A Bounding Quantifier

The logic MSOL+\(\mathbb{B}\) is defined, by extending monadic second-order logic on the infinite binary tree with a new bounding quantifier\(\mathbb{B}\). In this logic, a formula \(\mathbb{B}\)X. φ(X) states that there is a finite bound on the size of sets satisfying φ(X). Satisfiability is proved decidable for two fragments of MSOL+\(\mathbb{B}\): formulas of the form \(\neg\mathbb{B}\)X.φ(X), with φ a \(\mathbb{B}\)-free formula; and formulas built from \(\mathbb{B}\)-free formulas by nesting \(\mathbb{B}\), ∃, ∨ and ∧.