AMSNP: a tame fragment of existential second-order logic
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Amalgamation monotone SNP (AMSNP) is a fragment of existential second-order logic that strictly contains the logics (connected) MMSNP of Feder and Vardi and guarded monotone SNP of Bienvenu, ten Cate, Lutz, and Wolter; it is a promising candidate for an expressive subclass of NP that exhibits a complexity dichotomy. We show that AMSNP has a complexity dichotomy if and only if Constraint Satisfaction Problems for reducts of finitely bounded homogeneous structures have a complexity dichotomy. For such CSPs, powerful universal-algebraic hardness conditions are known that are conjectured to describe the border between NP-hard and polynomial-time tractable CSPs. The connection to CSPs also implies that every AMSNP sentence can be evaluated in polynomial time on classes of finite structures of bounded treewidth. We show that the syntax of AMSNP is decidable. The proof relies on the following fact, which we believe is of independent interest in model theory: for classes of finite structures given by finitely many forbidden substructures, the amalgamation property is decidable.
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