Towards Completeness via Proof Search in the Linear Time mu-Calculus

Amina Doumane (1) , David Baelde (2) , Lucca Hirschi (2) , Alexis Saurin (1, 3)

1 PPS - Preuves, Programmes et Systèmes
2 LSV - Laboratoire Spécification et Vérification [Cachan]
3 PI.R2 - Design, study and implementation of languages for proofs and programs

Modal µ-calculus is one of the central languages of logic and verification , whose study involves notoriously complex objects: au-tomata over infinite structures on the model-theoretical side; infinite proofs and proofs by (co)induction on the proof-theoretical side. Nevertheless, axiomatizations have been given for both linear and branching time µ-calculi, with quite involved completeness arguments. We come back to this central problem, considering it from a proof search viewpoint, and provide some new completeness arguments in the linear time µ-calculus. Our results only deal with restricted classes of formulas that closely correspond to (non-alternating) ω-automata but, compared to earlier proofs, our completeness arguments are direct and constructive. We first consider a natural circular proof system based on sequent calculus, and show that it is complete for inclusions of parity automata directly expressed as formulas, making use of Safra's construction directly in proof search. We then consider the corresponding finitary proof system, featuring (co)induction rules, and provide a partial translation result from circular to finitary proofs. This yields completeness of the finitary proof system for inclusions of sufficiently determin-istic parity automata, and finally for arbitrary Büchi automata.

Mots clés: circular proofs; proof theory; linear time mu calculus; parity automata; Buchi automata; Safra construction

Domaines: Logique en informatique [cs.LO]

Amina Doumane, David Baelde, Lucca Hirschi, Alexis Saurin. Towards Completeness via Proof Search in the Linear Time mu-Calculus: The case of Büchi inclusions. 2016. ⟨hal-01275289⟩