Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

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, à la MathCheck, can be used as an assistant by mathematicians to either find counterexamples or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS. 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 capabilities of several different CAS, namely the SAGE, Maple, and Magma systems. As case studies, we study three long-standing open mathematical conjectures, two from graph theory regarding properties of hypercubes, and one from combinatorics about Hadamard matrices. The first conjecture states that any matching of any d-dimensional hypercube can be extended to a Hamiltonian cycle; the second states that given an edge-antipodal coloring of a hypercube there always exists a monochromatic path between two antipodal vertices; the third states that Hadamard matrices exist for all orders divisible by 4. Previous results on the graph theory conjectures 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. Regarding Hadamard matrices, we demonstrate the advantages of SAT+CAS by constructing Williamson matrices up to order 42 (equivalently, Hadamard up to order \(4\times 42=168\)), improving the bounds up to which Williamson matrices of even order have been constructed. Prior state-of-the-art construction was only feasibly performed for odd numbers, where possible.

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Notes

  1. 1.

    All code and data is available at https://sites.google.com/site/uwmathcheck/.

  2. 2.

    http://www.jpl.nasa.gov/blog/2013/8/hadamard-matrix.

  3. 3.

    For notational convenience, we often use existential quantifiers when defining constraints; these are unrolled in the implementation. We only deal with finite graphs.

  4. 4.

    We were unable to find the original source of the results for \(d\le 4\), however the result is asserted in [22]. We also verified these results using our system.

  5. 5.

    In [63], the Hamiltonian cycle detection problem is reduced to SAT by encoding it as a permutation problem, such that if a Hamiltonian cycle exists, then the permutation extracted from the model corresponds to the cycle. In order to negate this existential check (as is needed by Formula 9), one must ensure that all permutations of the vertices do not correspond to a Hamiltonian cycle. While this can be succinctly expressed a quantified Boolean formula (QBF), a substantial number of universally quantified variables must be introduced. Hamiltonian cycle detection can also be succinctly expressed in answer set programming (ASP) [43], but similar search-space explosion problems still exist.

  6. 6.

    http://www.reduce-algebra.com/index.htm.

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Zulkoski, E., Bright, C., Heinle, A. et al. Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures. J Autom Reasoning 58, 313–339 (2017). https://doi.org/10.1007/s10817-016-9396-y

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Keywords

  • Travel Salesman Problem
  • Hamiltonian Cycle
  • Computer Algebra System
  • Conjunctive Normal Form
  • Satisfiability Modulo Theory