An improved homomorphism preservation theorem from lower bounds in circuit complexity

Previous work of the author [Rossman 2008a] 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 AC 0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence Φ of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence Ψ of quantifier-rank k O (1) . Quantitatively, this improves the result of [Rossman 2008a], where the upper bound on the quantifier-rank of Ψ is a non-elementary function of k.