A topological characterization of the stable and minimal model classes of propositional logic programs

We present many-valued disjunctive logic programs in which classical disjunctive logic program clauses are extended by a truth value that respects the material implication. Interestingly, these many-valued disjunctive logic programs have both a probabilistic semantics in probabilities over possible worlds and a truth-functional semantics. We then define minimal, perfect, and stable models and show that they have the same properties like their classical counterparts. In particular, perfect and stable models are always minimal models. Under local stratification, the perfect model semantics coincides with the stable model semantics. Finally, we show that some special cases of propositional many-valued disjunctive logic programming under minimal, perfect, and stable model semantics have the same complexity like their classical counterparts.

We show that stable models of logic programs may be viewed as minimal models of programs that satisfy certain additional constraints. To do so, we transform the normal programs into disjunctive logic programs and sets of integrity constraints. We show that the stable models of the normal program coincide with the minimal models of the disjunctive program thatsatisfy the integrity constraints. As a consequence, the stable model semantics can be characterized using theextended generalized closed world assumption for disjunctive logic programs. Using this result, we develop a bottomup algorithm for function-free logic programs to find all stable models of a normal program by computing the perfect models of a disjunctive stratified logic program and checking them for consistency with the integrity constraints. The integrity constraints provide a rationale as to why some normal logic programs have no stable models.

Each stable model of a logic program is computed in isolation. This does not allow one to reason in any stable model with information from other stable models. Such information interchange is needed when computing with full introspection, as performed by Gelfond's epistemic specifications, or when modeling multi-agent reasoning using stable models. In this paper, we define syntactic and semantic structures that allow the use of information from multiple stable models when computing one stable model. Hence a notion of second order stability is introduced and every computed model should be stable at that level. We define a concept of trans-epistemic (te-) logic programs that is reduced to a logic program using information from a trans-epistemic interpretation. The te-interpretation is checked for stability against the set of stable models of the logic program using a consensus function. We discus the properties of trans-epistemic stable models and motivate their use with examples.

Turner's logic of universal causation is a general logic for nonmonotonic reasoning. It has its origin in McCain and Turner's causal action theories which have been translated to propositional logic and logic programming with nested expressions. In this paper, we propose to do the same for Turner's logic, and show thatTurner's logic can actually be mapped to McCain and Turner's causal theories. These results can be used to construct a system for reasoning in Turner's logic.

The relationship between TMS and general logic programs is an important issue in non-monotonic logic programming. In this paper, we prove that, after we translate the TMS theory into a general logic program, the TMS's well-founded assignment (or extension) is equivalent to the corresponding general logic program's stable model. It means that TMS can be completely integrated into a non-monotonic logic programming environment.

Minimal Herbrand models of sets of first-order clauses are useful in several areas of computer science, for example, automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are inappropriate because they generate nonminimal Herbrand models and can be inefficient. This article describes an approach for generating the minimal Herbrand models of sets of first-order clauses. The approach builds upon i>positive unit hyperresolution (i>PUHR) i>tableaux, that are in general smaller than conventional tableaux. PUHR tableaux formalize the approach initially introduced with the theorem prover SATCHMO. Two minimal model generation procedures are described. The first one expands PUHR tableaux depth-first relying on a i>complement splitting expansion rule and on a form of backtracking involving constraints. A Prolog implementation, named MM-SATCHMO, of this procedure is given, and its performance on benchmark suites is reported. The second minimal model generation procedure performs a breadth-first, constrained expansion of PUHR (complement) tableaux. Both procedures are optimal in the sense that each minimal model is constructed only once, and the construction of nonminimal models is interrupted as soon as possible. They are complete in the following sense: The depth-first minimal model generation procedure computes all minimal Herbrand models of the considered clauses provided these models are all finite. The breadth-first minimal model generation procedure computes all finite minimal Herbrand models of the set of clauses under consideration. The proposed procedures are compared with related work in terms of both principles and performance on benchmark problems.

We study the relationship between argumentation (abduction) and disjunctive logic programming. Based on the paradigm of argumentation, an abductive semantic framework for disjunctive logic programming is presented, in which the disjunctions of negative literals are taken as possible assumptions rather than only negative literals as the case of non-disjunctive logic programming. In our framework, three semantics PDH, CDH and WFDH are defined by three kinds of acceptable hypotheses to represent credulous reasoning, moderate reasoning and skeptical reasoning in AI, respectively. On the other hand, our semantic framework could be established in a broader class than that of disjunctive programs (called bi-disjunctive logic programs) and, hence, the corresponding abductive framework is abbreviated as BDAS (Bi-Disjunctive Argumentation-theoretic Semantics). Besides its rich expressive power and nondeterminism, BDAS integrates and naturally extends many key semantics, such as the minimal models, EGCWA, the well-founded model, and the stable models. In particular, a novel and interesting argumentation-theoretic characterization of EGCWA is shown. Thus the framework in this paper does not only provides a new way of performing argumentation (abduction) in disjunctive logic programming, but also is a simple, intuitive and unifying semantic framework for disjunctive logic programming.KeywordsLogic ProgramLogic ProgrammingArgument FrameworkDisjunctive ProgramDisjunctive LogicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper studies the stable model semantics of logic programs with (abstract) constraint atoms and their properties. We introduce a succinct abstract representation of these constraint atoms in which a constraint atom is represented compactly. We show two applications. First, under this representation of constraint atoms, we generalize the Gelfond–Lifschitz transformation and apply it to define stable models (also called answer sets) for logic programs with arbitrary constraint atoms. The resulting semantics turns out to coincide with the one defined by Son et al. (2007), which is based on a fixpoint approach. One advantage of our approach is that it can be applied, in a natural way, to define stable models for disjunctive logic programs with constraint atoms, which may appear in the disjunctive head as well as in the body of a rule. As a result, our approach to the stable model semantics for logic programs with constraint atoms generalizes a number of previous approaches. Second, we show that our abstract representation of constraint atoms provides a means to characterize dependencies of atoms in a program with constraint atoms, so that some standard characterizations and properties relying on these dependencies in the past for logic programs with ordinary atoms can be extended to logic programs with constraint atoms.

In this paper, we focus on the problem of existence and computing of small and large stable models. We show that for every fixed integer k, there is a linear-time algorithm to decide the problem LSM (large stable models problem): does a logic program P have a stable model of size at least [mid ]P[mid ]−k? In contrast, we show that the problem SSM (small stable models problem) to decide whether a logic program P has a stable model of size at most k is much harder. We present two algorithms for this problem but their running time is given by polynomials of order depending on k. We show that the problem SSM is fixed-parameter intractable by demonstrating that it is W[2]-hard. This result implies that it is unlikely an algorithm exists to compute stable models of size at most k that would run in time O(mc), where m is the size of the program and c is a constant independent of k. We also provide an upper bound on the fixed-parameter complexity of the problem SSM by showing that it belongs to the class W[3].

In this paper, we focus on the problem of existence and computing of small and large stable models. We show that for every fixed integer k, there is a linear-time algorithm to decide the problem LSM (large stable models problem): does a logic program P have a stable model of size at least ∣P∣−k? In contrast, we show that the problem SSM (small stable models problem) to decide whether a logic program P has a stable model of size at most k is much harder. We present two algorithms for this problem but their running time is given by polynomials of order depending on k. We show that the problem SSM is fixed-parameter intractable by demonstrating that it is W[2]-hard. This result implies that it is unlikely an algorithm exists to compute stable models of size at most k that would run in time O(mc), where m is the size of the program and c is a constant independent of k. We also provide an upper bound on the fixed-parameter complexity of the problem SSM by showing that it belongs to the class W[3].

It is claimed that the notion of preference is a fundamental modality in computing and is a generalization of the notion of minimality. A logic of feasible preference is presented. The non-monotonic behavior of negation in logic programming is modeled as a symbolic optimization problem. As a case study, for the class of logic programs with one or more stable models, we give a preferential transformation of logic programs that identifies their stable models as the optimal worlds in the intended model of the corresponding preferential theory. Minimization and minimization orderings are given explicit syntactic representations and their due status in the model theory. Preference logics gives a very elegant model theory for defaults, without any mention of fixpoints. Further, nonmonotonic reasoning is carried out in a monotonic logic, since members of the optimal worlds are not identified with theorems of a preferential theory. Preference logics have great potential to bring the areas of Symbolic Computation, Knowledge Representation and Classical Optimization closer.

Logic programming: Laxness and saturation

The notion of minimal Herbrand models can be generalized to goal-generic models Α, which have the property that P ⊧ ∀(L1 ∧ … ∧ Ln) iff Α ⊧ ∀(L1 ∧ … ∧ Ln), where P is a program and ←L1, …, Ln is a goal. In the first part of this paper we characterize a large class of goal-generic models for definite Horn programs. These are minimal models which satisfy Clark's Equality Theory and which contain an infinite number of ‘unnamed’ elements. Predictably, the situation deteriorates when negation is allowed. There are simple stratified programs like Q = { p(0)←; q(0)←⌝p(x); r(0)→⌝q(0)} with no perfect model Α such that, for all normal goals ←L1, … , Ln, PERF(Q) ⊧ ∀(L1 ∧ … ∧ Ln) iff Α ⊧ ∀(L1 ∧ … ∧ Ln). However, there do exist two models A, B such that PERF(Q) ⊧ ∀(L1 ∧ … ∧ Ln) iff {A,B} ⊧ ∀(L1 ∧ … ∧ Ln, suggesting the conjecture that all stratified programs P have finite goal-generic models sets in PERF(P). In the second part of this paper we show that this conjecture is false. The proof depends on a characterization of model domain elements by way of Cauchy term sequences, a finitary treatment of infinite terms. A model A realizes a Cauchy term sequence T = ‘t1’, εN if there is some element in the domain of Α which is an instance of every term in T, and A omits T otherwise. Basic results from model theory show that the models of Clark's Equality Theory display a rich variety in their realization and omission of Cauchy term sequences. We give a program P which can determine if a model realizes any given computable Cauchy sequence, and show that P is a counterexample to the conjecture.

Translating Multi-Agent Autoepistemic Logic into Logic Program

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 programming, the concept of a stable model was extended to several new programming constructs, and for some of these extensions the minimality 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.