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Finite Model Theory and Proof Complexity Revisited: Distinguishing Graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus

Author Benedikt Pago

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Benedikt Pago
  • Mathematical Foundations of Computer Science, RWTH Aachen University, Germany

Acknowledgements

I would like to thank Daniel Wiebking for extensively answering my numerous questions on Deep Weisfeiler Leman and the properties of coherent configurations.

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Benedikt Pago. Finite Model Theory and Proof Complexity Revisited: Distinguishing Graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CSL.2023.31

Abstract

This paper extends prior work on the connections between logics from finite model theory and propositional/algebraic proof systems. We show that if all non-isomorphic graphs in a given graph class can be distinguished in the logic Choiceless Polynomial Time with counting (CPT), then they can also be distinguished in the bounded-degree extended polynomial calculus (EPC), and the refutations have roughly the same size as the resource consumption of the CPT-sentence. This allows to transfer lower bounds for EPC to CPT and thus constitutes a new potential approach towards better understanding the limits of CPT. A super-polynomial EPC lower bound for a Ptime-instance of the graph isomorphism problem would separate CPT from Ptime and thus solve a major open question in finite model theory. Further, using our result, we provide a model theoretic proof for the separation of bounded-degree polynomial calculus and bounded-degree extended polynomial calculus.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • finite model theory
  • proof complexity
  • graph isomorphism

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