Computation of the semantics of autoepistemic belief theories

Abstract

Recently, one of the authors introduced a simple and yet powerful non-monotonic knowledge representation framework, called the Autoepistemic Logic of Beliefs, AEB. Theories in AEB are called autoepistemic belief theories. Every belief theory T has been shown to have the least static expansionT which is computed by iterating a natural monotonic belief closure operator ΨT starting from T. This way, the least static expansion T of any belieftheory provides its natural non-monotonic semantics which is called the static semantics.It is easy to see that if a belief theory T is finite then the construction of its least static expansion T stops after countably many iterations. However, a somewhat surprising result obtained in this paper shows that the least static expansion of any finite belief theory T is in fact obtained by means of a single iteration of the belief closure operator ΨT (although this requires T to be of a special form, we also show that T can be always put in this form). This result eliminates the need for multiple iterations in the computation of static semantics and allows us to replace the fixed-point definition of static semantics by the equivalent explicit and straightforward definition given by T=ΨT(T).The second, closely related result establishes an intriguing relationship between static semantics T and Clark's completions comp(T) of finite belief theories. Here we use a slightly generalized version of comp(T) (see Definition 3.2). It shows that the static semantics T of T is obtained by augmenting T with the set Bcomp(T)={BF:F∈comp(T)} thus ensuring that all formulae that belong to Clark's completion comp(T) of T are believed to be true.Both results open the way for a more efficient implementation of static semantics: the first, because only one iteration is needed, and the second because reasoning in a non-standard logic (belief theories under static semantics) can be reduced to classical theorem proving involving Clark's completion.