Abstract
We present a method and an associated system, called MathCheck, 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 open-source 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 d-dimensional 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.
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Notes
- 1.
All code+data is available at https://bitbucket.org/ezulkosk/sagesat.
- 2.
For notational convenience, we often use existential quantifiers when defining constraints; these are unrolled in the implementation. We only deal with finite graphs.
- 3.
We were unable to find the original source of the results for \(d\le 4\), however the result is asserted in [10]. We also verified these results using our system.
- 4.
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Zulkoski, E., Ganesh, V., Czarnecki, K. (2015). MathCheck: A Math Assistant via a Combination of Computer Algebra Systems and SAT Solvers. In: Felty, A., Middeldorp, A. (eds) Automated Deduction - CADE-25. CADE 2015. Lecture Notes in Computer Science(), vol 9195. Springer, Cham. https://doi.org/10.1007/978-3-319-21401-6_41
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