Computer-aided proof of Erdős discrepancy properties

In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (xn) there exists a subsequence xd,x2d,x3d,…,xkd, for some positive integers k and d, such that | xd + x2d + ... + xid | > C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C=1 a human proof of the conjecture exists; for C=2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solvers, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C=2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. In the similar way, we obtain a precise bound of 127 645 on the maximal lengths of both multiplicative and completely multiplicative sequences of discrepancy 3. We also demonstrate that unrestricted discrepancy 3 sequences can be longer than 130 000

Computer-aided proof of Erdős discrepancy properties

In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (xn) there exists a subsequence xd, x2d, x3d, ..., xkd, for some positive integers k and d, such that |∑ki=1xi·d| >C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C = 1 a human proof of the conjecture exists; for C = 2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solvers, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C = 2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. In the similar way, we obtain a precise bound of 127 645 on the maximal lengths of both multiplicative and completely multiplicative sequences of discrepancy 3. We also demonstrate that unrestricted discrepancy 3 sequences can be longer than 130 000

Computer-aided proof of Erdős discrepancy properties

In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (xn) there exists a subsequence xd,x2d,x3d,…,xkd, for some positive integers k and d, such that | xd + x2d + ... + xid | > C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C=1 a human proof of the conjecture exists; for C=2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solvers, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C=2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. In the similar way, we obtain a precise bound of 127 645 on the maximal lengths of both multiplicative and completely multiplicative sequences of discrepancy 3. We also demonstrate that unrestricted discrepancy 3 sequences can be longer than 130 000

Verified Propagation Redundancy and Compositional UNSAT Checking in CakeML

Modern SAT solvers can emit independently-checkable proof certificates to validate their results. The state-of-the-art proof system that allows for compact proof certificates is propagation redundancy (PR). However, the only existing method to validate proofs in this system with a formally verified tool requires a transformation to a weaker proof system, which can result in a significant blowup in the size of the proof and increased proof validation time. This article describes the first approach to formally verify PR proofs on a succinct representation. We present (i) a new Linear PR (LPR) proof format, (ii) an extension of the DPR-trim tool to efficiently convert PR proofs into LPR format, and (iii) cake_lpr, a verified LPR proof checker developed in CakeML. We also enhance these tools with (iv) a new compositional proof format designed to enable separate (parallel) proof checking. The LPR format is backwards compatible with the existing LRAT format, but extends LRAT with support for the addition of PR clauses. Moreover, cake_lpr is verified using CakeML ’s binary code extraction toolchain, which yields correctness guarantees for its machine code (binary) implementation. This further distinguishes our clausal proof checker from existing checkers because unverified extraction and compilation tools are removed from its trusted computing base. We experimentally show that: LPR provides efficiency gains over existing proof formats; cake_lpr ’s strong correctness guarantees are obtained without significant sacrifice in its performance; and the compositional proof format enables scalable parallel proof checking for large proofs

Nonexistence Certificates for Ovals in a Projective Plane of Order Ten

In 1983, a computer search was performed for ovals in a projective plane of order ten. The search was exhaustive and negative, implying that such ovals do not exist. However, no nonexistence certificates were produced by this search, and to the best of our knowledge the search has never been independently verified. In this paper, we rerun the search for ovals in a projective plane of order ten and produce a collection of nonexistence certificates that, when taken together, imply that such ovals do not exist. Our search program uses the cube-and-conquer paradigm from the field of satisfiability (SAT) checking, coupled with a programmatic SAT solver and the nauty symbolic computation library for removing symmetries from the search.Comment: Appears in the Proceedings of the 31st International Workshop on Combinatorial Algorithms (IWOCA 2020

The Resolution of Keller's Conjecture

We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Keller's conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $\mathbb{R}^7$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $\mathbb{R}^8$ exists (which we also verify), this completely resolves Keller's conjecture.Comment: 25 pages, 9 figures, 3 tables; IJCAR 202

cake_lpr: Verified Propagation Redundancy Checking in CakeML

Modern SAT solvers can emit independently checkable proof certificates to validate their results. The state-of-the-art proof system that allows for compact proof certificates is propagation redundancy (PR). However, the only existing method to validate proofs in this system with a formally verified tool requires a transformation to a weaker proof system, which can result in a significant blowup in the size of the proof and increased proof validation time. This paper describes the first approach to formally verify PR proofs on a succinct representation; we present (i) a new Linear PR (LPR) proof format, (ii) a tool to efficiently convert PR proofs into LPR format, and (iii) cake_lpr, a verified LPR proof checker developed in CakeML. The LPR format is backwards compatible with the existing LRAT format, but extends the latter with support for the addition of PR clauses. Moreover, cake_lpr is verified using CakeML’s binary code extraction toolchain, which yields correctness guarantees for its machine code (binary) implementation. This further distinguishes our clausal proof checker from existing ones because unverified extraction and compilation tools are removed from its trusted computing base. We experimentally show that LPR provides efficiency gains over existing proof formats and that the strong correctness guarantees are obtained without significant sacrifice in the performance of the verified executable

Equation-regular sets and the Fox–Kleitman conjecture

Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2k-variable linear Diophantine equation ∑k i=1 (xi − yi) = b is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of the corresponding equation for all b ≥ 1. In particular, this independently confirms the conjecture for k = 3. We also briefly discuss the case k = 4