Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

Abstract

Abstract. The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set N = {1, 2, . . . } of natural numbers be divided into two parts, such that no part contains a triple (a, b, c) with a 2 + b 2 = c 2 ? A prize for the solution was offered by Ronald Graham over two decades ago. We solve this problem, proving in fact the impossibility, by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers. An important role is played by dedicated look-ahead heuristics, which indeed allowed to solve the problem on a cluster with 800 cores in about 2 days. Due to the general interest in this mathematical problem, our result requires a formal proof. Exploiting recent progress in unsatisfiability proofs of SAT solvers, we produced and verified a proof in the DRAT format, which is almost 200 terabytes in size. From this we extracted and made available a compressed certificate of 68 gigabytes, that allows anyone to reconstruct the DRAT proof for checking.

Efficient Certified RAT Verification

“…Furthermore, we have shown that our Coq checker's ability to check entailment and thereby transformation proofs has allowed us to check the transformation proof from [6], the only SAT-related step in the original proof of the Boolean Pythagorean Triples problem that we were unable to verify in [4].…”

“…We also used the possibility of verifying entailments to check the transformation proof from [6], the only SAT-related step in the original proof of the Boolean Pythagorean Triples problem that we were unable to verify in [4]. The certified LRAT checker in Coq was able to verify this proof in 8 minutes and 25 seconds, including approx.…”

“…The ACL2 checker is able to check the validity of adding each of the 68,667 clauses in the transformation proof from [6] in less than 9 seconds. The certified checking of this LRAT proof is almost as fast as non-certified checking and conversion of the DRAT proof into the LRAT proof by DRAT-trim.…”

Solving Very Hard Problems: Cube-and-Conquer, a Hybrid SAT Solving Method

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AbstractA recent success of SAT solving has been the solution of the boolean Pythagorean Triples problem [Heule et al, 2016], delivering the largest proof yet, of 200 terabytes in size. We present this and the underlying paradigm Cube-and-Conquer, a powerful general method to solve big SAT problems, based on integrating the "old" and "new" methods of SAT solving.

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Unsatisfiability Proofs for Parallel SAT Solver Portfolios with Clause Sharing and Inprocessing

“…A proof in the DRAT format is a sequence of clauses, which have been learned or deleted during the run in a sequential SAT solver, and includes all known formula simplification techniques. Recently, the DRAT format received media attention because SAT solvers solved the Pythagorean Triples Problem and its 200 Terabytes proof was expressed in the format [22]. However, parallel SAT solvers such as Plingeling cannot express their proofs in the DRAT format, since the proofs constructed from the sequential incarnations cannot be merged into a single DRAT proof.…”