MathCheck: A Math Assistant via a Combination of Computer Algebra Systems and SAT Solvers

Abstract

We present a method and an associated system, called Math-Check, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assistant by mathematicians to either counterexample or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS system. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL(T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the graph-theoretic capabilities of an opensource CAS, called SAGE. As case studies, we look at two long-standing open mathematical conjectures from graph theory regarding properties of hypercubes: the first conjecture states that any matching of any ddimensional hypercube can be extended to a Hamiltonian cycle; and the second states that given an edge-antipodal coloring of a hypercube, there always exists a monochromatic path between two antipodal vertices. Previous results have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. We provide detailed performance analysis and show an exponential reduction in search space via the SAT+CAS combination relative to finite brute-force search.

The SAT+CAS Method for Combinatorial Search with Applications to Best Matrices

“…The Ruskey-Savage conjecture (that every matching of a hypercube can be extended into a Hamiltonian cycle) was previously known to hold in the orders n = 2, 3, and 4 [29]. Using MathCheck we showed that the conjecture also holds in the order n = 5 for the first time [79].…”

“…1. The Ruskey-Savage conjecture (see [78,79]). This conjecture states that any matching of a hypercube graph can be extended to a Hamiltonian cycle.…”

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

“…The fact that satisfiability checking and symbolic computation had great potential synergy was first pointed out by Erika Ábrahám in an invited talk at the conference ISSAC in 2015 (Ábrahám 2015). 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).…”

Applying computer algebra systems with SAT solvers to the Williamson conjecture

“…The idea of combining SAT solvers with computer algebra systems originated independently in two works published in 2015: In a paper at the conference CADE entitled "MATHCHECK: A Math Assistant via a Combination of Computer Algebra Systems and SAT Solvers" by Zulkoski et al (2015) and in an invited talk at the conference ISSAC entitled "Building Bridges between Symbolic Computation and Satisfiability Checking" by Ábrahám (2015). The paradigm was also anticipated by Jovanović and de Moura (2012) who used CAS techniques in SAT-like search algorithms.…”

“…Secondly, similar in style to (Zulkoski et al, 2015) we incorporate functionality from computer algebra systems to increase the efficiency of the search in what we call the "SAT+CAS" paradigm. This approach of combining computer algebra systems with SAT or SMT solvers was also independently proposed at the conference ISSAC by Ábrahám (2015).…”