Maximal Decidable Fragments of Halpern and Shoham’s Modal Logic of Intervals

Abstract

In this paper, we focus our attention on the fragment of Halpern and Shoham’s modal logic of intervals (HS) that features four modal operators corresponding to the relations “meets”, “met by”, “begun by”, and “begins” of Allen’s interval algebra ($A\bar{A}B\bar{B}$ logic). $A\bar{A}B\bar{B}$ properly extends interesting interval temporal logics recently investigated in the literature, such as the logic $B\bar{B}$ of Allen’s “begun by/begins” relations and propositional neighborhood logic $A\bar{A}$, in its many variants (including metric ones). We prove that the satisfiability problem for $A\bar{A}B\bar{B}$, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, $A\bar{A}B\bar{B}$ turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when $A\bar{A}B\bar{B}$ is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows ℕ, ℤ, ℚ, and ℝ.