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.
@InProceedings{rossman:LIPIcs.ITCS.2017.27, author = {Rossman, Benjamin}, title = {{An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {27:1--27:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.27}, URN = {urn:nbn:de:0030-drops-81435}, doi = {10.4230/LIPIcs.ITCS.2017.27}, annote = {Keywords: circuit complexity, finite model theory} }
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