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**11 - 13**of**13**### "Syntactic" AC-Unification

, 1994

"... The rules for unification in a simple syntactic theory, using Kirchner's mutation [15, 16] do not terminate in the case of associativecommutative theories. We show that in the case of a linear equation, these rules terminate, yielding a complete set of solved forms, each variable introduced by the u ..."

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The rules for unification in a simple syntactic theory, using Kirchner's mutation [15, 16] do not terminate in the case of associativecommutative theories. We show that in the case of a linear equation, these rules terminate, yielding a complete set of solved forms, each variable introduced by the unifiers corresponding to a (trivial) minimal solution of the (trivial) Diophantine equation where all coefficients are 1. A nonlinear problem can be first treated as a linear one, that is considering two occurrences of a same variable as two different variables. After this step, one has to solve the equations between the different values that have been obtained for the different occurrences of a same variable. We show that one can restrict the search of the solutions of these latter equations to linear substitutions. This result is based on the analysis of how the minimal solutions of a linear Diophantine equation can be built-up using the solutions of the trivial Diophantine equation asso...

### On n-Syntactic Equational Theories

, 1992

"... We define the n-syntactic theories as a natural extension of the syntactic theories. A n-syntactic theory is an equational theory which admits a finite presentation in which every proof can be performed with at most n applications of an axiom at the root,but no finite presentation in which every pr ..."

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We define the n-syntactic theories as a natural extension of the syntactic theories. A n-syntactic theory is an equational theory which admits a finite presentation in which every proof can be performed with at most n applications of an axiom at the root,but no finite presentation in which every proof can be performed with at most n - 1 applications of an axiom at the root. The n-syntactic theories inherit the good properties of the syntactic theories for solving the word problem, or matching or unification problems. We show that for any integer n >= 1, there exists a n-syntactic theory.

### Recueil d’articles

"... Annexes du manuscrit d’HDR — Le calcul de réécritureiiSommaire Présentations du calcul de réécriture 1 Propriétés des calculs à motifs 57 Extensions du calcul de réécriture 75 Expressivité du calcul de réécriture 147 Systèmes de types pour le calcul de réécriture 195 Applications 225Présentations du ..."

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Annexes du manuscrit d’HDR — Le calcul de réécritureiiSommaire Présentations du calcul de réécriture 1 Propriétés des calculs à motifs 57 Extensions du calcul de réécriture 75 Expressivité du calcul de réécriture 147 Systèmes de types pour le calcul de réécriture 195 Applications 225Présentations du calcul de réécriture [CK01] [CKL01a]