Mycielski graphs are a family of trianglefree graphs 𝑀_𝑘 with arbitrarily high chromatic number. 𝑀_𝑘 has chromatic number k and there is a short informal proof of this fact, yet finding proofs of it via automated reasoning techniques has proved to be a challenging task. In this paper, we study the complexity of clausal proofs of the uncolorability of 𝑀_𝑘 with 𝑘−1 colors. In particular, we consider variants of the PR (propagation redundancy) proof system that are without new variables, and with or without deletion. These proof systems are of interest due to their potential uses for proof search. As our main result, we present a sublinearlength and constantwidth PR proof without new variables or deletion. We also implement a proof generator and verify the correctness of our proof. Furthermore, we consider formulas extended with clauses from the proof until a short resolution proof exists, and investigate the performance of CDCL in finding the short proof. This turns out to be difficult for CDCL with the standard heuristics. Finally, we describe an approach inspired by SAT sweeping to find proofs of these extended formulas.
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Strong ExtensionFree Proof Systems
We introduce proof systems for propositional logic that admit short proofs of hard formulas as well as the succinct expression of most techniques used by modern SAT solvers. Our proof systems allow the derivation of clauses that are not necessarily implied, but which are redundant in the sense that their addition preserves satisfiability. To guarantee that these added clauses are redundant, we consider various efficiently decidable redundancy criteria which we obtain by first characterizing clause redundancy in terms of a semantic implication relationship and then restricting this relationship so that it becomes decidable in polynomial time. As the restricted implication relation is based on unit propagationa core technique of SAT solversit allows efficient proof checking too. The resulting proof systems are surprisingly strong, even without the introduction of new variablesa key feature of short proofs presented in the proofcomplexity literature. We demonstrate the strength of our proof systems on the famous pigeon hole formulas by providing short clausal proofs without new variables.
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 Award ID(s):
 1526760
 NSFPAR ID:
 10120534
 Date Published:
 Journal Name:
 Journal of Automated Reasoning
 ISSN:
 01687433
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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