Succinctness of Cosafety Fragments of LTL via Combinatorial Proof Systems

Luca Geatti, Alessio Mansutti, Angelo Montanari

Abstract: This paper focuses on succinctness results for fragments of Linear Temporal Logic with Past (\(\textsf{LTL}\)) devoid of binary temporal operators like until, and provides methods to establish them. We prove that there is a family of cosafety languages \((\mathcal {L}_n)_{n \ge 1}\) such that \(\mathcal {L}_n\) can be expressed with a pure future formula of size \(\mathcal {O}(n)\), but it requires formulae of size \(2^{\varOmega (n)}\) 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].