Solving equations in the free algebra T (F; X) (i.e. unification) uses the two rules: f(~s) = f( ~ t) ! ~s = ~ t (decomposition) and s[x] = x !? (occur-check). These two rules are not correct in quotients of T (F; X) by a finitely generated congruence =E . Following C. Kirchner, we first define classes of equational theories (called syntactic and cycle syntactic respectively) for which it is possible to derive some rules replacing the two above ones. Then, we show that these abstract classes are relevant: all shallow theories, i.e. theories which can be generated by equations in which variables occur at depth at most one, are both syntactic and cycle syntactic. Moreover, the new set of unification rules is terminating, which proves that unification is decidable and finitary in shallow theories. We give still further extensions. If the set of equivalence classes is infinite, a problem which turns out to be decidable in shallow theories, then shallow theories fulfill Colmerauer's indep...

. We define a subclass of the class of linear equational theories, called finitely closable linear theories. We consider unification problems with no repeated variables. We show the decidability of this subclass, and give an algorithm in PSPACE. If all function symbols are monadic, then the running time is in NP, and quadratic for unitary monadic finitely closable linear theories. 1

Abstract. Most of the standard pleasant properties of term rewriting systems are undecidable; to wit: local confluence, confluence, normalization, termination, and completeness. Mere undecidability is insufficient to rule out a number of possibly useful properties: For instance, if the set of normalizing term rewriting systems were recursively enumerable, there would be a program yielding “yes ” in finite time if applied to any normalizing term rewriting system. The contribution of this paper is to show (the uniform version of) each member of the list of properties above (as well as the property of being a productive specification of a stream) complete for the class Π 0 2. Thus, there is neither a program that can enumerate the set of rewriting systems enjoying any one of the properties, nor is there a program enumerating the set of systems that do not. For normalization and termination we show both the ordinary version and the ground versions (where rules may contain variables, but only

We give a set of inference rules for solving E-unification. We prove the completeness with Eager Merging for linear theories and goals with no repeated variables. If the theory is further restricted to have no repeated variables, we show that E-unification is decidable and has linear complexity when the theory is considered constant, and cubic when it is part of the input. For any E-unification problem, equations can be transformed into this class and the problem can be quickly approximated.

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...

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.