Trans-epistemic semantics for logic programs

In terms of the arithmetic hierarchy, the complexity of the set of minimal models and of the set of stable models of a propositional general logic program has previously been described. However, not every set of interpretations of this level of complexity is obtained as such a set. In this paper we identify the sets of interpretations which are minimal or stable model classes by their properties in an appropriate topology on the space of interpretations. Closely connected with the topological characterization, in parallel with results previously known for stable model classes we obtain for minimal model classes both a normal-form representation as the set of minimal models of a prerequisite-free program and a logical description in terms of formulas. Our approach centers on the relation which we establish between stable and minimal model classes. We include examples of calculations which can be performed by these methods. Let P be a propositional general logic program over a finite or countably infinite set U of atoms. Recall that an interpretation for P is a subset of U. By a model of P we mean an interpretation S c_ U which is closed under Te; by a minimal model of P we mean a model S of P such that no proper subset S' ~ S is a model of P; and by a stable model of P we mean a set S _C U such that S = least fixpoint of T~Ls(t,), where GLs(P) is the Gelfond-Lifschit z transform of program P with respect to S defined in [3]. Let Mod(P), Min(P), and Stab(P) denote, respectively, the set of models, minimal models, and stable models of P. For a given program P, we see that these classes are related by

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.

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.

M. Bezem defined an extensional semantics for positive higher-order logic programs. Recently, it was demonstrated by Rondogiannis and Symeonidou that Bezem's technique can be extended to higher-order logic programs with negation, retaining its extensional properties, provided that it is interpreted under a logic with an infinite number of truth values. Rondogiannis and Symeonidou also demonstrated that Bezem's technique, when extended under the stable model semantics, does not in general lead to extensional stable models. In this paper, we consider the problem of extending Bezem's technique under the well-founded semantics. We demonstrate that the well-founded extensionfailsto retain extensionality in the general case. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting cannot distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.

Over recent years, various semantics have been proposed for dealing with updates in the setting of logic programs. The availability of different semantics naturally raises the question of which are most adequate to model updates. A systematic approach to face this question is to identify general principles against which such semantics could be evaluated. In this paper we motivate and introduce a new such principle – the refined extension principle – which is complied with by the stable model semantics for (single) logic programs. It turns out that none of the existing semantics for logic program updates, even though based on stable models, complies with this principle. For this reason, we define a refinement of the dynamic stable model semantics for Dynamic Logic Programs that complies with the principle.

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].

We propose C4, a four-valued semantics for normal, logic programs. Using this semantics, we define two types of entailment: strong and weak. We show that a normal, logic program strongly entails a sentence under C4 if, and only if, the program entails that sentence under the well founded semantics and it weakly entails a sentence if, and only if, the program entails that sentence under the two-valued stable model semantics in case the program has any stable models. We argue that this shows that the difference between the well founded semantics and the stable model semantics can be characterized in terms of their attitude to what we call contested information. We use this insight to propose a general theory of contested reasoning.

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.

A decision problem is called parameterized if its input is a pair of strings. One of these strings is referred to as a parameter . The following problem is an example of a parameterized decision problem with k serving as a parameter: given a propositional logic program P and a nonnegative integer k , decide whether P has a stable model of size no more than k . Parameterized problems that are NP-complete often become solvable in polynomial time if the parameter is fixed. The problem to decide whether a program P has a stable model of size no more than k , where k is fixed and not a part of input, can be solved in time O ( mn k ), where m is the size of P and n is the number of atoms in P . Thus, this problem is in the class P. However, algorithms with the running time given by a polynomial of order k are not satisfactory even for relatively small values of k .The key question then is whether significantly better algorithms (with the degree of the polynomial not dependent on k ) exist. To tackle it, we use the framework of fixed-parameter complexity. We establish the fixed-parameter complexity for several parameterized decision problems involving models, supported models, and stable models of logic programs. We also establish the fixed-parameter complexity for variants of these problems resulting from restricting attention to definite Horn programs and to purely negative programs. Most of the problems considered in the paper have high fixed-parameter complexity. Thus, it is unlikely that fixing bounds on models (supported models, stable models) will lead to fast algorithms to decide the existence of such models.

Translating Multi-Agent Autoepistemic Logic into Logic Program

Although negative conclusions are presented implicitly in Normal Hybrid Probabilistic Programs (NHPP) [26] through the closed world assumption, representing and reasoning with explicit negation is needed in NHPP to allow the ability to reason with incomplete knowledge. In this paper we extend the language of NHPP to explicitly encode classical negation in addition to non-monotonic negation. The semantics of the extended language is based on the answer set semantics of traditional logic programming [9]. We show that the proposed semantics is a natural extension to the answer set semantics of traditional logic programming [9]. In addition, the proposed semantics is reduced to stable probabilistic model semantics of NHPP [26]. The importance of that is computational methods developed for NHPP can be applied to the proposed language. Furthermore, we show that some commonsense probabilistic knowledge can be easily represented in the proposed language.KeywordsLogic ProgramLogic ProgrammingClassical NegationIncomplete KnowledgeProbability IntervalThese 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.

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.

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.

In his paper [12] Ross extends the well-founded semantics for normal logic programs [16] to disjunctive logic programs. His definition is top-down and it is closer to a procedural semantics than to the elegant fixpoint definition of the well-founded semantics for normal logic programs. In the present paper, we propose a declarative, bottom-up fixpoint definition of the well-founded semantics for disjunctive logic programs. Our construction of the greatest unfounded set of extended literals is similar with the construction of the greatest unfounded set for normal programs. As a consequence, the connection between the well-founded semantics for normal programs and the well-founded semantics for disjunctive programs is made clearer.KeywordsLogic ProgramDeductive DatabaseNormal ProgramDisjunctive ProgramExtended AtomThese 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.