Succinctness of Cosafety Fragments of LTL via Combinatorial Proof Systems

Abstract

AbstractThis paper focuses on succinctness results for fragments of Linear Temporal Logic with Past ($$\textsf{LTL}$$ LTL ) devoid of binary temporal operators like until, and provides methods to establish them. We prove that there is a family of cosafety languages $$(\mathcal {L}_n)_{n \ge 1}$$ ( L n ) n ≥ 1 such that $$\mathcal {L}_n$$ L n can be expressed with a pure future formula of size $$\mathcal {O}(n)$$ O ( n ) , but it requires formulae of size $$2^{\varOmega (n)}$$ 2 Ω ( n ) to be captured with past formulae. As a by-product, such a succinctness result shows the optimality of the pastification algorithm proposed in [Artale et al., KR, 2023].We show that, in the considered case, succinctness cannot be proven by relying on the classical automata-based method introduced in [Markey, Bull. EATCS, 2003]. In place of this method, we devise and apply a combinatorial proof system whose deduction trees represent $$\textsf{LTL}$$ LTL formulae. The system can be seen as a proof-centric (one-player) view on the games used by Adler and Immerman to study the succinctness of $$\textsf{CTL}$$ CTL .