The Complexity of One-Agent Refinement Modal Logic

Abstract

AbstractWe 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).KeywordsTree StructureModal LogicTuring MachineConstraint SystemAtomic PropositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.