Fuzzy autoepistemic logic and its relation to fuzzy answer set programming

Abstract

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.

On the relationship between fuzzy autoepistemic logic and fuzzy modal logics of belief

“…As was done for (fuzzy) autoepistemic logic [1,9], we will sometimes consider a formula φ in L B as a B-free formula φ * by treating subformulas of the form Bψ as new propositional variables. For instance, B(a ∧ Bb) is then seen as a fresh variable p B(a∧Bb) and has in principle no connection to Bb.…”

“…It can be seen as the closed set of beliefs of an ideal rational agent reflecting on his own beliefs. In this section we first consider a many-valued generalization of the notion of stable expansions from [9]. Then we introduce a possible world semantics for the resulting many-valued stable expansions and we show their relation to the fuzzy modal logics studied in Section 3.…”

“…Hence, answer sets also correspond to valid sentences in logic of only knowing. Recently, a fuzzy generalization of autoepistemic logic has been defined in [9]. In the same paper, it has also been shown that the important relation between autoepistemic logic and answer set programming is preserved: the answer sets of a fuzzy answer set program (e.g.…”

“…, k−1 k , 1}. Indeed, this assumption ensures that two key properties are satisfied: Ł c k is complete for deductions from an arbitrary set of premises, [10], (b): [8], (c)-(d): [9], (e): [6]. and the Baaz-Monteiro projection operator is definable in Ł c k .…”

“…In Section 4, we deal with the complexity of these logics and prove NP-completeness for two variants of the satisfiability problem. Then, in Section 5, we consider possible world semantics for the fuzzy autoepistemic logic of [9] and provide a characterization of stable fuzzy expansions both in terms of many-valued K45 belief sets, and in terms of proper generalizations of stable sets. In Section 6, we generalize the propositional fragment of Levesque's "only knowing" logic, we provide a sound and complete axiomatization, and we characterize stable fuzzy expansions in terms of the belief sets involving the "only knowing" operator O.…”