It is shown that order-invariance of two-variable first-logic is decidable in the finite. This is an immediate consequence of a decision procedure obtained for the finite satisfiability problem for existential second-order logic with two first-order variables (ESO 2 ) on structures with two linear orders and one induced successor. We also show that finite satisfiability is decidable on structures with two successors and one induced linear order. In both cases, so far only decidability for monadic ESO 2 has been known. In addition, the finite satisfiability problem for ESO 2 on structures with one linear order and its induced successor relation is shown to be decidable in non-deterministic exponential time.

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 study the expressive power and succinctness of orderinvariant sentences of first-order (FO) and monadic second-order (MSO) logic on graphs of bounded tree-depth. Order-invariance is undecidable in general and, therefore, in finite model theory, one strives for logics with a decidable syntax that have the same expressive power as orderinvariant sentences. We show that on graphs of bounded tree-depth, order-invariant FO has the same expressive power as FO, and orderinvariant MSO has the same expressive power as the extension of FO with modulo-counting quantifiers. Our proof techniques allow for a finegrained analysis of the succinctness of these translations. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. Our techniques can be adapted to obtain a similar quantitative variant of a known result that the expressive power of MSO and FO coincides on graphs of bounded tree-depth.

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.

Abstract. We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MOD p , for short) with arbitrary numerical predicates. We require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas.