Undecidable Propositional Bimodal Logics and One-Variable First-Order Linear Temporal Logics with Counting

Abstract

First-order temporal logics are notorious for their bad computational behavior. It is known that even the two-variable monadic fragment is highly undecidable over various linear timelines, and over branching time even one-variable fragments might be undecidable. However, there have been several attempts at finding well-behaved fragments of first-order temporal logics and related temporal description logics, mostly either by restricting the available quantifier patterns or by considering sub-Boolean languages. Here we analyze seemingly “mild” extensions of decidable one-variable fragments with counting capabilities, interpreted in models with constant, decreasing, and expanding first-order domains. We show that over most classes of linear orders, these logics are (sometimes highly) undecidable, even without constant and function symbols, and with the sole temporal operator “eventually.” We establish connections with bimodal logics over 2D product structures having linear and “difference” (inequality) component relations and prove our results in this bimodal setting. We show a general result saying that satisfiability over many classes of bimodal models with commuting “unbounded” linear and difference relations is undecidable. As a byproduct, we also obtain new examples of finitely axiomatizable but Kripke incomplete bimodal logics. Our results generalize similar lower bounds on bimodal logics over products of two linear relations, and our proof methods are quite different from the known proofs of these results. Unlike previous proofs that first “diagonally encode” an infinite grid and then use reductions of tiling or Turing machine problems, here we make direct use of the grid-like structure of product frames and obtain lower-complexity bounds by reductions of counter (Minsky) machine problems. Representing counter machine runs apparently requires less control over neighboring grid points than tilings or Turing machine runs, and so this technique is possibly more versatile, even if one component of the underlying product structures is “close to” being the universal relation.