Preservation and decomposition theorems for bounded degree structures

Abstract

Abstract. We provide elementary algorithms for two preservation theorems for firstorder sentences (FO) on the class C d of all finite structures of degree at most d: For each FO-sentence that is preserved under extensions (homomorphisms) on C d , a C d -equivalent existential (existential-positive) FO-sentence can be constructed in 5-fold (4-fold) exponential time. This is complemented by lower bounds showing that a 3-fold exponential blow-up of the computed existential (existential-positive) sentence is unavoidable. Both algorithms can be extended (while maintaining the upper and lower bounds on their time complexity) to input first-order sentences with modulo m counting quantifiers (FO+MODm).Furthermore, we show that for an input FO-formula, a C d -equivalent Feferman-Vaught decomposition can be computed in 3-fold exponential time. We also provide a matching lower bound.

When Locality Meets Preservation

“…Maybe surprisingly, the study of this seemingly arbitrary notion of positive locality has a direct application in the study of preservation under extensions over classes of finite structures. In the case of finite structures, our notion of local elementary embeddings describes exactly disjoint unions, and might explain why they featured so prominently in the study of preservation under extensions (Atserias et al, 2006(Atserias et al, , 2008Rossman, 2008;Harwath et al, 2015).…”

“…For example, Lyndon's Positivity Theorem fails even over the class of finite words Kuperberg (2021). Nevertheless, there are a few known instances of classes of finite structures where some preservation theorems hold Atserias et al (2006); Rossman (2008Rossman ( , 2016; Harwath et al (2015), and this type of question is still actively investigated (e.g. Chen and Flum, 2021;Kuperberg, 2021;Dawar and Sankaran, 2021).…”

“…This can be thought of as FO[σ] being limited to describing the local behaviour of structures. In the study of preservation theorems such as the Loś-Tarski Theorem over finite structures, a first step is often to use Gaifman's normal form and rely on the structural properties of models as a substitute for compactness Atserias et al (2008Atserias et al ( , 2006; Harwath et al (2015); Dawar and Sankaran (2021).…”

When Locality Meets Preservation

“…For example, Lyndon's Positivity Theorem fails even over the class of finite words [13]. Nevertheless, there are a few known instances of classes of finite structures where some preservation theorems hold [2,12,20,21], and this type of question is still actively investigated [e.g. 4,7,13].…”

“…This can be thought of as FO[σ] being limited to describing the local behaviour of structures. In the study of preservation theorems such as the Łoś-Tarski Theorem over finite structures, a first step is often to use Gaifman's normal form and rely on the structural properties of models as a substitute for compactness [1,2,7,12].…”