Preservation and decomposition theorems for bounded degree structures

Abstract

We provide elementary algorithms for two preservation theorems for first-order sentences with modulo m counting quantifiers (FO+MODm) on the class Cd of all finite structures of degree at most d: For each FO+MODm-sentence that is preserved under extensions (homomorphisms) on Cd, a Cd-equivalent existential (existential-positive) FO-sentence can be constructed in 6-fold (4-fold) exponential time. For FO-sentences, the algorithm has 5-fold (4-fold) exponential time complexity. This is complemented by lower bounds showing that for FO-sentences a 3-fold exponential blow-up of the computed existential (existential-positive) sentence is unavoidable. Furthermore, we show that for an input FO-formula, a Cd-equivalent Feferman-Vaught decomposition can be computed in 3-fold exponential time. We also provide a matching lower bound.