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.
flp04 developed the theory of disjunctive logic programs with aggregates. Similarly as p04 and pdb06 , flp04 do not allow for aggregates to appear in the heads of program clauses. This is one of the differences between that approach and programs with monotone (convex) constraints we studied here. The other major difference is related to the postulate of the minimality of stable models (called answer sets in the context of the formalism considered by flp04 ). In keeping with the spirit of the original answer-set semantics @cite_24 , answer sets of disjunctive programs with aggregates, as defined by flp04 , are minimal models. Stable models of programs with abstract constraints do not have this property. However, for the class of programs with abstract monotone constraints with the heads of rules of the form @math the semantics of answer sets defined by flp04 coincides with the semantics of stable models by mt04 and mnt03,mnt06 .
@cite_24
2064864283 50549667 2085084839 1638553575 2050890691 2087432892 1598820892 2010968193 2106500031 2063061546 1508718221 2048082592 49730540 2169808787 1520574003 1800801472 80590317 1516535258 1984529395 2170232725
The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modelling power of ASP in terms of natural and concise problem representations. Previous semantic definitions typically agree in the case of non-recursive aggregates, but the picture is less clear for aggregates involved in recursion. Some proposals explicitly avoid recursive aggregates, most others differ, and many of them do not satisfy desirable criteria, such as minimality or coincidence with answer sets in the aggregate-free case. In this paper we define a semantics for programs with arbitrary aggregates (including monotone, antimonotone, and nonmonotone aggregates) in the full ASP language allowing also for disjunction in the head (disjunctive logic programming - DLP). This semantics is a genuine generalization of the answer set semantics for DLP, it is defined by a natural variant of the Gelfond-Lifschitz transformation, and treats aggregate and non-aggregate literals in a uniform way. This novel transformation is interesting per se also in the aggregate-free case, since it is simpler than the original transformation and does not need to differentiate between positive and negative literals. We prove that our semantics guarantees the minimality (and therefore the incomparability) of answer sets, and we demonstrate that it coincides with the standard answer set semantics on aggregate-free programs. Moreover, we carry out an in-depth study of the computational complexity of the language. The analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of aggregates, disjunction, and negation. While the addition of aggregates does not affect the complexity of the full DLP language, it turns out that their presence does increase the complexity of normal (i.e., non-disjunctive) ASP programs up to the second level of the polynomial hierarchy. However, we show that there are large classes of aggregates the addition of which does not cause any complexity gap even for normal programs, including the fragment allowing for arbitrary monotone, arbitrary antimonotone, and stratified (i.e., non-recursive) nonmonotone aggregates. The analysis provides some useful indications on the possibility to implement aggregates in existing reasoning engines. In recent years, Answer Set Programming ASP, logic programming under the stable model or answer set semantics, has seen several extensions by generalizing the notion of an atom in these programs: be it aggregate atoms, HEX atoms, generalized quantifiers, or abstract constraints, the idea is to have more complicated satisfaction patterns in the lattice of Herbrand interpretations than traditional, simple atoms. In this paper we refer to any of these constructs as generalized atoms. It is known that programs with generalized atoms that have monotonic or antimonotonic satisfaction patterns do not increase complexity with respect to programs with simple atoms if satisfaction of the generalized atoms can be decided in polynomial time under most semantics. It is also known that generalized atoms that are nonmonotonic being neither monotonic nor antimonotonic can, but need not, increase the complexity by one level in the polynomial hierarchy if non-disjunctive programs under the FLP semantics are considered. In this paper we provide the precise boundary of this complexity gap: programs with convex generalized atom never increase complexity, while allowing a single non-convex generalized atom under reasonable conditions always does. We also discuss several implications of this result in practice. We introduce the stable model semantics fordisjunctive logic programs and deductive databases, which generalizes the stable model semantics, defined earlier for normal (i.e., non-disjunctive) programs. Depending on whether only total (2-valued) or all partial (3-valued) models are used we obtain thedisjunctive stable semantics or thepartial disjunctive stable semantics, respectively. The proposed semantics are shown to have the following properties: • For normal programs, the disjunctive (respectively, partial disjunctive) stable semantics coincides with thestable (respectively,partial stable) semantics. • For normal programs, the partial disjunctive stable semantics also coincides with thewell-founded semantics. • For locally stratified disjunctive programs both (total and partial) disjunctive stable semantics coincide with theperfect model semantics. • The partial disjunctive stable semantics can be generalized to the class ofall disjunctive logic programs. • Both (total and partial) disjunctive stable semantics can be naturally extended to a broader class of disjunctive programs that permit the use ofclassical negation. • After translation of the programP into a suitable autoepistemic theory ( P ) the disjunctive (respectively, partial disjunctive) stable semantics ofP coincides with the autoepistemic (respectively, 3-valued autoepistemic) semantics of ( P ) . We investigate mca-programs, that is, logic programs with clauses built of monotone cardinality atoms of the form kX, where k is a non-negative integer and X is a finite set of propositional atoms. We develop a theory of mca-programs. We demonstrate that the operational concept of the one-step provability operator generalizes to mca-programs, but the generalization involves nondeterminism. Our main results show that the formalism of mca-programs is a common generalization of (1) normal logic programming with its semantics of models, supported models and stable models, (2) logic programming with cardinality atoms and with the semantics of stable models, as defined by Niemela, Simons and Soininen, and (3) of disjunctive logic programming with the possible-model semantics of Sakama and Inoue. The answer set semantics presented by [27] has been widely used to define so called FLP answer sets for different types of logic programs. However, it was recently observed that when being extended from normal to more general classes of logic programs, this approach may produce answer sets with circular justifications that are caused by self-supporting loops. The main reason for this behavior is that the FLP answer set semantics is not fully constructive by a bottom up construction of answer sets. In this paper, we overcome this problem by enhancing the FLP answer set semantics with a level mapping formalism such that every answer set I can be built by fixpoint iteration of a one-step provability operator (more precisely, an extended van Emden-Kowalski operator for the FLP reduct fΠI). This is inspired by the fact that under the standard answer set semantics, each answer set I of a normal logic program Π is obtainable by fixpoint iteration of the standard van Emden-Kowalski one-step provability operator for the Gelfond-Lifschitz reduct ΠI, which induces a level mapping. The enhanced FLP answer sets, which we call well-justified FLP answer sets, are thanks to the level mapping free of circular justifications. As a general framework, the well-justified FLP answer set semantics applies to logic programs with first-order formulas, logic programs with aggregates, description logic programs, hex-programs etc., provided that the rule satisfaction is properly extended to such general logic programs. We study in depth the computational complexity of FLP and well-justified FLP answer sets for general classes of logic programs. Our results show that the level mapping does not increase the worst-case complexity of FLP answer sets. Furthermore, we describe an implementation of the well-justified FLP answer set semantics, and report about an experimental evaluation, which indicates a potential for performance improvements by the level mapping in practice. This paper addresses complexity issues for important problems arising with disjunctive logic programming. In particular, the complexity of deciding whether a disjunctive logic program is consistent is investigated for a variety of well-known semantics, as well as the complexity of deciding whether a propositional formula is satisfied by all models according to a given semantics. We concentrate on finite propositional disjunctive programs with as well as without integrity constraints, i.e., clauses with empty heads; the problems are located in appropriate slots of the polynomial hierarchy. In particular, we show that the consistency check is Σ 2 p -complete for the disjunctive stable model semantics (in the total as well as partial version), the iterated closed world assumption, and the perfect model semantics, and we show that the inference problem for these semantics is Π 2 p -complete; analogous results are derived for the answer sets semantics of extended disjunctive logic programs. Besides, we generalize previously derived complexity results for the generalized closed world assumption and other more sophisticated variants of the closed world assumption. Furthermore, we use the close ties between the logic programming framework and other nonmonotonic formalisms to provide new complexity results for disjunctive default theories and disjunctive autoepistemic literal theories. In this paper, we propose an extension of the well-founded and stable model semantics for logic programs with aggregates. Our approach uses Approximation Theory, a fixpoint theory of stable and well-founded fixpoints of non-monotone operators in a complete lattice. We define the syntax of logic programs with aggregates and define the immediate consequence operator of such programs. We investigate the well-founded and stable semantics generated by Approximation Theory. We show that our approach extends logic programs with stratified aggregation and that it correctly deals with well-known benchmark problems such as the shortest path program and the company control problem. Let P be an arbitrary theory and let X be any given logic. Let M be a set of atoms. We say that M is a X-stable model of P if M is a classical model of P and P∪¬M proves in logic X all atoms in M, this is denoted by P∪¬M ⊩xM. We prove that being an X-stable model is an invariant property for disjunctive programmes under a large class of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that replaces ¬a with ¬□a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning are equivalent for disjunctive programmes. In addition, we show that such semantics can be characterized by a fixed-point operator in terms of classical logic. We also present a simple translation of a disjunctive programme D into a normal programme N, such that the PStable model semantics of N corresponds to the stable semantics of D over the common language. We present the formal proof of this statement. Disjunctive logic programming (DLP) is a very expressive formalism. It allows for expressing every property of finite structures that is decidable in the complexity class ΣP2(=NPNP). Despite this high expressiveness, there are some simple properties, often arising in real-world applications, which cannot be encoded in a simple and natural manner. Especially properties that require the use of arithmetic operators (like sum, times, or count) on a set or multiset of elements, which satisfy some conditions, cannot be naturally expressed in classic DLP. To overcome this deficiency, we extend DLP by aggregate functions in a conservative way. In particular, we avoid the introduction of constructs with disputed semantics, by requiring aggregates to be stratified. We formally define the semantics of the extended language (called ), and illustrate how it can be profitably used for representing knowledge. Furthermore, we analyze the computational complexity of , showing that the addition of aggregates does not bring a higher cost in that respect. Finally, we provide an implementation of in DLVa state-of-the-art DLP systemand report on experiments which confirm the usefulness of the proposed extension also for the efficiency of computation. We propose a semantics for aggregates in deductive databases based on a notion of minimality. Unlike some previous approaches, we form a minimal model of a program component including aggregate operators, rather than insisting that the aggregate apply to atoms that have been fully determined or that aggregate functions are rewritten in terms of negation. In order to guarantee the existence of such a minimal model we need to insist that the domains over which we are aggregating are complete lattices and that the program is in a sense mono- tonic. Our approach generalizes previous approaches based on the well-founded semantics and various forms of stratification. We are also able to handle a large variety of monotonic (or pseudo-monotonic) aggregate functions. Disjunctive Logic Programming (DLP) is a very expressive formalism: it allows to express every property of finite structures that is decidable in the complexity class ΣP2 (NPNP). Despite the high expressiveness of DLP, there are some simple properties, often arising in real-world applications, which cannot be encoded in a simple and natural manner. Among these, properties requiring to apply some arithmetic operators (like sum, times, count) on a set of elements satisfying some conditions, cannot be naturally expressed in DLP. To overcome this deficiency, in this paper we extend DLP by aggregate functions. We formally define the semantics of the new language, named DLPA. We show the usefulness of the new constructs on relevant knowledge-based problems. We analyze the computational complexity of DLPA, showing that the addition of aggregates does not bring a higher cost in that respect. We provide an implementation of the DLPA language in DLV- the state-of-the-art DLP system - and report on experiments which confirm the usefulness of the proposed extension also for the efficiency of the computation. We present a novel combination of disjunctive programs under the answer set semantics with description logics for the Semantic Web. The combination is based on a well-balanced interface between disjunctive programs and description logics, which guarantees the decidability of the resulting formalism without assuming syntactic restrictions. We show that the new formalism has very nice semantic properties. In particular, it faithfully extends both disjunctive programs and description logics. Furthermore, we describe algorithms for reasoning in the new formalism, and we give a precise picture of its computational complexity. We also define the well-founded semantics for the normal case, where normal programs are combined with tractable description logics, and we explore its semantic and computational properties. In particular, we show that the well-founded semantics approximates the answer set semantics. We also describe algorithms for the problems of consistency checking and literal entailment under the well-founded semantics, and we give a precise picture of their computational complexity. As a crucial property, in the normal case, consistency checking and literal entailment under the well-founded semantics are both tractable in the data complexity, and even first-order rewritable (and thus can be done in LogSpace in the data complexity) in a special case that is especially useful for representing mappings between ontologies. Recently Ferraris, Lee and Lifschitz proposed a new definition of stable models that does not refer to grounding, which applies to the syntax of arbitrary first-order sentences. We show its relation to the idea of loop formulas with variables by Chen, Lin, Wang and Zhang, and generalize their loop formulas to disjunctive programs and to arbitrary first-order sentences. We also extend the syntax of logic programs to allow explicit quantifiers, and define its semantics as a subclass of the new language of stable models by Such programs inherit from the general language the ability to handle nonmonotonic reasoning under the stable model semantics even in the absence of the unique name and the domain closure assumptions, while yielding more succinct loop formulas than the general language due to the restricted syntax. We also show certain syntactic conditions under which query answering for an extended program can be reduced to entailment checking in first-order logic, providing a way to apply first-order theorem provers to reasoning about non-Herbrand stable models. Answer set programming (ASP) with disjunction offers a powerful tool for declaratively representing and solving hard problems. Many NP-complete problems can be encoded in the answer set semantics of logic programs in a very concise and intuitive way, where the encoding reflects the typical “guess and check” nature of NP problems: The property is encoded in a way such that polynomial size certificates for it correspond to stable models of a program. However, the problem-solving capacity of full disjunctive logic programs (DLPs) is beyond NP, and captures a class of problems at the second level of the polynomial hierarchy. While these problems also have a clear “guess and check” structure, finding an encoding in a DLP reflecting this structure may sometimes be a non-obvious task, in particular if the “check” itself is a co-NP-complete problem; usually, such problems are solved by interleaving separate guess and check programs, where the check is expressed by inconsistency of the check program. In this paper, we present general transformations of head-cycle free (extended) disjunctive logic programs into stratified and positive (extended) disjunctive logic programs based on meta-interpretation techniques. The answer sets of the original and the transformed program are in simple correspondence, and, moreover, inconsistency of the original program is indicated by a designated answer set of the transformed program. Our transformations facilitate the integration of separate “guess” and “check” programs, which are often easy to obtain, automatically into a single disjunctive logic program. Our results complement recent results on meta-interpretation in ASP, and extend methods and techniques for a declarative “guess and check” problem solving paradigm through ASP. We introduce a family of partial stable model semantics for logic programs with arbitrary aggregate relations. The semantics are parametrized by the interpretation of aggregate relations in three-valued logic. Any semantics in this family satisfies two important properties: (i) it extends the partial stable semantics for normal logic programs and (ii) total stable models are always minimal. We also give a specific instance of the semantics and show that it has several attractive features. Equilibrium logic, introduced by David Pearce, extends the concept of an answer set from logic programs to arbitrary sets of formulas. Logic programs correspond to the special case in which every formula is a “rule” — an implication that has no implications in the antecedent (body) and consequent (head). The semantics of equilibrium logic looks very different from the usual definitions of an answer set in logic programming, as it is based on Kripke models. In this paper we propose a new definition of equilibrium logic which uses the concept of a reduct, as in the standard definition of an answer set. Second, we apply the generalized concept of an answer set to the problem of defining the semantics of aggregates in answer set programming. We propose, in particular, a semantics for weight constraints that covers the problematic case of negative weights. Our semantics of aggregates is an extension of the approach due to Faber, Leone, and Pfeifer to a language with choice rules and, more generally, arbitrary rules with nested expressions. Dept. of Computer Science, K.U.LeuvenCelestijnenlaan 200A, B-3001 Heverlee, BelgiumE-mail: pelov,marcd,maurice @cs.kuleuven.ac.beAbstract. We deﬁne a translation of aggregate programs to normal lo-gic programs which preserves the set of partial stable models. We thendeﬁne the classes of deﬁnite and stratiﬁed aggregate programs and showthat the translation of such programs are, respectively, deﬁnite and strat-iﬁed logic programs. Consequently these two classes of programs havea single partial stable model which is two-valued and is also the well-founded model. Our deﬁnition of stratiﬁcation is more general than theexisting one and covers a strictly larger class of programs. Partial stable models for deductive databases, i.e., normal function-free logic programs (also called datalog programs), have two equivalent definitions: one based on 3-valued logics and another based on the notion of unfounded set. The notion of partial stable model has been extended to disjunctive deductive databases using 3-valued logics. In this paper, a characterization of partial stable models for disjunctive datalog programs is given using a suitable extension of the notion of unfounded set. Two interesting sub-classes of partial stable models, M-stable (Maximal-stable) (also called regular models, preferred extension,and maximal stable classes) and L-stable (Least undefined-stable) models, are then extended from normal to disjunctive datalog programs. On the one hand, L-stable models are shown to be the natural relaxation of the notion of total stable model; on the other hand the less strict M-stable models, endowed with a nice modularity property, may be appealing from the programming and computational point of view. M-stable and L-stable models are also compared with the regular models for disjunctive datalog programs recently proposed in the literature. Logic programs with ordered disjunction (LPODs) combine ideas underlying Qualitative Choice Logic (Brewka, Benferhat, & Le Berre 2002) and answer set programming. Logic programming under answer set semantics is extended with a new connective called ordered disjunction. The new connective allows us to represent alternative, ranked options for problem solutions in the heads of rules: A × B intuitively means: if possible A, but if A is not possible then at least B. The semantics of logic programs with ordered disjunction is based on a preference relation on answer sets. LPODs are useful for applications in design and configuration and can serve as a basis for qualitative decision making. This paper relates the Defeasible Logic Programming (DeLP) framework and its semantics SEMDeLP to classical logic programming frameworks. In DeLP, we distinguish between two different sorts of rules: strict and defeasible rules. Negative literals (∼A) in these rules are considered to represent classical negation. In contrast to this, in normal logic programming (NLP), there is only one kind of rules, but the meaning of negative literals (not A) is different: they represent a kind of negation as failure, and thereby introduce defeasibility. Various semantics have been defined for NLP, notably the well-founded semantics (WFS) (van , Proceedings of the Seventh Symposium on Principles of Database Systems, 1988, pp. 221-230; J. ACM 38 (3) (1991) 620) and the stable semantics Stable (Gelfond and Lifschitz, Fifth Conference on Logic Programming, MIT Press, Cambridge, MA, 1988, pp. 1070-1080; Proceedings of the Seventh International Conference on Logical Programming, Jerusalem, MIT Press, Cambridge, MA, 1991, pp. 579-597).In this paper we consider the transformation properties for NLP introduced by Brass and Dix (J. Logic Programming 38(3) (1999) 167) and suitably adjusted for the DeLP framework. We show which transformation properties are satisfied, thereby identifying aspects in which NLP and DeLP differ. We contend that the transformation rules presented in this paper can help to gain a better understanding of the relationship of DeLP semantics with respect to more traditional logic programming approaches. As a byproduct, we obtain the result that DeLP is a proper extension of NLP.