What is Uni cation ? A Categorical View of Substitution , Equation and Solution

From a general perspective, a substitution is a transformation from one space to another, an equation is a pair of such substitutions, and a solution to an equation is a substitution that yields the same value when composed with (i.e., when substituted into) the substitutions that constitute the given equation. In some special cases, solutions are called uniiers. Other examples include Scott domain equations, uniication grammars, type inference, and diierential equations. The intuition that the composition of substitutions should be associative when deened, and should have identities, motivates a general concept of substitution system based on category theory. Notions of morphism, congruence, and quotient are given for substitution systems, each with the expected properties, and some general cardinality bounds are proved for most general solution sets (which are minimal sets of solutions with the property that any other solution is a substitution instance of one in the set). The notions of equation and solution are also generalized to systems of equations, i.e., to constraint solving, and applied to clarify the notions of \compositionality" and \uniication" in linguistic uniication grammar. This paper is self-contained as regards category theory, and indeed, could be used as an introductory tutorial on that subject.