Answer Sets in a Fuzzy Equilibrium Logic

Lecture Notes in Computer Science, 2009

A number of generalizations of answer set programming have been proposed in the literature to deal with vagueness, uncertainty, and partial rule satisfaction. We introduce a unifying framework that entails most of the existing approaches to fuzzy answer set programming. In this framework, rule bodies are defined using arbitrary fuzzy connectives with monotone partial mappings. As an approximation of full answer sets, k-answer sets are introduced to deal with conflicting information, yielding a flexible framework that encompasses, among others, existing work on valued constraint satisfaction and answer set optimization.

Fuzzy Sets and Systems

Autoepistemic logic is an important formalism for nonmonotonic reasoning. It extends propositional logic by offering the ability to reason about an agent's (lack of) beliefs. Moreover, it is well known to generalize the stable model semantics of answer set programming. Fuzzy logics on the other hand are multi-valued logics, which allow to model the intensity to which properties are satisfied. We combine these ideas to a fuzzy autoepistemic logic which can be used to reason about one's beliefs in the degrees to which properties are satisfied. We show that many properties from classical autoepistemic logic, e.g. the equivalence between autoepistemic models and stable expansions, remain valid under this generalization. In this paper, we consider a version of fuzzy answer set programming and show that its answer sets can be equivalently described as models in fuzzy autoepistemic logic. We also define a fuzzy logic of minimal belief and negation-as-failure and use this as a tool ...

Theory and Practice of Logic Programming, 2013

Fuzzy answer set programming (FASP) is a recent formalism for knowledge representation that enriches the declarativity of answer set programming by allowing propositions to be graded. To now, no implementations of FASP solvers are available and all current proposals are based on compilations of logic programs into different paradigms, like mixed integer programs or bilevel programs. These approaches introduce many auxiliary variables which might affect the performance of a solver negatively. To limit this downside, operators for approximating fuzzy answer sets can be introduced: Given a FASP program, these operators compute lower and upper bounds for all atoms in the program such that all answer sets are between these bounds. This paper analyzes several operators of this kind which are based on linear programming, fuzzy unfounded sets and source pointers. Furthermore, the paper reports on a prototypical implementation, also describing strategies for avoiding computations of these operators when they are guaranteed to not improve current bounds. The operators and their implementation can be used to obtain more constrained mixed integer or bilevel programs, or even for providing a basis for implementing a native FASP solver. Interestingly, the semantics of relevant classes of programs with unique answer sets, like positive programs and programs with stratified negation, can be already computed by the prototype without the need for an external tool.

International Journal of Approximate Reasoning

Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules – a syntactic generalization which does not affect the expressivity of ASP in the classical case – the picture changes drast...

2012

Possibilistic answer set programming (PASP) extends answer set programming (ASP) by attaching to each rule a degree of certainty. While such an extension is important from an application point of view, existing semantics are not well-motivated, and do not always yield intuitive results. To develop a more suitable semantics, we first introduce a characterization of answer sets of classical ASP programs in terms of possibilistic logic where an ASP program specifies a set of constraints on possibility distributions. This characterization is then naturally generalized to define answer sets of PASP programs. We furthermore provide a syntactic counterpart, leading to a possibilistic generalization of the well-known Gelfond-Lifschitz reduct, and we show how our framework can readily be implemented using standard ASP solvers.

2004

Abstract. The study of strong equivalence between logic programs or nonmonotonic theories under answer set semantics, begun in [18], is extended to the case where the programs or theories concerned are formulated in different languages. We suggest that theories in different languages be considered equivalent in the strong sense or synonymous if and only if each is bijectively interpretable (hence translatable) into the other.

Fuzzy answer set programming (FASP) is a generaliza-tion of answer set programming (ASP) in which propositions are al-lowed to be graded. Little is known about the computational com-plexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we first present an overview of previous results on the computational complexity of FASP under Łukasiewicz semantics, after which we show NP-completeness for normal and disjunctive FASP programs. Moreover, for this type of FASP programs we will show a reduction to bilevel linear programming, thus opening the door to practical applications.

Annals of Mathematics and Artificial Intelligence

In this paper, we show how the formalism of Logic Programs with Ordered Disjunction (LPODs) and Possibilistic Answer Set Programming (PASP) can be merged into the single framework of Logic Programs with Possibilistic Ordered Disjunction (LPPODs). The LPPODs framework embeds in a unified way several aspects of common-sense reasoning, nonmonotonocity, preferences, and uncertainty, where each part is underpinned by a well established formalism. On one hand, from LPODs it inherits the distinctive feature of expressing context-dependent qualitative preferences among different alternatives (modeled as the atoms of a logic program). On the other hand, PASP allows for qualitative certainty statements about the rules themselves (modeled as necessity values according to possibilistic logic) to be captured. In this way, the LPPODs framework supports a reasoning which is nonmonotonic, preference- and uncertainty-aware. The LPPODs syntax allows for the specification of (1) preferences among the ...

2011

Abstract Interpolation is an important property of classical and many non-classical logics that has been shown to have interesting applications in computer science and AI. Here we study the Interpolation Property for the the non-monotonic system of equilibrium logic, establishing weaker or stronger forms of interpolation depending on the precise interpretation of the inference relation. These results also yield a form of interpolation for ground logic programs under the answer sets semantics.

Fuzzy answer set programming (FASP) has recently been proposed as a generalization of answer set programming in which propo-sitions are allowed to be graded. Little is known about its computational complexity. In this paper we present some results and reveal a connection to an open problem about integer equations, suggesting that character-izing the complexity of FASP may not be straightforward.