# Towards parametrizing word equations

• Volume: 35, Issue: 4, page 331-350
• ISSN: 0988-3754

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## Abstract

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Classically, in order to resolve an equation $u\approx v$ over a free monoid ${X}^{*}$, we reduce it by a suitable family $ℱ$ of substitutions to a family of equations $uf\approx vf$, $f\in ℱ$, each involving less variables than $u\approx v$, and then combine solutions of $uf\approx vf$ into solutions of $u\approx v$. The problem is to get $ℱ$ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to $u\approx v$. We carry out such a parametrization in the case the prime equations in the graph involve at most three variables.

## How to cite

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Abdulrab, H., Goralčík, P., and Makanin, G. S.. "Towards parametrizing word equations." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 35.4 (2001): 331-350. <http://eudml.org/doc/92669>.

@article{Abdulrab2001,
abstract = {Classically, in order to resolve an equation $u\approx v$ over a free monoid $X^*$, we reduce it by a suitable family $\mathcal \{F\}$ of substitutions to a family of equations $uf\approx vf$, $f\in \mathcal \{F\}$, each involving less variables than $u\approx v$, and then combine solutions of $uf\approx vf$ into solutions of $u\approx v$. The problem is to get $\mathcal \{F\}$ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to $u\approx v$. We carry out such a parametrization in the case the prime equations in the graph involve at most three variables.},
author = {Abdulrab, H., Goralčík, P., Makanin, G. S.},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {equation; free monoid; parametrization; universal family},
language = {eng},
number = {4},
pages = {331-350},
publisher = {EDP-Sciences},
title = {Towards parametrizing word equations},
url = {http://eudml.org/doc/92669},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Abdulrab, H.
AU - Goralčík, P.
AU - Makanin, G. S.
TI - Towards parametrizing word equations
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 4
SP - 331
EP - 350
AB - Classically, in order to resolve an equation $u\approx v$ over a free monoid $X^*$, we reduce it by a suitable family $\mathcal {F}$ of substitutions to a family of equations $uf\approx vf$, $f\in \mathcal {F}$, each involving less variables than $u\approx v$, and then combine solutions of $uf\approx vf$ into solutions of $u\approx v$. The problem is to get $\mathcal {F}$ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to $u\approx v$. We carry out such a parametrization in the case the prime equations in the graph involve at most three variables.
LA - eng
KW - equation; free monoid; parametrization; universal family
UR - http://eudml.org/doc/92669
ER -

## References

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1. [1] J. Jaffar, Minimal and Complete Word Unification. J. ACM 37 (1990) 47-85. Zbl0697.68052MR1072430
2. [2] Yu.I. Hmelevskiĭ, Equations in free semigroups. Trudy Mat. Inst. Steklova 107 (1971); English Translation in Proc. Steklov Inst. Math. 107 (1971) 1976. Zbl0326.02032MR393284
3. [3] A. Lentin, Équations dans les monoïdes libres. Gautier-Villars, Paris (1972). Zbl0258.20058MR333034
4. [4] M. Lothaire, Combinatorics on Words. Addison-Wesley (1983). Zbl0514.20045MR675953
5. [5] G.S. Makanin, The problem of solvability of equations in a free semigroup. Mat. Sbornik 103 (1977) 147-236 (in Russian); English Translation in Math. USSR Sbornik 32 (1977) 128-198. Zbl0396.20037MR470107
6. [6] G.S. Makanin, On general solution of equations in free semigroups, in Proc. of IWWERT’91, edited by H. Abdulrab and J.P. Pécuchet. Springer, Lecture Notes in Comput. Sci. 677, 1-5. Zbl0925.20067
7. [7] G. Plotkin, Building-in Equational Theories. Machine intelligence 7 (1972) 73-90. Zbl0262.68036

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