Note: An assertion concerning functionally complete algebras and NP-completeness

Authors:
Gábor Horváth;Chrystopher L. Nehaniv;Csaba Szabó
Affiliations:
Centre for Computer Science & Informatics Research, University of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, United Kingdom;Centre for Computer Science & Informatics Research, University of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, United Kingdom;Department of Algebra and Number Theory, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/c, 1117, Hungary
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

In a paper published in J. ACM in 1990, Tobias Nipkow asserted that the problem of deciding whether or not an equation over a nontrivial functionally complete algebra has a solution is NP-complete. However, close examination of the reduction used shows that only a weaker theorem follows from his proof, namely that deciding whether or not a system of equations has a solution is NP-complete over such an algebra. Nevertheless, the statement of Nipkow is true as shown here. As a corollary of the proof we obtain that it is coNP-complete to decide whether or not an equation is an identity over a nontrivial functionally complete algebra.