The Power of the Weak

Abstract

A landmark result in the study of logics for formal verification is Janin and Walukiewicz’s theorem, stating that the modal μ-calculus (μML) is equivalent modulo bisimilarity to standard monadic second-order logic (here abbreviated as SMSO) over the class of labelled transition systems (LTSs for short). Our work proves two results of the same kind, one for the alternation-free or noetherian fragment μ N ML of μML on the modal side and one for WMSO, weak monadic second-order logic, on the second-order side. In the setting of binary trees, with explicit functions accessing the left and right successor of a node, it was known that WMSO is equivalent to the appropriate version of alternation-free μ-calculus. Our analysis shows that the picture changes radically once we consider, as Janin and Walukiewicz did, the standard modal μ-calculus, interpreted over arbitrary LTSs. The first theorem that we prove is that, over LTSs, μ N ML is equivalent modulo bisimilarity to noetherian MSO (NMSO), a newly introduced variant of SMSO where second-order quantification ranges over “conversely well-founded” subsets only. Our second theorem starts from WMSO and proves it equivalent modulo bisimilarity to a fragment of μ N ML defined by a notion of continuity. Analogously to Janin and Walukiewicz’s result, our proofs are automata-theoretic in nature: As another contribution, we introduce classes of parity automata characterising the expressiveness of WMSO and NMSO (on tree models) and of μ C ML and μ N ML (for all transition systems).

Model theory of monadic predicate logic with the infinity quantifier

“…), so that by (10) we find that D × ω, U ) | τ B S (d, k) indeed. Finally, by monotonicity it directly follows from…”

“…It is not difficult to see that the above grammar produces the same formulas as the one in Definition 17. The latter presentation, however, is more convenient in the context of our companion paper [10], and in line with the definition of the fragments Con B (ME ∞ (A)) studied in the next section.…”

“…In our companion paper [10], among other things we provide a Janin-Walukiewicz type characterisation result for weak monadic second-order logic. Analogous to the case of full monadic second-order logic discussed previously, our proof crucially employs normal form results and characterisation theorems for ME ∞ , as listed in the Tables 1 and 2.…”

“…For instance, this is the topic of Lyndon's Theorem [23] which characterises the formulas that are preserved under a weaker notion of homomorphism as the ones that are positive in all predicates occurring in the formula. Our preference for the notion of invariance under quotients stems from the fact that the property of invariance under quotients plays a key role in characterising the bisimulation-invariant fragments of various monadic second-order logics, as is explained in our companion paper [10].…”

“…In our companion paper [10] on automata, we need versions of these results for the monotone and the continuous fragment. For this purpose we define some slight modifications of the translations (•) • and (•) • which restricts to positive and syntactically continuous sentences.…”