Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures
We 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.
KeywordsExpressivity succinctness first-order logic monadic second-order logic order-invariance tree-depth
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- 5.Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic – A Language-Theoretic Approach. Cambridge University Press (2012)Google Scholar
- 6.Elberfeld, M., Grohe, M., Tantau, T.: Where first-order and monadic second-order logic coincide. In: Proc. LICS 2012, pp. 265–274. IEEE Computer Society (2012)Google Scholar
- 7.Gajarský, J., Hliněný, P.: Faster deciding MSO properties of trees of fixed height, and some consequences. In: Proc. FSTTCS 2012, pp. 112–123 (2012)Google Scholar
- 9.Libkin, L.: Elements of Finite Model Theory. Springer (2004)Google Scholar
- 10.Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms. Springer, Heidelberg (2012)Google Scholar