints), the stable-model semantics, in its broadly accepted generalization stemming from the work by Pearce, Ferraris and Lifschitz, has a competitor: the semantics proposed by Faber, Leone and Pfeifer, which seems to be essentially different. Our goal is to explain the rela- tionship between the two semantics. Pearce, Ferraris and Lifschitz’s extension of the stable-model semantics is best viewed in the setting of arbitrary propositional theories. We propose here an extension of the Faber-Leone-Pfeifer semantics, or FLP semantics, for short, to the full propositional language, which reveals both common threads and differences between the FLP and stable-model semantics. We use our characterizations of FLP-stable models to derive corresponding re- sults on strong equivalence and on normal forms of theories under the FLP se- mantics. We apply a similar approach to define supported models for arbitrary propositional theories, and to study their properties. 1 Introduction The stable-model semantics, introduced by Gelfond and Lifschitz [1], is the founda- tion of answer-set programming [2–4], a paradigm for modeling and solving search problems. From its inception, developing theoretical underpinnings of the stable-model semantics has been a major research objective. In particular, a contribution by Pearce [5] explained the stable-model semantics in terms of models of theories in the logic here-and-there (HT, for short), introduced by Heyting [6].
    [Show full text]
  • Stable Models Stable Models (‘Answer Sets’) for a Normal Logic Program, a Stable Model Is a Set of Atoms
    491 Knowledge Representation Stable models Stable models (‘Answer sets’) For a normal logic program, a stable model is a set of atoms. Let P be a ground normal logic program, i.e., one without variables. If P is not ground Marek Sergot (contains variables) then replace it by all the ground instances of its clauses. (The resulting Department of Computing set of clauses may not be finite.) Imperial College, London Notation When r is a (ground) clause of the form: February 2006 v1.2, November 2016 v1.2c A B ,...,B , not C ,..., not C ← 1 m 1 n + head(r) = A, body (r) = B1,...,Bm , body−(r) = C1,...,Cn . When P is a set of A normal logic program (sometimes a ‘general logic program’) is a set of clauses of the clauses, heads(P ) = head({r) r P .} { } form: { | ∈ } Suppose we have a set X of atoms from the language of P . The idea is that we use X to A L ,...,L (n 0) ← 1 n ≥ simplify P by ‘partially evaluating’ all clauses with nbf-literals against X, and then we see whether the simplified program P X we are left with (the ‘reduct’) has a least Herbrand where A is an atom and each Li is either an atom or a nbf-literal of the form not A where A is an atom. not denotes negation by failure. model that coincides with X. A model of a normal logic program P is a set of atoms X such that T (X) X, or P ⊆ equivalently, such that X is an (ordinary, classical) model of the clauses P ¬ obtained by Definition replacing every occurrence of not in P by ordinary, truth-functional negation .
    [Show full text]
  • Functional Stable Model Semantics, Answer Set Programming Modulo Theories, and Action Languages
    Functional Stable Model Semantics, Answer Set Programming Modulo Theories, and Action Languages Joohyung Lee Automated Reasoning Group Arizona State University, USA NRAC 2013 Invited Talk 1 Abstract "Answer Set Programming Modulo Theories (ASPMT)" is a recently proposed framework which tightly integrates answer set programming (ASP) and satisfiability modulo theories (SMT). Its mathematical foundation is the functional stable model semantics, an enhancement of the traditional stable model semantics to allow defaults involving functions as well as predicates. This talk will discuss how ASPMT can provide a way to overcome limitations of the propositional setting of ASP, how action language C+ can be reformulated in terms of ASPMT, and how it can be implemented based on the reformulation. 2 Introduction 3 Answer Set Programming (ASP) Declarative programming paradigm. Suitable for knowledge intensive and combinatorial search problems. Theoretical basis: answer set semantics (Gelfond & Lifschitz, 1988). Expressive representation language: defaults, recursive definitions, aggregates, preferences, etc. ASP solvers: smodels (Helsinki University of Technology, 1996) dlv (Vienna University of Technology, 1997) cmodels (University of Texas at Austin, 2002) pbmodels (University of Kentucky, 2005) clasp (University of Potsdam, 2006) { winning first places at ASP'07/09/11/12, PB'09/11/12, and SAT'09/11/12 dlv-hex computing HEX programs. oClingo for reactive answer set programming. 4 Declarative Problem Solving using ASP The basic idea is to represent the given problem by a set of rules, to find answer sets for the program using an ASP solver, and to extract the solutions from the answer sets. 5 N-Queens Puzzle in the Language of clingo number(1..n).
    [Show full text]
  • Extending Prolog with Nonmonotonic Reasoning*
    NORTH- HOLLAND TECHNICAL NOTE EXTENDING PROLOG WITH NONMONOTONIC REASONING* WEIDONG CHEN C> Nonmonotonic reasoning has been developed to capture common sense inferences. This paper considers nonmonotonic reasoning in logic pro- grams with negation, specifically its implementation using Prolog and its integration with Prolog execution. Even within logic programming frame- works, a variety of inferencing methods, model-theoretic semantics, and language features have been proposed for different forms of nonmonotonic reasoning. A challenging problem is to incorporate them into an integrated system. This paper describes an implementation of such a system in Pro- log and different reasoning capabilities that are supported. By combining Prolog technology with various mechanisms of nonmonotonic reasoning, the resulting system offers a more realistic testbed for applications Of common sense reasoning. <~ 1. INTRODUCTION In logic programming, a nonmonotonic inference rule, namely, negation as failure, is normally used to establish negative atoms. Extensive research has been conducted on the declarative semantics of logic programs with negation. The connections between logic programming and nonmonotonic reasoning have attracted strong in- terests [12, 16, 18]. In particular, stable models [12] for normal logic programs correspond closely to extensions in nonmonotonic reasoning. This paper considers nonmonotonic reasoning in logic programming frameworks, which share with Prolog [15] a core rule-based language. Prolog is by far the most *Work supported in part by the National Science Foundation under Grants IRI-92-12074 and IRI-93-14897. Address correspondence to Weidong Chen, Department of Computer Science and Engineering, Southern Methodist University, Dallas, TX 75275-0122. Received June 1995; accepted November 1995. THE JOURNAL OF LOGIC PROGRAMMING (~) Elsevier Science Inc., 1996 0743-1066/96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0743-1066(95)00151-4 170 w.
    [Show full text]
  • First-Order Stable Model Semantics and First-Order Loop Formulas
    Journal of Artificial Intelligence Research 42 (2011) 125-180 Submitted 03/11; published 10/11 First-Order Stable Model Semantics and First-Order Loop Formulas Joohyung Lee [email protected] Yunsong Meng [email protected] School of Computing, Informatics, and Decision Systems Engineering Arizona State University Tempe, AZ 85287, USA Abstract Lin and Zhao's theorem on loop formulas states that in the propositional case the stable model semantics of a logic program can be completely characterized by propositional loop formulas, but this result does not fully carry over to the first-order case. We investigate the precise relationship between the first-order stable model semantics and first-order loop formulas, and study conditions under which the former can be represented by the latter. In order to facilitate the comparison, we extend the definition of a first-order loop formula which was limited to a nondisjunctive program, to a disjunctive program and to an arbitrary first-order theory. Based on the studied relationship we extend the syntax of a logic program with explicit quantifiers, which allows us to do reasoning involving non-Herbrand stable models using first-order reasoners. Such programs can be viewed as a special class of first- order theories under the stable model semantics, which yields more succinct loop formulas than the general language due to their restricted syntax. 1. Introduction According to the theorem on loop formulas (Lin & Zhao, 2004), the stable models of a logic program (Gelfond & Lifschitz, 1988) can be characterized as the models of the logic program that satisfy all its loop formulas.
    [Show full text]
  • Negation-As-Failure Considered Harmful
    Negation-as-failure considered harmful Pablo R. Fillottrani Departamento de Ciencias e Ingenier´ıa de la Computacion´ Universidad Nacional del Sur Av. Alem 1253, 8000 Bah´ıa Blanca, Argentina [email protected] Abstract In logic programs, negation-as-failure has been used both for representing negative information and for provid- ing default nonmonotonic inference. In this paper we argue that this twofold role is not only unnecessary for the expressiveness of the language, but it also plays against declarative programming, especially if further negation symbols such as strong negation are also available. We therefore propose a new logic programming approach in which negation and default inference are independent, orthogonal concepts. Semantical characterization of this approach is given in the style of answer sets, but other approaches are also possible. Finally, we compare them with the semantics for logic programs with two kinds of negation. Keywords: negation as failure, logic programming, knowledge representation 1 INTRODUCTION The utility of a language as a tool for practical development of knowledge representation systems is grounded on a simple syntax with intuitive semantics, and efficient proof procedures. So, in order to keep a broad scope of users and applications, the language should be both powerful and simple at the same time. The absence of any of these conditions produces languages with theoretical interest, but difficult to use in real applications. We think the original success of Logic Programming was due to the fulfillment of these requirements, so any extension must also preserve them. Negation-as-failure partially satisfied these properties at the beginning, despite its lack of declara- tive meaning in all programs.
    [Show full text]
  • Epic Series in Computing 1 Answer Set Programming
    EPiC Series in Computing Volume 51, 2017, Pages 64{68 ARCADE 2017. 1st International Workshop on Automated Reasoning: Challenges, Appli- cations, Directions, Exemplary Achievements A Case for Query-driven Predicate Answer Set Programming Gopal Gupta Elmer Salazar, Kyle Marple, Zhuo Chen, Farhad Shakerin Department of Computer Science The University of Texas at Dallas 1 Answer Set Programming Answer Set Programming (ASP) [1] has emerged as a successful paradigm for developing in- telligent applications. ASP is based on adding negation as failure to logic programming under the stable model semantics regime [2]. ASP allows for sophisticated reasoning mechanisms that are employed by humans (and that can be clubbed under the rubric of common sense reason- ing) to be modeled elegantly. The main characteristic of answer set programming is that it supports non-monotonic logic-based reasoning mechanisms that can model default reasoning, counterfactual reasoning, abductive reasoning, etc.|mechanisms that humans use for everyday reasoning. ASP is particularly good at modeling reasoning based on default rules (and exceptions to these default rules) [1]. Humans simplify their burden of reasoning by relying on default rules. For example, we will automatically assume that if Tweety is a bird, it flies. In general, we discount the possibility that Tweety might be a penguin, an ostrich, or it may be wounded, etc. We may not consider these exceptional situations right in the beginning. However, once we learn that that one of the exceptional situation holds, we will withdraw our earlier conclusion that Tweety can fly. Likewise, if I call a very close friend of mine on his mobile phone in the morning on a weekday, and he doesn't pick it up, then I would immediately conclude that he is busy getting ready for work (because normally he picks up the phone immediately).
    [Show full text]
  • On the Minimality of Stable Models
    On the Minimality of Stable Models Paolo Ferraris1 and Vladimir Lifschitz2 1Google, USA 2Department of Computer Science, University of Texas at Austin, USA Abstract. The class of logic programs covered by the original definition of a stable model has the property that all stable models of a program in this class are minimal. In the course of research on answer set pro- gramming, the concept of a stable model was extended to several new programming constructs, and for some of these extensions the minimal- ity property does not hold. We are interested in syntactic conditions on a logic program that guarantee the minimality of its stable models. This question is addressed here in the context of the general theory of stable models of first-order sentences. 1 Introduction A Prolog program with negation, viewed as a logical formula, usually has several minimal Herbrand models, and only one of them may reflect the actual behavior of Prolog. For instance, the propositional rule p not q; viewed as the formula :q ! p (1) written in logic programming notation, has two minimal models, fpg and fqg; the first of them is the \intended" model. Early research on the semantics of nega- tion as failure [Bidoit and Froidevaux, 1987, Gelfond, 1987, Apt et al., 1988, Van Gelder, 1988, Van Gelder et al., 1988] was motivated by the need to dis- tinguish between the intended model of a logic program and its other minimal models. The definition of a stable model proposed in [Gelfond and Lifschitz, 1988] had a similar motivation. According to Theorem 1 from that paper, every stable model of a logic program is minimal.
    [Show full text]
  • History of Logic Programming
    LOGIC PROGRAMMING Robert Kowalski 1 INTRODUCTION The driving force behind logic programming is the idea that a single formalism suffices for both logic and computation, and that logic subsumes computation. But logic, as this series of volumes proves, is a broad church, with many denomi- nations and communities, coexisting in varying degrees of harmony. Computing is, similarly, made up of many competing approaches and divided into largely disjoint areas, such as programming, databases, and artificial intelligence. On the surface, it might seem that both logic and computing suffer from a similar lack of cohesion. But logic is in better shape, with well-understood relationships between different formalisms. For example, first-order logic extends propositional logic, higher-order logic extends first-order logic, and modal logic extends classical logic. In contrast, in Computing, there is hardly any relationship between, for example, Turing machines as a model of computation and relational algebra as a model of database queries. Logic programming aims to remedy this deficiency and to unify different areas of computing by exploiting the greater generality of logic. It does so by building upon and extending one of the simplest, yet most powerful logics imaginable, namely the logic of Horn clauses. In this paper, which extends a shorter history of logic programming (LP) in the 1970s [Kowalski, 2013], I present a personal view of the history of LP, focusing on logical, rather than on technological issues. I assume that the reader has some background in logic, but not necessarily in LP. As a consequence, this paper might also serve a secondary function, as a survey of some of the main developments in the logic of LP.
    [Show full text]
  • A New Approach to Conceive ASP Solvers
    A new approach to conceive ASP solvers Tarek Khaled Supervisor : Bela¨ıd Benhamou Aix Marseille Universite,´ LIS, Marseille, France. ftarek.khaled,[email protected] Abstract. The Answer set programming (ASP) is a non-monotonic declarative programming paradigm that is widely used for the formulation of problems in artificial intelligence. The ASP paradigm provides also a general framework for the resolution of decision and optimization problems. The idea behind ASP is to represent a problem as a logic program and solve that problem by computing sta- ble models. In our work, we propose a new method for searching stable models of logical programs. This method is based on a relatively new semantic that had not been exploited yet. This semantic captures and extends that one of the stable models. The method performs a DPLL enumerative process only on a restricted set of literals called the strong back-door (STB). This method has the advantage to use a Horn clause representation having the same size as the input logic pro- gram and has constant spatial complexity. It avoids the heaviness induced by the loop management from which suffer most of the ASP solvers based on the Clark completion. 1 Introduction The ASP is increasingly used, and this is due to the availability of several efficient ASP solvers like smodels [1], Clasp [2] and those based on SAT solvers like ASSAT [3] and Cmodels [4]. The natural way to use ASP is to express a problem as a logic program with default negation. To get a concise expression of the problem, the logic program is expressed in First Order Logic (FOL).
    [Show full text]