A Certain Loss of Identity

For pragmatic reasons it is useful to exclude the identity relation from thèimplementable subset' of Ruby. However there are many expressions in the relational calculus whose natural meaning is just this identity relation. This note gives an identity-free account of some of these expressions, and shows that there is no satisfactory identity-free account of some others. This is an exercise in writing about Ruby without drawing any pictures, in part because it is about those expressions which would correspond to blank pictures. What there is when there is nothing there In Ruby one uses relations to represent circuit components, and the composition R ; S of relations corresponds to some connection of two components in which the parts of the R represented by its range are connected to the parts of the S represented by its domain. With this interpretation, the repeated composition R n naturally represents a `pipeline' of n components, each an R, connected in a linear array. At least, this is the natural interpretation for n > 0. For strictly positive n and m it is clearly the case (and can be shown by induction) that R n+m is the composition R n ; R m of smaller powers of R. This suggests that it might be desirable for any deenition of R 0 to be consistent with R n = R n ; R 0 = R 0 ; R n. One possible choice of a value for R 0 consistent with these equations would be the identity (or equality) relation, id, for which x id x for all possible x. Sub-expressions whose values are as`big' as the identity relation are generally undesirable in Ruby expressions which might be classed asìmplementa-tions'. These expressions are the results of reenements, ones which we expect to be able to pass to an implementor, perhaps a silicon manufacturer, with the expectation that they will be accurately represented by some (probably nite) machine operating on a physical data representation. For example, in reening to a design to be implemented in digital electronics, we would not want to say that we had completed the design until all the components in the expressions were relations between tuples of binary signal values. This paper is not meant to be self-contained, and should be read in the context of an introduction to Ruby, such as reference 1].