Mycielski graphs are a family of triangle-free 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 sublinear-length and constant-width 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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Clausal Proofs of Mutilated Chessboards
Mutilated chessboard problems have been called a “tough nut to crack” for automated reasoning. They are, for instance, hard for resolution, resulting in exponential runtime of current SAT solvers. Although there exists a well-known short argument for solving mutilated chessboard problems, this argument is based on an abstraction that is challenging to discover by automated-reasoning techniques. In this paper, we present another short argument that is much easier to compute and that can be expressed within the recent (clausal) PR proof system for propositional logic. We construct short clausal proofs of mutilated chessboard problems using this new argument and validate them using a formally-verified proof checker.
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- Award ID(s):
- 1813993
- NSF-PAR ID:
- 10121126
- Date Published:
- Journal Name:
- NASA Formal Methods (NFM 2019)
- Volume:
- 11460
- Page Range / eLocation ID:
- 204-210
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-20652-9_13
