Higher order unification via explicit substitutions
Higher order unification is equational unification for beta eta-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus, the methods for equational unification (such as narrowing) built upon grafting (i.e,, substitution without renaming) cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory. This is achieved by replacing substitution by grafting, but this replacement is not straightforward as it raises two major problems. First, some unification problems have solutions with grafting but no solution with substitution. Then equational unification algorithms rest upon the fact that grafting and reduction commute. But grafting and beta eta-reduction do not commute in lambda-calculus and reducing an equation may change the set of its solutions. This difficulty comes from the interaction between the substitutions initiated by prl-reduction and the ones initiated by the unification process. Two kinds of Variables are involved: those of beta eta-conversion and those of unification. So, we need to set up a calculus which distinguishes between these two kinds of variables and such that reduction and grafting commute. For this purpose, the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated. Such a separation and delay are provided by a calculus of explicit substitutions. Unification in such a calculus can be performed by well-known algorithms such as narrowing, but we present a specialised algorithm for greater efficiency. At last we show how to relate unification in lambda-calculus and in a calculus with explicit substitutions. Thus, we come up with a new higher order unification algorithm which eliminates some burdens of the previous algorithms, in particular the functional handling of scopes. Huet's algorithm can be seen as a specific strategy for our algorithm, since each of its steps can be decomposed into elementary ones, leading to a more atomic description of the unification process. Also, solved forms in lambda-calculus can easily be computed from solved forms in lambda sigma-calculus. (C) 2000 Academic Press.
higher order unification explicit substitutions equational unification