An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity [article]

Benjamin Rossman
2016 arXiv   pre-print
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 Φ of quantifier-rank k is preserved under homomorphisms on finite
more » ... s, 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 [39], where the upper bound on the quantifier-rank of Ψ is a non-elementary function of k.
arXiv:1612.08192v1 fatcat:y23tkqrz6bhwxmslgtmswmew2a