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arXiv · 1511.05888

Preservation and decomposition theorems for bounded degree structures

Frederik Harwath·Lucas Heimberg·Nicole Schweikardt
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Abstract

We provide elementary algorithms for two preservation theorems for first-order sentences (FO) on the class âd of all finite structures of degree at most d: For each FO-sentence that is preserved under extensions (homomorphisms) on âd, a âd-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 âd-equivalent Feferman-Vaught decomposition can be computed in 3-fold exponential time. We also provide a matching lower bound.

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Frederik Harwath, Lucas Heimberg, Nicole Schweikardt. 2015-12-27. Preservation and decomposition theorems for bounded degree structures. https://doi.org/10.2168/lmcs-11(4%3A17)2015

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