Journal of the Association for Computing Machinery, Vol. 43, No. 4, July 1996, pp. 670–684.

Complexity of Makanin’s Algorithm

ANTONI KOŚCIELSKI

AND

LESZEK PACHOLSKI

Abstract. The exponent of periodicity is an important factor in estimates of complexity of word-unification algorithms. We prove that the exponent of periodicity of a minimal solution of a word equation is of order 2^{1.07d}, where d is the length of the equation. We also give a lower bound 2^{0.29d}, so our upper bound is almost optimal and exponentially better

than the original bound (6d)^{2^{2d^{4}}} + 2. Consequently, our result implies an exponential improvement of known upper bounds on complexity of word-unification algorithms.

Categories and Subject Descriptors. F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—computations on discrete structures; I.1.2 [Algebraic Manipulation]: Algorithms—algebraic algorithms

General Terms. Algorithms, complexity, equations

Additional Key words and Phrases. Diophantine equation, periodicity, semantic unification, semigroups, word equations

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