From Non-punctuality to Non-adjacency: A Quest for Decidability of Timed Temporal Logics with Quantifiers

Abstract

Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are prominent real-time extensions of Linear Temporal Logic (LTL). In general, the satisfiability checking problem for these extensions is undecidable when both the future (Until, U) and the past (Since, S) modalities are used (denoted by MTL[U,S] and TPTL[U,S]). In a classical result, the satisfiability checking for Metric Interval Temporal Logic (MITL[U,S]), a non-punctual fragment of MTL[U,S], is shown to be decidable with EXPSPACE complete complexity. A straightforward adoption of non-punctuality does not recover decidability in the case of TPTL[U,S]. Hence, we propose a more refined notion called non-adjacency for TPTL[U,S] and focus on its 1-variable fragment, 1-TPTL[U,S]. We show that non-adjacent 1-TPTL[U,S] is strictly more expressive than MITL. As one of our main results, we show that the satisfiability checking problem for non-adjacent 1-TPTL[U,S] is decidable with EXPSPACE complete complexity. Our decidability proof relies on a novel technique of anchored interval word abstraction and its reduction to a non-adjacent version of the newly proposed logic called PnEMTL. We further propose an extension of MSO [<] (Monadic Second Order Logic of Orders) with Guarded Metric Quantifiers (GQMSO) and show that it characterizes the expressiveness of PnEMTL. That apart, we introduce the notion of non-adjacency in the context of GQMSO (NA-GQMSO), which is a syntactic generalization of logic Q2MLO due to Hirshfeld and Rabinovich and show the decidability of satisfiability checking for NA-GQMSO.