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Abstract-This paper's main result presents a 3-fold exponential algorithm that transforms a first-order formula ϕ together with a number d into a formula in Gaifman normal form that is equivalent to ϕ on the class of structures of degree at most d. For structures of polynomial growth, we even get a 2-fold exponential algorithm.These results are complemented by matching lower bounds: We show that for structures of degree 2, a 2-fold exponential blow-up in the size of formulas cannot be avoided. And for structures of degree 3, a 3-fold exponential blow-up is unavoidable.As a result of independent interest we obtain a 1-fold exponential algorithm which transforms a given first-order sentence ϕ of a very restricted shape into a sentence in Gaifman normal form that is equivalent to ϕ on all structures.

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.

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.