Abstract
We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.
Highlights
One branch of model theory engages with the characterisation of semantical fragments, which typically are undecidable, as syntactical fragments of the logics under consideration
We provide a syntactical characterisation of all formulae of existential second-order logic obeying this property via a normal form called myopic-Σ11, a notion based on ideas of Galliani and Hella [GH13]
We construct myopic-Σ11 formulae that can define the winning regions of those inclusion-exclusion games that are closed under unions. Such games are eligible for any Σ11-formula, but since our interest lies in those formulae that are closed under unions, we introduce a restricted version of such games, called union games, that precisely correspond to the model-checking games of union closed Σ11-formulae
Summary
One branch of model theory engages with the characterisation of semantical fragments, which typically are undecidable, as syntactical fragments of the logics under consideration. It is known that both independence logic FO(⊥) and inclusion-exclusion logic FO(⊆, | ) have the same expressive power as full existential second-order logic Σ11 [Gal12] The team in such logics corresponds to the free relational variable in existential second-order formulae, enabling us to ask the same questions about fragments with certain closure properties in both frameworks. Galliani and Hella have shown that inclusion logic FO(⊆) corresponds to greatest fixed-point-logic GFP+ and, by using the Immerman-Vardi Theorem, it captures all Ptime computable queries on ordered structures [GH13] They proved that every union closed dependency notion that itself is first-order definable (where the formula has access to a predicate for the team) is already definable in inclusion logic.
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