Towards a \geometry" for Logical Programming

1. Extended Abstract This talk will describe a general framework which has been established after several years of work within which it is possible to show that the solution of a general uniication problem may be viewed as that of calculating the intersection of a certain family of aane varieties ((6]). In this way, the usual realm, in which the problem of symbolic expression uniication is formulated, that of monoids, is replaced by the richer and more familiar setup of linear algebra and aane geometry. In this regard, it is shown that to any expression there is associated an aane function within a certain special family. These functions, in turn, deene a family of aane subspaces. Then, within this family, it may be shown that the uniier of two given expressions is given by the parametric equations of the intersection variety, which again belongs to the family and thus represents an expression. Contrary to usual aane geometry, it may be shown that these parametric equations are unique, except for the order in the vectors involved and thus the uniier, if it exists, is the unique expression which solves the uniication problem. The framework we describe is based on the construction of the ring of p-tangles introduced in 3] which generalises the notion of innnite tree as given in ((1]). It is a characteristic two, non commutative ring with unit, with no zero divisors over which a certain module is constructed. This module is, in turn, used to represent symbolic expressions through a family of aane maps in the module: those for which the coeecients involved in the parametric equations have a special structure which we have called "list". These list structures are fundamental to our work and represent a decomposition of a tree as a sum of disjoint p-tangles. The main objective of the talk is to show with examples several applications of the ring of tangles. The structure of the talk is as follows: First we summarize from 3], 4],and 5] those deenitions and results for the ring of p-tangles which are relevant to this work. In order to highlight those ideas explored in the following sections, we thoroughly examine a practical example. Replacing a variable by an expression is one of the operations which may be done in an automatic fashion within our model. To do so we shall associate to each symbolic expression a tree-like structure, which …