Model Theory of Monadic Predicate Logic with the Infinity Quantifier
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This paper establishes model-theoretic properties of FOE^∞, a variation of monadic first-order logic that features the generalised quantifier ∃^∞ (`there are infinitely many'). We provide syntactically defined fragments of FOE^∞ characterising four different semantic properties of FOE^∞-sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence φ to a sentence φ^p belonging to the corresponding syntactic fragment, with the property that φ is equivalent to φ^p precisely when it has the associated semantic property. Our methodology is first to provide these results in the simpler setting of monadic first-order logic with (FOE) and without (FO) equality, and then move to FOE^∞ by including the generalised quantifier ∃^∞ into the picture. As a corollary of our developments, we obtain that the four semantic properties above are decidable for FOE^∞-sentences. Moreover, our results are directly relevant to the characterisation of automata and expressiveness modulo bisimilirity for variants of monadic second-order logic. This application is developed in a companion paper.
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