An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

Previous work of the author [39] 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 $\Phi$ of quantifier-rank $k$ is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence $\Psi$ of quantifier-rank $k^{O(1)}$. Quantitatively, this improves the result of [39], where the upper bound on the quantifier-rank of $\Psi$ is a non-elementary function of $k$.