The Complexity of One-Agent Refinement Modal Logic

Laura Bozzelli, Hans van Ditmarsch, Sophie Pinchinat

Abstract: We investigate the complexity of satisfiability for one-agent Refinement Modal Logic (\text{\sffamily RML}), a known extension of basic modal logic (\text{\sffamily ML}) obtained by adding refinement quantifiers on structures. It is known that \text{\sffamily RML} has the same expressiveness as \text{\sffamily ML}, but the translation of \text{\sffamily RML} into \text{\sffamily ML} is of non-elementary complexity, and \text{\sffamily RML} is at least doubly exponentially more succinct than \text{\sffamily ML}. In this paper, we show that \text{\sffamily RML}-satisfiability is 'only' singly exponentially harder than \text{\sffamily ML}-satisfiability, the latter being a well-known PSPACE-complete problem. More precisely, we establish that \text{\sffamily RML}-satisfiability is complete for the complexity class AEXP_{\text{\sffamily pol}}, i.e., the class of problems solvable by alternating Turing machines running in single exponential time but only with a polynomial number of alternations (note that NEXPTIME⊆ AEXP_{\text{\sffamily pol}}⊆ EXPSPACE).