Completeness and Decidability of Converse PDL in the Constructive Type Theory of Coq

Christian Doczkal (1) , Joachim Bard (2)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 Saarland University [Saarbrücken]

The completeness proofs for Propositional Dynamic Logic (PDL) in the literature are non-constructive and usually presented in an informal manner. We obtain a formal and constructive completeness proof for Converse PDL by recasting a completeness proof by Kozen and Parikh into our constructive setting. We base our proof on a Pratt-style decision method for satisfiability constructing finite models for satisfiable formulas and pruning refutations for unsatisfiable formulas. Completeness of Segerberg's axiomatization of PDL is then obtained by translating pruning refutations to derivations in the Hilbert system. We first treat PDL without converse and then extend the proofs to Converse PDL. All results are formalized in Coq/Ssreflect.

Domaines: Logique en informatique [cs.LO]

Christian Doczkal, Joachim Bard. Completeness and Decidability of Converse PDL in the Constructive Type Theory of Coq. Certified Programs and Proofs, Jan 2018, Los Angeles, United States. ⟨hal-01646782⟩