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An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

Author Benjamin Rossman

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Benjamin Rossman

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Benjamin Rossman. An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 27:1-27:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


Previous work of the author [Rossmann'08] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann'08], where the upper bound on quantifier-rank is a non-elementary function of k.
  • circuit complexity
  • finite model theory


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