An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

TL;DR

This paper proves a refined version of the Homomorphism Preservation Theorem for finite structures by linking it to circuit complexity lower bounds, specifically on the AC^0 formula size of the colored subgraph isomorphism problem.

Contribution

It provides a new proof connecting model theory and circuit complexity, improving the quantifier-rank bounds from non-elementary to polynomial in the original rank.

Findings

Homomorphism Preservation Theorem holds for finite structures with improved bounds.

Quantifier-rank of equivalent existential-positive sentences is polynomially related to original.

Connects finite model theory with circuit complexity lower bounds.

Abstract

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 $k$.