Complexity and Succinctness Issues for Linear-Time Hybrid Logics

Abstract

Full linear-time hybrid logic (HL) is a non-elementary and equally expressive extension of standard LTL + past obtained by adding the well-known binder operators ↓ and ∃. We investigate complexity and succinctness issues for HL in terms of the number of variables and nesting depth of binder modalities. First, we present direct automata-theoretic decision procedures for satisfiability and model-checking of HL, which require space of exponential height equal to the nesting depth of binder modalities. The proposed algorithms are proved to be asymptotically optimal by providing matching lower bounds. Second, we show that for the one-variable fragment of HL, the considered problems are elementary and, precisely, Expspace-complete. Finally, we show that for all 0 ≤ h < k, there is a succinctness gap between the fragments HL ^k and HL ^h with binder nesting depth at most k and h, respectively, of exponential height equal to k − h.