Higher Order Unification via Explicit Substitutions

Higher order uniication is equational uniication for-conversion. But it is not rst order equational uniication, as substitution has to avoid capture. Thus the methods for equational uniication (such as narrowing) built upon grafting (i.e. substitution without renaming), cannot be used for higher order uniication, which needs speciic algorithms. Our goal in this paper is to reduce higher order uniication to rst order equational uniication in a suitable theory. This is achived by replacing substitution by grafting, but this replacement is not straightforward as it raises two major problems. First, some uniication problems have solutions with grafting but no solution with substitution. Then equational uniication algorithms rest upon the fact that grafting and reduction commute. But grafting and-reduction do not commute in-calculus and reducing an equation may change the set of its solutions. This diiculty comes from the interaction between the substitutions initiated by-reduction and the ones initiated by the uniication process. The diierence is at the variable level. Two kinds of variables are involved: those of-conversion and those of uniication. So, we need to set up a calculus which distinguishes these two kinds of variables and such that reduction and grafting commute. For that, the application of a substitution of a reduction variable to a uniication one must be delayed until this variable is instantiated. Such a separation and a delay are provided by a calculus of explicit substitutions. Uniication in such a calculus can be performed by well-known algorithms such as narrowing, but we present a specialized algorithm for a greater eeciency. At last we show how to relate uniication in-calculus and in a calculus with explicit substitutions. Thus we come up with a new higher order uniication algorithm which eliminates some burdens of the previous algorithms, in particular the functional handling of scopes. Huet's algorithm, can be seen as a speciic strategy for our algorithm, since each of its step is decomposed in elementary ones, giving a more atomic description of the uniication process. Also, solved forms in-calculus can easily be computed from solved forms in-calculus. RRsumm : L'uniication d'ordre suprieur consiste uniier modulo-conversion. Ce n'est donc pas de l'uniication quationnelle modulo-conversion car les substitutions doivent viter les captures. Les mmthodes (comme la surrrduction) permettant d'uniier modulo une thhorie quationnelle sont basses sur l'opration de greee (qui consiste substituer sans renomer) et ne peuvent donc pas s'appliquer directement l'uniication d'ordre suprieur. Ce papier a pour but de rrduire …