On the Complexity of Computation Maximal Exponent of Periodicity of Word Equations and Expressible Relations (Note)

Denote by SWE the satisfiability problem for word equations. Consider the problem of computability of the maximal exponent of periodicity for word equations MaxExpProblem ( e , k ) in which given positive integer k and a word equation e one is to check whether maximal exponent of periodicity of a solution of a word equation is at least k . We present both a deterministic polynomial time reduction of the MaxExpProblem ( e , k ) problem to the satisfiability problem for word equations and a deterministic polynomial time reduction in the opposite direction. Consequently, if the satisfiability problem is in NP then the other problem is NP-complete. As a simple consequence we get that computation of maximal exponent of periodicity of a solution set for a word equation is in P SWE (SWE is an oracle) and, consequently in PSPACE. Let ϕ be an existential boolean formula built on word equations expressing a relation on words by means of its unknowns. Consider the problem of checking the maximal exponent of periodicity of a component of a relation CMaxExpProblem ( ϕ , k ) in which given positive integer k and a formula ϕ one is to check whether maximal exponent of periodicity of a component of the relation is at least k . We give nondeterministic polynomial time reduction of the described above CMaxExpProblem ( ϕ , k ) problem to the satisfiability problem for word equations and deterministic polynomial time reduction in the other direction. Consequently, if the satisfiability problem is in NP then the other problem is NP-complete. Denote by SAT ( WE ) the satisfiability problem of closed existential formula built on word equations. As a simple consequence of our considerations we get that computation of maximal exponent of periodicity of a component of expressible relation is in P SAT ( WE ) (SAT ( WE ) is an oracle), and, consequently, in PSPACE. Another consequence of that is that if SWE is in NP, then the computability problem is in P N P. The above results hold also for semigroups with involution. If the expressible relation is defined by a formula using word equations and regular constraints we prove that the problem is PSPACE-complete. The same holds for free semigroups with involution and free groups.