Abstract
The Boolean satisfiability problem (SAT) is the problem of determining whether the variables of a given Boolean formula can be consistently replaced by true or false in such a way that the formula evaluates to true. In fact, SAT was the first known NP-complete problem. In recent years, SAT has found numerous industrial applications, particularly in model checking tools. In the current work, three approaches to Boolean satisfiability based on Clifford subalgebras are presented. In the first approach, an “idem-Clifford” algebraic test for satisfiability is presented. This test is straightforward to implement symbolically (e.g., using Mathematica), but does not yield the specific solution sets for a given formula. In the second approach, nilpotent adjacency matrix methods are extended to Boolean formulas in order to determine not only whether or not a Boolean formula is satisfiable but to explicitly obtain all solutions. This approach requires the construction of a graph associated with a given Boolean formula. Finally, a “new” algebraic framework is developed that combines the convenience of the first approach with the power of the second, recovering explicit solutions without the need to construct graphs. The algebraic formalism presented here readily lends itself to symbolic computations and provides the theoretical basis of a Clifford-algebraic SAT solver.
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Notes
Often, the negation of \(x_i\) is denoted by \(\lnot x_i\).
Note that \(1\le |\mathfrak {c}_i|\le k\) for each i.
In other words, \(\mathcal {N}(v)\) represents the “edge-neighborhood” of v.
References
Ben Slimane, J., Schott, R., Song, Y.-Q., Staples, G.S., Tsiontsiou, E.: Operator calculus algorithms for multi-constrained paths. Int. J. Math. Comput. Sci. 10, 69–104 (2015). http://ijmcs.future-in-tech.net/10.1/R-Jamila.pdf
Biere, A., Biere, A., Heule, M., van Maaren, H., Walsh, T.: Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications. IOS Press, Amsterdam (2009). ISBN:9781586039295
Budinich, M., Budinich, P.: A spinorial formulation of the maximum clique problem of a graph. J. Math. Phys. 47, 043502 (2006)
Budinich, M.: The Boolean satisfiability problem in Clifford algebra. Theoretical Computer Science. In press. (2019) https://doi.org/10.1016/j.tcs.2019.03.027
Cassiday, C., Staples, G.S.: On representations of semigroups having hypercube-lik Cayley graphs. Clifford Anal. Clifford Algebras Appl. 4, 111–130 (2015)
Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, pp. 151–158 (1971). https://doi.org/10.1145/800157.805047
Cruz-Sánchez, H., Staples, G.S., Schott, R., Song, Y-Q.: Operator calculus approach to minimal paths: precomputed routing in a store-and-forward satellite constellation. In: Proceedings of IEEE Globecom 2012, Anaheim, CA, USA, December 3–7, pp. 3455–3460 (2012). ISBN: 978-1-4673-0919-6
Davis, A.: Boolean Satisfiability, Graph Problems, and Zeons. M.S. Thesis. Southern Illinois University Edwardsville, Edwardsville (2018)
Harris, G., Staples, G.S.: Spinorial formulations of graph problems. Adv. Appl. Clifford Algebras 22, 59–77 (2012). https://doi.org/10.1007/s00006-011-0298-0
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations: Proceedings of a Symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pp. 85–103. Plenum Press, New York (1972)
Nefzi, B., Schott, R., Song, Y.-Q., Staples, G.S., Tsiontsiou, E.: An operator calculus approach for multi-constrained routing in wireless sensor networks, Proceedings of ACM MobiHoc, Hangzhou, CHINA, ACM New York, NY, USA, pp. 367–376 (2015)
Schott, R., Staples, G.S.: Nilpotent adjacency matrices and random graphs. Ars Combinatoria 98, 225–239 (2011)
Schott, R., Staples, G.S.: Operator Calculus on Graphs (Theory and Applications in Computer Science). Imperial College Press, London (2012)
Staples, G.S., Stellhorn, T.: Zeons, orthozeons, and graph colorings. Adv. Appl. Clifford Algebras 27, 1825–1845 (2017). https://doi.org/10.1007/s00006-016-0732-4
Staples, G.S.: CliffMath: Clifford algebra computations in Mathematica, 2008–2019. http://www.siue.edu/~sstaple/index_files/research.htm. Accessed 14 Apr (2019)
Staples, G.S.: A new adjacency matrix for finite graphs. Adv. Appl. Clifford Algebras 18, 979–991 (2008). https://doi.org/10.1007/s00006-008-0116-5
Vizel, Y., Weissenbacher, G., Malik, S.: Boolean satisfiability solvers and their applications in model checking. Proc. IEEE 103, 2021–2035 (2015). https://doi.org/10.1109/JPROC.2015.2455034
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This article is part of the ENGAGE 2019 Topical Collection on Geometric Algebra for Computing, Graphics and Engineering edited by Linwang Yuan (EiC), Werner Benger, Dietmar Hildenbrand, and Eckhard Hitzer.
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Davis, A., Staples, G.S. Zeon and Idem-Clifford Formulations of Boolean Satisfiability. Adv. Appl. Clifford Algebras 29, 60 (2019). https://doi.org/10.1007/s00006-019-0978-8
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DOI: https://doi.org/10.1007/s00006-019-0978-8