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 Lukasiewicz 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 drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully

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 proporties 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 to show that fuzzy autoepistemic logic generalizes fuzzy answer set programming.

In this chapter, we present a tutorial about fuzzy answer set programming (FASP); we give a gentle introduction to its basic ideas and definitions. FASP is a combination of answer set programming and fuzzy logics which has recently been proposed. From the answer set semantics, FASP inherits the declarative nonmonotonic reasoning capabilities, while fuzzy logic adds the power to model continuous problems. FASP can be tailored towards different applications since fuzzy logics gives a great flexibility, e.g. by the possibility to use different generalizations of the classical connectives. In this chapter, we consider a rather general form of FASP programs; the connectives can in principal be interpreted by arbitrary [0, 1] n → [0, 1]-mappings. Despite that very general connectives are allowed, the presented framework turns out to be an intuitive extension of answer set programming.

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Abstract. Autoepistemic logic is one of the principal formalisms for nonmonotonic reasoning. It extends propositional logic by offering the ability to reason about an agent's (lack of) knowledge or 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 with which a property is satisfied. We combine these ideas to a fuzzy autoepistemic logic which can be used to reason about one's knowledge about the degrees to which proporties are satisfied. In this paper we show that many properties from classical autoepistemic logic remain valid under this generalization and that the important relation between autoepistemic logic and answer set programming is preserved in the sense that fuzzy autoepistemic logic generalizes fuzzy answer set programming.