Curtis Bright

TCS Journal 2026 Journal Article

Computing the base-b representation of quadratic irrationals using automata

• Aaron Barnoff
• Curtis Bright
• Jeffrey Shallit

IJCAI Conference 2025 Conference Paper

Verified Certificates via SAT and Computer Algebra Systems for the Ramsey R(3, 8) and R(3, 9) Problems

• Zhengyu Li
• Conor Duggan
• Curtis Bright
• Vijay Ganesh

The Ramsey problem R(3, k) seeks to determine the smallest value of n such that any red/blue edge coloring of the complete graph on n vertices must either contain a blue triangle (3-clique) or a red clique of size k. Despite its significance, many previous computational results for the Ramsey R(3, k) problem such as R(3, 8) and R(3, 9) lack formal verification. To address this issue, we use the software MathCheck to generate certificates for Ramsey problems R(3, 8) and R(3, 9) (and symmetrically R(8, 3) and R(9, 3)) by integrating a Boolean satisfiability (SAT) solver with a computer algebra system (CAS). Our SAT+CAS approach significantly outperforms traditional SAT-only methods, demonstrating an improvement of several orders of magnitude in runtime. For instance, our SAT+CAS approach solves R(3, 8) (resp. , R(8, 3)) sequentially in 59 hours (resp. , in 11 hours), while a SAT-only approach using state-of-the-art CaDiCaL solver times out after 7 days. Additionally, in order to be able to scale to harder Ramsey problems R(3, 9) and R(9, 3) we further optimized our SAT+CAS tool using a parallelized cube-and-conquer approach. Our results provide the first independently verifiable certificates for these Ramsey numbers, ensuring both correctness and completeness of the exhaustive search process of our SAT+CAS tool.

AAAI Conference 2024 Short Paper

A SAT + Computer Algebra System Verification of the Ramsey Problem R(3, 8) (Student Abstract)

• Conor Duggan
• Zhengyu Li
• Curtis Bright
• Vijay Ganesh

The Ramsey problem R(3,8) asks for the smallest n such that every red/blue coloring of the complete graph on n vertices must contain either a blue triangle or a red 8-clique. We provide the first certifiable proof that R(3,8) = 28, automatically generated by a combination of Boolean satisfiability (SAT) solver and a computer algebra system (CAS). This SAT+CAS combination is significantly faster than a SAT-only approach. While the R(3,8) problem was first computationally solved by McKay and Min in 1992, it was not a verifiable proof. The SAT+CAS method that we use for our proof is very general and can be applied to a wide variety of combinatorial problems.

IJCAI Conference 2024 Conference Paper

A SAT Solver + Computer Algebra Attack on the Minimum Kochen–Specker Problem

• Zhengyu Li
• Curtis Bright
• Vijay Ganesh

One of the fundamental results in quantum foundations is the Kochen–Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i. e. , a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known, the problem of finding the minimum KS vector system in three dimensions (3D) has remained stubbornly open for over 55 years. To address the minimum KS problem, we present a new verifiable proof-producing method based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS) that uses an isomorph-free orderly generation technique that is very effective in pruning away large parts of the search space. Our method shows that a KS system in 3D must contain at least 24 vectors. We show that our sequential and parallel Cube-and-Conquer (CnC) SAT+CAS methods are significantly faster than SAT-only, CAS-only, and a prior CAS-based method of Uijlen and Westerbaan. Further, while our parallel pipeline is somewhat slower than the parallel CnC version of the recently introduced Satisfiability Modulo Theories (SMS) method, this is in part due to the overhead of proof generation. Finally, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a size of 40. 3 TiB in order 23.

AAAI Conference 2024 Short Paper

A SAT Solver and Computer Algebra Attack on the Minimum Kochen-Specker Problem (Student Abstract)

• Zhengyu Li
• Curtis Bright
• Vijay Ganesh

The problem of finding the minimum three-dimensional Kochen–Specker (KS) vector system, an important problem in quantum foundations, has remained open for over 55 years. We present a new method to address this problem based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS). Our approach improved the lower bound on the size of a KS system from 22 to 24. More importantly, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a proof size of 41.6 TiB for order 23. The efficiency is due to the powerful combination of SAT solvers and CAS-based orderly generation.

AAAI Conference 2022 Short Paper

Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares (Student Abstract)

• Noah Rubin
• Curtis Bright
• Brett Stevens
• Kevin Cheung

We use integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). We improve the performance of the solvers by formulating an extended symmetry breaking method and provide an alternative CP encoding which performs much better in practice. Using state-of-the-art solvers we are able to quickly find pairs of MOLS (or prove their nonexistence) in all orders up to and including eleven. We also analyze the effectiveness of using CP and IP solvers to search for triples of MOLS and estimate the running time of using this approach to resolve the longstanding open problem of determining the existence of a triple of MOLS of order ten.

AAAI Conference 2021 Conference Paper

A SAT-based Resolution of Lam’s Problem

• Curtis Bright
• Kevin K. H. Cheung
• Brett Stevens
• Ilias Kotsireas
• Vijay Ganesh

In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam’s problem from projective geometry—the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam’s problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches—highlighting the difficulty of relying on specialpurpose search code for nonexistence results.

IJCAI Conference 2020 Conference Paper

Unsatisfiability Proofs for Weight 16 Codewords in Lam's Problem

• Curtis Bright
• Kevin K. H. Cheung
• Brett Stevens
• Ilias Kotsireas
• Vijay Ganesh

In the 1970s and 1980s, searches performed by L. Carter, C. Lam, L. Thiel, and S. Swiercz showed that projective planes of order ten with weight 16 codewords do not exist. These searches required highly specialized and optimized computer programs and required about 2, 000 hours of computing time on mainframe and supermini computers. In 2010, these searches were verified by D. Roy using an optimized C program and 16, 000 hours on a cluster of desktop machines. We performed a verification of these searches by reducing the problem to the Boolean satisfiability problem (SAT). Our verification uses the cube-and-conquer SAT solving paradigm, symmetry breaking techniques using the computer algebra system Maple, and a result of Carter that there are ten nonisomorphic cases to check. Our searches completed in about 30 hours on a desktop machine and produced nonexistence proofs of about 1 terabyte in the DRAT (deletion resolution asymmetric tautology) format.

AAAI Conference 2019 Conference Paper

A SAT+CAS Approach to Finding Good Matrices: New Examples and Counterexamples

• Curtis Bright
• Dragomir Ž. Ðoković
• Ilias Kotsireas
• Vijay Ganesh

We enumerate all circulant good matrices with odd orders divisible by 3 up to order 70. As a consequence of this we find a previously overlooked set of good matrices of order 27 and a new set of good matrices of order 57. We also find that circulant good matrices do not exist in the orders 51, 63, and 69, thereby finding three new counterexamples to the conjecture that such matrices exist in all odd orders. Additionally, we prove a new relationship between the entries of good matrices and exploit this relationship in our enumeration algorithm. Our method applies the SAT+CAS paradigm of combining computer algebra functionality with modern SAT solvers to efficiently search large spaces which are specified by both algebraic and logical constraints.

AAAI Conference 2018 Conference Paper

A SAT+CAS Method for Enumerating Williamson Matrices of Even Order

• Curtis Bright
• Ilias Kotsireas
• Vijay Ganesh

We present for the first time an exhaustive enumeration of Williamson matrices of even order n < 65. The search method relies on the novel SAT+CAS paradigm of coupling SAT solvers with computer algebra systems so as to take advantage of the advances made in both the field of satisfiability checking and the field of symbolic computation. Additionally, we use a programmatic SAT solver which allows conflict clauses to be learned programmatically, through a piece of code specifically tailored to the domain area. Prior to our work, Williamson matrices had only been enumerated for odd orders n < 60, so our work increases the bounds that Williamson matrices have been enumerated up to and provides the first enumeration of Williamson matrices of even order. Our results show that Williamson matrices of even order tend to be much more abundant than those of odd orders. In particular, Williamson matrices exist for every even order n < 65 but do not exist in orders 35, 47, 53, and 59.