Hex Semantics Via Approximation Fixpoint Theory

Approximation Fixpoint Theory (AFT) is an algebraic framework for studying fixpoints of possibly nonmonotone lattice operators, and thus extends the fixpoint theory of Tarski and Knaster. In this paper, we uniformly define 2-, and 3-valued (ultimate) answer-set semantics, and well-founded semantics of disjunction-free HEX programs by applying AFT. In the case of disjunctive HEX programs, AFT is not directly applicable. However, we provide a definition of 2-valued (ultimate) answer-set semantics based on non-deterministic approximations and show that answer sets are minimal, supported, and derivable in terms of bottom-up computations. Finally, we extensively compare our semantics to closely related semantics, including constructive dl-program semantics. Since HEX programs are a generic formalism, our results are applicable to a wide range of formalisms.