Abstract
AbstractWe study the expressive power and succinctness of order-invariant 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 order-invariant sentences. We show that on graphs of bounded tree-depth, order-invariant FO has the same expressive power as FO, and order-invariant MSO has the same expressive power as the extension of FO with modulo-counting quantifiers. Our proof techniques allow for a fine-grained 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.KeywordsExpressivitysuccinctnessfirst-order logicmonadic second-order logicorder-invariancetree-depth
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