Clark's Completion Research Articles

Aspmc: New frontiers of algebraic answer set counting

In the last decade, there has been increasing interest in extensions of answer set programming (ASP) that cater for quantitative information such as weights or probabilities. A wide range of quantitative reasoning tasks for ASP and logic programming, among them probabilistic inference and parameter learning in the neuro-symbolic setting, can be expressed as algebraic answer set counting (AASC) tasks, i.e., weighted model counting for ASP with weights calculated over some semiring, which makes efficient solvers for AASC desirable. In this article, we present ▪, a new solver for AASC that pushes the limits of efficient solvability. Notably, ▪ provides improved performance compared to the state of the art in probabilistic inference by exploiting three insights gained from thorough theoretical investigations in our work. Namely, we consider the knowledge compilation step in the AASC pipeline, where the underlying logical theory specified by the answer set program is converted into a tractable circuit representation, on which AASC is feasible in polynomial time. First, we provide a detailed comparison of different approaches to knowledge compilation for programs, revealing that translation to propositional formulas followed by compilation to sd-DNNF seems favorable. Second, we study how the translation to propositional formulas should proceed to result in efficient compilation. This leads to the second and third insight, namely a novel way of breaking the positive cyclic dependencies in a program, called TP-Unfolding, and an improvement to the Clark Completion, the procedure used to transform programs without positive cyclic dependencies into propositional formulas. Both improvements are tailored towards efficient knowledge compilation. Our empirical evaluation reveals that while all three advancements contribute to the success of ▪, TP-Unfolding improves performance significantly by allowing us to handle cyclic instances better.

Decidability of the Clark's completion semantics for monadic programs and queries

AbstractThere are many different semantics for general logic programs (i.e. programs that use negation in the bodies of clauses). Most of these semantics are Turing complete (in a sense that can be made precise), implying that they are undecidable. To obtain decidability one needs to put additional restrictions on programs and queries. In logic programming it is natural to put restrictions on the underlying first-order language. In this note, we show the decidability of the Clark's completion semantics for monadic general programs and queries.

Preferred Extensions as Minimal Models of Clark's Completion Semantics

Dung established the connections between several logic pro- gramming semantics and various argumentation framework semantics. In this paper we present a characterization of the preferred semantics of argumentation frameworks (which is dened in terms of a maximal admissible set w.r.t. set inclusion) in terms of minimal logic models of Clark's completion. Additionally, we make use of integer programming for computing preferred extensions by a method dened by Bell et al. (3), which translates a logic program into an integer program which in turn can be solved by an ad hoc solver.

Monotonic Answer Set Programming

Answer set programming (ASP) does not allow for incrementally constructing answer sets or locally validating constructions like proofs by only looking at a part of the given program. In this article, we elaborate upon an alternative approach to ASP that allows for incremental constructions. Our approach draws its basic intuitions from the area of default logics. We investigate the feasibility of the concept of semi-monotonicity known from default logics as a basis of incrementality. On the one hand, every logic program has at least one answer set in our alternative setting, which moreover can be constructed incrementally based on generating rules. On the other hand, the approach may produce answer sets lacking characteristic properties of standard answer sets, such as being a model of the given program. We show how integrity constraints can be used to re-establish such properties, even up to correspondence with standard answer sets. Furthermore, we develop an SLD-like proof procedure for our incremental approach to ASP, which allows for query-oriented computations. Also, we provide a characterization of our definition of answer sets via a modification of Clark's completion. Based on this notion of program completion, we present an algorithm for computing the answer sets of a logic program in our approach.

Open answer set programming with guarded programs

Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the program's constants. We define a fixed-point logic (FPL) extension of Clark's completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called ( loosely ) guarded programs . Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed-point logic (μ(L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling, for the first time, a characterization of an answer set semantics by μLGF formulas. We further extend the open answer set semantics for programs with generalized literals. Such generalized programs (gPs) have interesting properties, for example, the ability to express infinity axioms. We restrict the syntax of gPs such that both rules and generalized literals are guarded . Via a translation to guarded fixed-point logic, we deduce 2-EXPTIME-completeness of satisfiability checking in such guarded gPs (GgPs). Bound GgPs are restricted GgPs with EXPTIME-complete satisfiability checking, but still sufficiently expressive to optimally simulate computation tree logic (CTL). We translate Datalog lite programs to GgPs, establishing equivalence of GgPs under an open answer set semantics, alternation-free μGF, and Datalog LITE.

Loop formulas for circumscription

Clark's completion is a simple nonmonotonic formalism and a special case of several nonmonotonic logics. Recently there has been work on extending completion with “loop formulas” so that general cases of nonmonotonic logics such as logic programs (under the answer set semantics) and McCain–Turner causal logic can be characterized by propositional logic in the form of “completion + loop formulas”. In this paper, we show that the idea is applicable to McCarthy's circumscription in the propositional case, with Lifschitz's pointwise circumscription playing the role of completion. We also show how to embed propositional circumscription in logic programs and in causal logic, inspired by the uniform characterization of “completion + loop formulas”.

Computation of the semantics of autoepistemic belief theories

Recently, one of the authors introduced a simple and yet powerful non-monotonic knowledge representation framework, called the Autoepistemic Logic of Beliefs, AEB. Theories in AEB are called autoepistemic belief theories. Every belief theory T has been shown to have the least static expansionT which is computed by iterating a natural monotonic belief closure operator ΨT starting from T. This way, the least static expansion T of any belieftheory provides its natural non-monotonic semantics which is called the static semantics.It is easy to see that if a belief theory T is finite then the construction of its least static expansion T stops after countably many iterations. However, a somewhat surprising result obtained in this paper shows that the least static expansion of any finite belief theory T is in fact obtained by means of a single iteration of the belief closure operator ΨT (although this requires T to be of a special form, we also show that T can be always put in this form). This result eliminates the need for multiple iterations in the computation of static semantics and allows us to replace the fixed-point definition of static semantics by the equivalent explicit and straightforward definition given by T=ΨT(T).The second, closely related result establishes an intriguing relationship between static semantics T and Clark's completions comp(T) of finite belief theories. Here we use a slightly generalized version of comp(T) (see Definition 3.2). It shows that the static semantics T of T is obtained by augmenting T with the set Bcomp(T)={BF:F∈comp(T)} thus ensuring that all formulae that belong to Clark's completion comp(T) of T are believed to be true.Both results open the way for a more efficient implementation of static semantics: the first, because only one iteration is needed, and the second because reasoning in a non-standard logic (belief theories under static semantics) can be reduced to classical theorem proving involving Clark's completion.

A rewrite mechanism for logic programs with negation

In several familiar cases of logic programs and queries with infinitely many answers, the usual operational mechanisms based on SLD-resolution and its extensions go into an infinite loop. It may therefore be desirable to look for mechanisms which synthesize possible structural relations on sets of answers. One such mechanism is known for pure logic programs: interpret them as rewrite programs, and execute them with a version of the Knuth-Bendix algorithm, called linear completion. W.r.t. positive queries for such programs, in most cases of productive loops, linear completion yields a finite set of answers and a finite set of rewrite rules involving just one predicate, from which the remaining answers can be deduced. It is thus natural to ask how much of the rewrite mechanism carries through for deducing negative information from pure programs, and more generally from normal logic programs. In this paper, we show that such an extension can be built in a natural way, with ideas from the Clark completion, and the domain of constrained rewriting. The correctness and completeness of this extended mechanism are proved w.r.t. the 3-valued declarative semantics of Künen for normal programs. Due to the synthesizing property and the acceptance of non ground negatives queries, our mechanism is in a sense constructive w.r.t. negation. An interesting side-effect of our approach seems to be its ability to parametrize the semantics of a normal program in terms of a meta-reduction rule set: in particular, we can operationally go beyond Künen's semantics, if desired.

Rules as actions: A situation calculus semantics for logic programs

We propose a novel semantics for logic programs with negation by viewing the application of a clause in a derivation as an action in the situation calculus. Program clauses are then identified with situation calculus effect axioms as they are understood in axiomatic theories of actions. We then solve the frame problem for these effect axioms using a recent approach of Reiter [21], and identify the resulting collection of axioms with the semantics of the original logic problem. An interesting consequence of this approach is that the logic programming negation-as-failure operator inherits its nonmonotonicity from the nonmonotonicity associated with the frame problem. One advantage of our proposal is that like Clark's completion semantics, ours is also formulated explicitly in classical logic. To illustrate the usefulness of our semantics, we prove sufficient conditions for two logic programs to be equivalent, and use this to verify the correctness of the well-known unfolding program transformation operator. We also discuss applications of this framework to formalizing search control operators in logic programming.

A CLASSIFICATION THEORY OF SEMANTICS OF NORMAL LOGIC PROGRAMS: I. STRONG PROPERTIES

Our aim in this article is to present a method for classifying and characterizing the various different semantics of logic programs with negation that have been considered in the last years. Instead of appealing to more or less questionable intuitions, we take a more structural view: our starting point is the observation that all semantics induce in a natural way non-monotonic entailment relations “ |˜ ”. The novel idea of our approach is to ask for the properties of these |˜ -relations and to use them for describing all possible semantics. The main properties discussed in this paper are adaptations of rules that play a fundamental role in general non-monotonic reasoning: Cumulativity and Rationality. They were introduced and investigated by Gabbay, Kraus, Lehmann, Magidor and Makinson. We show that the 3-valued version COMP 3 of Clark's completion, the stratified semantics M supp P as well as the well-founded semantics WFS and two extensions of it behave very regular: they are cumulative, rational and one of them is even supraclassical. While Pereira's recently proposed semantics O-SEM is not rational it is still cumulative. Cumulativity fails for the regular semantics REG-SEM of You/Yuan (recently shown to be equivalent to three other proposals). In a second article we will supplement these strong rules with a set of weak rules and consider the problem of uniquely describing a given semantics by its strong and weak properties together.

Propositional semantics for disjunctive logic programs

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion.

A class of programs for which SLDNF resolution and NAF rule are complete

In this paper, we prove completeness of SLDNF resolution and NAF rule for the class of allowed and locally stratified programs satisfying the further condition of well-behavedness. Well-behavedness imposes a computational restriction on programs whose aim is to ensure that the 3- and 2-valued consequences of Clark's completion of a program belonging to the above-mentioned class do coincide. Then one can apply Kunen's 3-valued completeness theorem in order to get a 2-valued completeness result

Tight, consistent, and computable completions for unrestricted logic programs

Clark's program completion offers an intuitive first-order semantics for logic programs. Unfortunately, it does not fully capture the “tight” bottom- up semantics for recursive programs since it does not express enough negative information. Therefore, various canonical model semantics have been proposed which offer the required tight semantics for increasingly general classes of programs. Canonical models are defined in terms of fixpoint operators on program interpretations. The resulting semantics is hard to specify, harder to understand, and impossible to compute. In this paper, we propose to rehabilitate the program completion. We show how it can be extended to capture the tight semantics, and how a consistent completion can be derived for all programs, without resorting to three-valued logic. Finally, by lifting the restriction to Herbrand domains, we exhibit a tight, consistent, computable semantics for unrestricted logic programs. As with any completion, the meaning of a program is a set of axioms directly constructed from the program text.

On the relations between stable and well-founded semantics of logic programs

We study the relations between stable and well-founded semantics of logic programs. 1. We show that stable semantics can be defined in the same way as well-founded semantics based on the basic notion of unfounded sets. Hence, stable semantics can be considered as “two-valued well-founded semantics”. 2. An axiomatic characterization of stable and well-founded semantics of logic programs is given by a new completion theory, called strong completion. Similar to the Clark's completion, the strong completion can be interpreted in either two-valued or three-valued logic. We show that ◦ Two-valued strong completion specifies the stable semantics. ◦ Three-valued strong completion specifies the well-founded semantics. 3. We study the equivalence between stable semantics and well-founded semantics. At first, we prove the equivalence between the two semantics for strict programs. Then we introduce the bottom-stratified and top-strict condition generalizing both the stratifiability and the strictness, and show that the new condition is sufficient for the equivalence between stable and well-founded semantics. Further, we show that the call-consistency condition is sufficient for the existence of at least one stable model.

The well-founded semantics for general logic programs

A general logic (abbreviated to hereafter) is a set of roles that have both positive and negative subgoals. It is common to view a deductive database as a general logic consisting of rules (IDB) slttmg above elementary relations (EDB, facts). It is desirable to associate one Herbrand model with a and think of that model as the meaning of the program, or Its declarative semantics. Ideally, queries directed to the would be answered in accordance with this model. Recent research indicates that some programs do not have a satisfactory total model; for such programs, the question of an appropriate partial model arises. Unfounded sets and well-founded partial are introduced and the well-founded semantics of a are defined to be its well-founded partial model. If the well-founded partial model is m fact a total model. it is called the well-founded model. It n shown that the class of programs possessing a total well-founded model properly includes previously studied classes of and locally stratified programs, The method in this paper is also compared with other proposals in the literature, including Clark's program completion, Fitting's and Kunen's 3-vahred interpretations of it, and the stable models of Gelfond and Lifschitz.

Unsolvable problems for SLDNF resolution

Shepherdson showed that there are maximal safe computation rules R m for SLDNF resolution, such that if a query succeeds under any safe rule it succeeds under R m , and if it fails under any safe rule it fails under R m . Later he showed that such maximal rules also get all possible answers, i.e., if a query succeeds with answer θ under any safe rule, then it succeeds with answer equivalent to θ under R m . The question was raised there, whether there were always recursive maximal rules. We answer this negatively. We also show the unsolvability of the problems of deciding whether a query leads to a dead end, whether the Clark completion comp( P) of a program P is consistent, and whether a query which is known to be a logical consequence of the program succeeds in PROLOG.

Computing protected circumscription

This paper deals with computing circumscription in the case of Horn data with additional protection (indefinite data), an intermediate investigation between Reiter's result on predicate completion and Lifschitz's efforts to make general (formula) circumscription more efficient as a computational tool. Reiter has shown a close tie between McCarthy's circumscription and Clark's predicate completion. Here we investigate a similar tie between an extended version of circumscription involving protected data, and an extended version of predicate completion. When we have a fully ground atomic protected theory, we show that an extension to the relational algebra can be used to obtain all (and only) correct answers. When general Horn axioms are added to the protected theory, we show that Horn axioms also can be used to compute sound answers; however, some correct answers will not be found.