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Show CHAPTER 3 ON THE COMPLETENESS OF FIRST-ORDER UNIFICATION IN EQUATIONAL THEORIES This chapter is devoted to the first result of this thesis: we prove that unification in equational theories is complete. That is, given an arbitrary equational theory, there exists an E-unification algorithm which generates complete sets of unifiers for all E-unifiable terms. The completeness of first-order unification in equational theories is shown by constructing a "universal" Eunification algorithm such that it is complete for any equational theory. The key observation is that, given a set of function symbols and their arities, any term can be constructed in a finite number of steps. This result is primarily of theoretical interest, because such universal algorithms tend to be very inefficient in that too many (indeed, usually infinite) irrelevant search paths need to be pursued. The narrowing process and term rewriting systems are useful tools for constructing complete unification algorithms for useful classes of equational theories. Term rewriting systems provide a computationally effective formalism. Narrowing can be viewed as extended reduction where the matching process (or one-way unification) is placed by two-way unification. A natural equation arises: when does narrowing yield a complete unification algorithm? That is, we are interested in exploring the power of narrowing. Although the narrowing method has been shown to be complete for the class of canonical term rewriting systems, the generality and limitations of narrowing are not yet fully understood. In this chapter, we present E-unification algorithms that are based on a fair interleaving of the narrowing process, a term construction process and the |
