Forcing MSO on Infinite Words in Weak MSO

Colin Riba (1)

1 LIP - Laboratoire de l'Informatique du Parallélisme

We propose a forcing-based interpretation of monadic second-order logic (MSO) on omega-words in Weak MSO (WMSO). The interpretation is purely syntactic and compositional. We show that a formula with parameters is true in MSO if and only if its interpretation is true in WMSO. We also show that a closed formula is true in MSO if and only if its interpretation is provable under some axioms which hold for WMSO, but without axiomatizing it. We use model-theoretic arguments. Our approach is inspired from point-free topology: infinite words, seen a topological points, are approximated by filters of bounded segments. We devise forcing conditions such that the corresponding generic filters approximate Ramseyan factorizations of infinite words modulo satisfaction of formulas of a given quantifier depth. Our interpretation parallels some approaches to McNaughton's Theorem (equivalence between non-deterministic Buchi automata and deterministic Rabin automata) but the obtained formulas do not by themselves describe deterministic automata.

Mots clés: Monadic Second Order Logic; Forcing; Buchi automata; McNaughton Theorem; Feferman-Vaught Composition Theorem; Henkin Completeness

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Colin Riba. Forcing MSO on Infinite Words in Weak MSO. 2013. ⟨hal-00776921v1⟩