Forcing MSO on Infinite Words in Weak MSO
Résumé
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.
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