# Solving string equations with constant restrictions

Conference paper

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## Abstract

We present a variant and extension of Makanin's [5] decision algorithm for string equations: Given are sets

*C*= {*c*_{1},...,*c*_{m}} of constants,*V*= {*x*_{1},...,*x*_{n}} of variables and a string equation*s*_{1}*s*_{2}where*s*_{1},*s*_{2}ε (*C*∪*V*)^{+}. Furthermore sets*r*(*x*_{ i }) ∈*C*, 1 ≤*i*<-*n*, are given which are called constant restrictions. A substitution*σ*solves the equation s_{1}*s*_{2}and satisfies the constant restrictions*R(x*_{ i }), 1 ≤*i<-n*, if σ(*s*_{1}) =*σ*(*s*_{2}) and σ(*x*_{ i })*ε*((*C-R(x*_{ i })) ∪*V*)^{+}for all*x*_{ i }ε*V*. I.e. we consider solutions of string equations such that certain constants do not appear in the substitutions of some variables.Modifying the decision algorithm of Makanin we obtain an algorithm which decides whether or not a given string equation has a solution satisfying the constant restrictions. Furthermore we think that we have, as a by-product, a very nice presentation of Makanin's algorithm.

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## References

- [1]Peter Auer. Unification in the combination of disjoint theories. 1991. To be published in the proceedings of the IWWERT'91.Google Scholar
- [2]Peter Auer.
*Unification with associative functions*. PhD thesis, Technical University Vienna, 1992.Google Scholar - [3]Franz Baader and Klaus Schulz. Unification in the union of disjoint equational theories: combining decision procedures. 1991. To be published in the proceedings of the IWWERT'91.Google Scholar
- [4]Joxan Jaffer. Minimal and complete word unification.
*Journal of the Association for Computing Machinery*, 37(1):47–85, January 1990.Google Scholar - [5]G. S. Makanin. The problem of solvability of equations in a free semiproup.
*Math. USSR Sbornik*, 32(2):129–198, 1977. English translation.Google Scholar - [6]G. S. Makanin. Recognition of the rank of equations in a free semigroup.
*Math. USSR Izvestija*, 14(3):499–545, 1980. English translation.Google Scholar - [7]J. P. Pecuchet.
*Equations avec constantes et algorithme de Makanin*. PhD thesis, Laboratoire d' informatique, Rouen, 1981.Google Scholar - [8]Klaus U. Schulz.
*Makanin's Algorithm — Two Improvements and a Generalization*. Technical Report 91-39, CIS—UniversitÄt München, 1991.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1993