Complexity of Unification in Free Groups and Free Semi-groups

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 at most 2, where n is the length of the equation. Since the best known lower bound is 2 our upper bound is almost optimal and exponentially better than the original bound (6n) 2n4 + 2. Thus our result implies exponential improvement of known upper bounds on complexity of word-unification algorithms. Moreover we give some evidence that, contrary to the common belief, the algorithm deciding satisfiability of equations in free groups, given by Makanin in not primitive recursive. The proofs are only sketched here. More details will be given in the full version. 0 Introduction. In this note we improve the known upper bound on the exponent of periodicity thus obtaining exponential speed-up of several word unification algorithms. We also comment on the complexity of the Makanin’s algorithm deciding satisfiability of equations in free groups. By N we denote the set of non-negative integers, N is the set of positive integers. Given any non empty set Σ by Σ∗ we denote the set of all words in Σ. Σ is the set of non-empty words in Σ. If W is a word, then |W | denotes the length of W . ε is the empty word. Let Σ, Ξ be two disjoint, nonempty, finite alphabets. Σ = {a1, . . . , an} is the set of (constant) letters and Ξ = {x1, . . . , xn} is the set of variable letters. A word equation in (Σ, Ξ) is a pair E = (W1,W2) of words in (Σ ∪ Ξ)∗, also denoted by W1 = W2. A solution of E is a function v : Σ → Ξ such that W1(v(x1)/x1, . . . , v(xm)/xm) = W2(v(x1)/x1, . . . , v(xm)/xm), where W (v(xi)/xi) denotes the word obtained from W by replacing each occurrence of xi by v(xi) . The length of a solution v is ∑m i=1 v(xi) . A solution is minimal if it has minimal length. It was shown by Makanin [MA1], that the problem if a word equation has a solution is decidable. Later related variants of word-unification problem, namely ∗Institute of Computes Science, WrocÃlaw University, Przesmyckiego 20, 51-151 WrocÃlaw, Poland †Institute of Mathematics, Polish Academy of Sciences, Kopernika 18, 51-617 WrocÃlaw, Poland the problems of finding a solution, finding a minimal solution and finding all minimal solutions, were studied by various researchers (see e.g. [APE], [PEC], [JAF]). Moreover, some variants of Makanin’s algorithm have been implemented (see [ABD]). In [JAF] Jaffar gave a procedure generating for a word equation E the minimal and complete collection of unifiers. This procedure stops with a positive answer when E is satisfiable. To stop the procedure in the case when E is not satisfiable a bound B, depending on the size of E , is placed on the length of each path of the reduction tree, B being an increasing function of the exponent of periodicity of a minimal solution of E . Thus the number of steps of the generation procedure of Jaffar will, in the case of unsatisfiable equation depend on the known bounds on the periodicity exponent. In spite of the fact that the algorithm of Makanin and its variants seem to have important applications and have been intensively studied, no serious investigations of their complexity have been undertaken. It seems that the understanding of the nature of the algorithm of Makanin is still very low. This paper contains a report on an attempt to understand the complexity of the Makanin’s algorithm for semi-groups and the complexity of the problem of solvability of word equations. An important factor in estimates of the complexity of the Makanin’s algorithm is the periodicity exponent of a minimal solution of a word equation. A periodicity exponent of a word W is the maximal integer p such that W = U1UU2 for some non-empty word U . An important fact used in the Makanin’s algorithm and its variants is that the periodicity exponent can be bounded by a recursive function of the length of an equation. In fact V.K.Bulitko [BUL] proved that if n is the length of an equation, then the index of periodicity of its minimal solution does not exceed (6n) 2n4 + 2. In [KPA] we forced this bound down to n 4 . The method, we have used, was based on Makanin’s reduction lemma and consisted of obtaining better bounds on the size of minimal positive integer solutions of sets of linear diophantine equations. The bounds we have obtained are close to the ones obtained recently by E.Bombieri and J.Vaaler [BVA] for minimal absolute values of integer solutions (not necessarily positive) of such equations, and seem to be close to optimal. On