Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

Journal of Symbolic Computation

Kotsireas

2020

Self Cite

We employ tools from the fields of symbolic computation and satisfiability checkingnamely, computer algebra systems and SAT solvers-to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of even order up to and including 70 which gives us deeper insight into the behaviour and distribution of Williamson matrices. We find that, in contrast to the case when the order is odd, Williamson matrices of even order are quite plentiful and exist in every even order up to and including 70. As a consequence of this and a new construction for 8-Williamson matrices we construct 8-Williamson matrices in all odd orders up to and including 35. We additionally enumerate all Williamson matrices whose orders are divisible by 3 and less than 70, finding one previously unknown set of Williamson matrices of order 63.

Section: The Williamson Conjecturementioning

confidence: 99%

Applying computer algebra systems with SAT solvers to the Williamson conjecture

Journal of Symbolic Computation

Kotsireas

2020

Self Cite

“…Some of the first successes were computing van der Waerden numbers by Kouril and Paul [35] and Ahmed, Kullmann, and Snevily [2], computing Green-Tao numbers by Kullmann [36], as well as solving a special case of the Erdős discrepancy conjecture by Konev and Lisitsa [34]. Other more recent combinatorial applications include proving the Boolean Pythagorean triples conjecture [28] and a new case of the Ruskey-Savage conjecture [52], as well as computing Ramsey numbers [17], Williamson matrices [8], complex Golay sequences [10], and Schur numbers [26].…”

Section: Related Workmentioning

confidence: 99%

A nonexistence certificate for projective planes of order ten with weight 15 codewords

Cheung

Stevens

et al. 2020

AAECC

Self Cite

Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no projective planes of order ten that generate codewords of weight fifteen, a result first shown in 1973 via an exhaustive computer search. We provide a simple satisfiability (SAT) instance and a certificate of unsatisfiability that can be used to automatically verify this result for the first time. All previous demonstrations of this result have relied on search programs that are difficult or impossible to verify-in fact, our search found partial projective planes that were missed by previous searches due to previously undiscovered bugs. Furthermore, we show how the performance of the SAT solver can be dramatically increased by employing functionality from a computer algebra system (CAS). Our SAT+CAS search runs significantly faster than all other published searches verifying this result.

“…At almost the same time this synergy was demonstrated by the system MATHCHECK presented at the conference CADE (Zulkoski, Ganesh, and Czarnecki 2015). The system MATHCHECK coupled a SAT solver with a computer algebra system and solved open cases of two conjectures in graph theory and was later extended to solve open cases in combinatorial conjectures (Zulkoski et al 2017). Since then, the SC 2 project (Ábrahám et al 2016) has organized an annual workshop on this topic for the last three years.…”

Section: The Sat+cas Paradigmmentioning

confidence: 99%

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

Djoković

Kotsireas

2019

AAAI

Self Cite

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.2. We use state-of-the-art programmatic SAT solvers, computer algebra systems, and mathematical libraries to very efficiently search spaces specified by both logical and algebraic constraints.