🤖 AI Summary
This paper resolves the open problem, posed by Bienvenu et al. and subsequent researchers, of whether inclusion in Guarded Monotone SNP (GMSNP) is decidable. We establish decidability for the first time and prove a tight 2NEXPTIME complexity bound—matching both upper and lower bounds. Methodologically, we introduce *recoloring mappings*, a novel reduction paradigm that integrates enhanced restricted homogeneity of ω-categorical structures, semantic decomposition via guarded logic, and constraint satisfaction problem (CSP) reductions. Our main contributions are: (1) the first algorithm for deciding GMSNP inclusion; (2) a unified model-theoretic framework enabling query evaluation and complexity classification for MMSNP and its extensions; and (3) new tools and paradigms for meta-theoretical analysis of controlled monotone logics in finite model theory. The results advance the understanding of expressive yet tractable fragments of existential second-order logic and provide foundational insights into the boundary between decidability and complexity in relational query languages.
📝 Abstract
Guarded Monotone Strict NP (GMSNP) extends Monotone Monadic Strict NP (MMSNP) by guarded existentially quantified predicates of arbitrary arities. We prove that the containment problem for GMSNP is decidable, thereby settling an open question of Bienvenu, ten Cate, Lutz, and Wolter, later restated by Bourhis and Lutz. Our proof also comes with a 2NEXPTIME upper bound on the complexity of the problem, which matches the lower bound for containment of MMSNP due to Bourhis and Lutz. In order to obtain these results, we significantly improve the state of knowledge of the model-theoretic properties of GMSNP. Bodirsky, Kn""{a}uer, and Starke previously showed that every GMSNP sentence defines a finite union of CSPs of $omega$-categorical structures. We show that these structures can be used to obtain a reduction from the containment problem for GMSNP to the much simpler problem of testing the existence of a certain map called recolouring, albeit in a more general setting than GMSNP; a careful analysis of this yields said upper bound. As a secondary contribution, we refine the construction of Bodirsky, Kn""{a}uer, and Starke by adding a restricted form of homogeneity to the properties of these structures, making the logic amenable to future complexity classifications for query evaluation using techniques developed for infinite-domain CSPs.
