ASSAT: computing answer sets of a logic program by SAT solvers

A consequence of a logic program under answer set semantics is one that is true for all answer sets. This article considers using loop formulas to compute some of these consequences in order to increase the efficiency of answer set solvers. Since computing loop formulas are in general intractable, we consider only loops with either no external support or at most one external support, as their loop formulas are either unit or binary clauses. We show that for disjunctive logic programs, loop formulas of loops with no external support can be computed in polynomial time, and that an iterative procedure using unit propagation on these formulas and the program completion computes the well-founded models in the case of normal logic programs and the least fixed point of a simplification operator used by DLV for disjunctive logic programs. For loops with at most one external support, their loop formulas can be computed in polynomial time for normal logic programs, but are NP-hard for disjunctive programs. So for normal logic programs, we have a procedure similar to the iterative one for loops without any external support, but for disjunctive logic programs, we present a polynomial approximation algorithm. All these algorithms have been implemented, and our experiments show that for certain logic programs, the consequences computed by our algorithms can significantly speed up current ASP solvers cmodels, clasp, and DLV.

In this article, we consider Answer Set Programming (ASP). It is a declarative problem solving paradigm that can be used to encode a problem as a logic program whose answer sets correspond to the solutions of the problem. It has been widely applied in various domains in AI and beyond. Given that answer sets are supposed to yield solutions to the original problem, the question of “why a set of atoms is an answer set” becomes important for both semantics understanding and program debugging. It has been well investigated for normal logic programs. However, for the class of disjunctive logic programs, which is a substantial extension of that of normal logic programs, this question has not been addressed much. In this article, we propose a notion of reduct for disjunctive logic programs and show how it can provide answers to the aforementioned question. First, we show that for each answer set, its reduct provides a resolution proof for each atom in it. We then further consider minimal sets of rules that will be sufficient to provide resolution proofs for sets of atoms. Such sets of rules will be called witnesses and are the focus of this article. We study complexity issues of computing various witnesses and provide algorithms for computing them. In particular, we show that the problem is tractable for normal and headcycle-free disjunctive logic programs, but intractable for general disjunctive logic programs. We also conducted some experiments and found that for many well-known ASP and SAT benchmarks, computing a minimal witness for an atom of an answer set is often feasible.

Answer Set Programming (ASP) is a constraint-based programming paradigm proposed recently by AI researchers. In this paradigm one solves a problem by encoding constraints as logic program rules in such a way that the solutions to the original problem correspond to the so-called answer sets of the logic program. This new programming paradigm has found applications in problems ranging from combinatorial search problems, such as the graph coloring and Hamiltonian circuit problems, to product configuration and data encryption problems. In this dissertation, we make the following two contributions to the new programming paradigm ASP. First, we propose and implement a new ASP system using traditional SAT (propositional satisfiability) solvers. This is done by a translation from logic programs to sets of clauses in propositional logic. Our experimental results show that for many benchmark domains, our system performs better than existing specialized ASP systems. Secondly, we study the phase transition phenomenon in randomly generated logic programs, and identify a region in a class of randomly generated logic programs where the logic programs are hard for all existing ASP systems.

Answer Set Programming (ASP) provides a powerful declarative paradigm for knowledge representation and reasoning. Recently, counting answer sets has emerged as an important computational problem with applications in probabilistic reasoning, network reliability analysis, and other domains. This has motivated significant research into designing efficient ASP counters. While substantial progress has been made for normal logic programs, the development of practical counters for disjunctive logic programs remains challenging. We present $\mathsf{sharpASP}$ - $\mathcal{SR}$ , a novel framework for counting answer sets of disjunctive logic programs based on subtractive reduction to projected propositional model counting. Our approach introduces an alternative characterization of answer sets that enables efficient reduction while ensuring the intermediate representations remain polynomial in size. This allows $\mathsf{sharpASP}$ - $\mathcal{SR}$ to leverage recent advances in projected model counting technology. Through extensive experimental evaluation on diverse benchmarks, we demonstrate that $\mathsf{sharpASP}$ - $\mathcal{SR}$ significantly outperforms existing counters on instances with large answer set counts. Building on these results, we develop a hybrid counting approach that combines enumeration techniques with $\mathsf{sharpASP}$ - $\mathcal{SR}$ to achieve state-of-the-art performance across the full spectrum of disjunctive programs. The extended version of the paper is available at: https://arxiv.org/abs/2507.11655 .

Relating defeasible and normal logic programming through transformation properties

Answer set programming (ASP) (see, for instance, [22]) is a new declarative programming paradigm suitable for solving a large range of problems related to knowledge representation and search. The paradigm is rooted in recent developments in several areas of artificial intelligence. ASP starts by encoding relevant domain knowledge as a (possibly disjunctive) logic program, Π. The connectives of this program are normally understood in accordance with the answer set (stable model) semantics [12,13]. The corresponding language is frequently referred to as A-Prolog (or ANS-Prolog). The language’s ability to express defaults, i.e. statements of the form “normally, objects of class C have property P”, coupled with its natural treatment of recursion, and other useful features, often leads to a comparatively concise and clear representation of knowledge. Insights on the nature of causality and its relationship with the answer sets of logic programs [14,21,25] allows for the description of the effects of actions which solves the frame, ramification, and qualification problems, which for a long time have caused difficulties in modeling knowledge about dynamic domains.

This paper extends fuzzy logic programs [12, 24] to allow the explicit representation of classical negation as well as non-monotonic negation, by introducing the notion of extended fuzzy logic programs. We present the fuzzy answer set semantics for the extended fuzzy logic programs, which is based on the classical answer set semantics of classical extended logic programs [7]. We show that the proposed semantics is a natural extension to the classical answer set semantics of classical extended logic programs [7]. Furthermore, we define fixpoint semantics for extended fuzzy logic programs with and without non-monotonic negation, and study their relationship to the fuzzy answer set semantics. In addition, we show that the fuzzy answer set semantics is reduced to the stable fuzzy model semantics for normal fuzzy logic programs introduced in [42]. The importance of that is computational methods developed for normal fuzzy logic programs can be applied to the extended fuzzy logic programs. Moreover, we show that extended fuzzy logic programs can be intuitively used for representing and reasoning about actions in fuzzy environment.

In this paper we introduce a formalism for solving Hierarchical Task Network (HTN) Planning using Answer Set Programming (ASP). We consider the formulation of HTN planning as described in the SHOP planning system and define a systematic translation method from SHOP’s representation of the planning problem into logic programs with negation. We show that our translation is sound and complete: answer sets of the logic program obtained by our translation correspond exactly to the solutions of the planning problem. We compare our method to (1) similar approaches based on non-HTN planning and (2) SHOP, a dedicated planning system. We show that our approach outperforms non-HTN methods and that its performance is better with ASP systems that allow for nonground programs than with ASP systems that require ground programs.

Answer Set Programming (ASP) is a framework in artificial intelligence and knowledge representation for declarative modeling and problem solving. Modern ASP solvers focus on the computation or enumeration of answer sets. However, a variety of probabilistic applications in reasoning or logic programming require counting answer sets. While counting can be done by enumeration, simple enumeration becomes immediately infeasible if the number of solutions is high. On the other hand, approaches to exact counting are of high worst-case complexity. In fact, in propositional model counting, exact counting becomes impractical. In this work, we present a scalable approach to approximate counting for answer set programming. Our approach is based on systematically adding XOR constraints to ASP programs, which divide the search space. We prove that adding random XOR constraints partitions the answer sets of an ASP program. In practice, we use a Gaussian elimination-based approach by lifting ideas from SAT to ASP and integrating it into a state of the art ASP solver, which we call ApproxASP. Finally, our experimental evaluation shows the scalability of our approach over the existing ASP systems.

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.

In recent years, Answer Set Programming has gained popularity as a viable paradigm for applications in knowledge representation and reasoning. This paper presents a novel methodology to compute answer sets of an answer set program. The proposed methodology maintains a bottom-up approach to the computation of answer sets (as in existing systems), but it makes use of a novel structuring of the computation, that originates from the non-ground version of the program. Grounding is lazily performed during the computation of the answer sets. The implementation has been realized using Constraint Logic Programming over finite domains.

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.

Loop formulas for circumscription

Recently Modular Nonmonotonic Logic Programs (MLP) have been introduced which incorporate a call-by-value mechanism and allow for unrestricted calls between modules, including mutual and self recursion, as an approach to provide module constructs akin to those in conventional programming in Nonmonotonic Logic Programming under Answer Set Semantics. This paper considers MLPs in a Datalog setting and provides characterizations of their answers sets in terms of classical (Herbrand) models of a first-order formula, extending a line of research for ordinary logic programs. To this end, we lift the well-known loop formulas method to MLPs, and we also consider the recent ordered completion approach that avoids explicit construction of loop formulas using auxiliary predicates. Independent of computational perspectives, the novel characterizations widen our understanding of MLPs and they may prove useful for semantic investigations.

In answer set programming (ASP), a problem at hand is solved by (i) writing a logic program whose answer sets correspond to the solutions of the problem, and by (ii) computing the answer sets of the program using ananswer set solveras a search engine. Typically, a programmer creates a series of gradually improving logic programs for a particular problem when optimizing program length and execution time on a particular solver. This leads the programmer to a meta-level problem of ensuring that the programs are equivalent, i.e., they give rise to the same answer sets. To ease answer set programming at methodological level, we propose a translation-based method for verifying the equivalence of logic programs. The basic idea is to translate logic programsPandQunder consideration into a single logic program EQT(P,Q) whose answer sets (if such exist) yield counter-examples to the equivalence ofPandQ. The method is developed here in a slightly more general setting by taking thevisibilityof atoms properly into account when comparing answer sets. The translation-based approach presented in the paper has been implemented as a translator calledlpeqthat enables the verification of weak equivalence within thesmodelssystem using the same search engine as for the search of models. Our experiments withlpeqandsmodelssuggest that establishing the equivalence of logic programs in this way is in certain cases much faster than naive cross-checking of answer sets.