Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Kord Eickmeyer; Michael Elberfeld; Frederik Harwath

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arXiv · 1603.09055

Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Kord Eickmeyer·Michael Elberfeld·Frederik Harwath

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Abstract

We study the expressive power and succinctness of order-invariant sentences of first-order (FO) and monadic second-order (MSO) logic on structures of bounded tree-depth. Order- invariance is undecidable in general and, thus, one strives for logics with a decidable syntax that have the same expressive power as order-invariant sentences. We show that on structures of bounded tree-depth, order-invariant FO has the same expressive power as FO. Our proof technique allows for a fine-grained analysis of the succinctness of this translation. 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. We obtain similar results for MSO. It is known that the expressive power of MSO and FO coincide on structures of bounded tree-depth. We provide a translation from MSO to FO and we show that this translation is essentially optimal regarding the formula size. As a further result, we show that order-invariant MSO has the same expressive power as FO with modulo-counting quantifiers on bounded tree-depth structures.

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Kord Eickmeyer, Michael Elberfeld, Frederik Harwath. 2016-03-30. Succinctness of Order-Invariant Logics on Depth-Bounded Structures. https://arxiv.org/abs/1603.09055

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