SAT Solvers and Computer Algebra Systems: A Powerful Combination for Mathematics

Abstract

Over the last few decades, many distinct lines of research aimed at automating mathematics have been developed, including computer algebra systems (CASs) for mathematical modelling, automated theorem provers for rst-order logic, SAT/SMT solvers aimed at program veri cation, and higher-order proof assistants for checking mathematical proofs. More recently, some of these lines of research have started to converge in complementary ways. One success story is the combination of SAT solvers and CASs (SAT+CAS) aimed at resolving mathematical conjectures.Many conjectures in pure and applied mathematics are not amenable to traditional proof methods. Instead, they are best addressed via computational methods that involve very large combinatorial search spaces. SAT solvers are powerful methods to search through such large combinatorial spaces-consequently, many problems from a variety of mathematical domains have been reduced to SAT in an a empt to resolve them. However, solvers traditionally lack deep repositories of mathematical domain knowledge that can be crucial to pruning such large search spaces. By contrast, CASs are deep repositories of mathematical knowledge but lack e cient general search capabilities. By combining the search power of SAT with the deep mathematical knowledge in CASs we can solve many problems in mathematics that no other known methods seem capable of solving.We demonstrate the success of the SAT+CAS paradigm by highlighting many conjectures that have been disproven, veri ed, or partially veri ed using our tool MathCheck. ese successes indicate that the paradigm is positioned to become a standard method for solving problems requiring both a signi cant amount of search and deep mathematical reasoning. For example, the SAT+CAS paradigm has recently been used by Heule, Kauers, and Seidl to nd many new algorithms for 3 × 3 matrix multiplication.