A Domain-theoretic Semantics of Lax Generic Functions

AbstractThe semantic structure of a calculus λm is studied. λm is a polymorphic calculus defined over a hierarchical type structure, and a function in this calculus, called a generic function, can be composed from more than one lambda expression and the ways it behaves on each type are weakly related in that it lax commutes with coercion functions.Since laxness is intermediate between ad-hocness and coherentness, λm has syntactic properties lying between those of calculi with ad-hoc generic functions and coherent generic functions studied in [Tsu95]. That is, though λm allows self application and thus is not normalizing, it does not have an unsolvable term. For this reason, all the semantic domains are connected by infinitely many mutually recursive equations and, at the same time, they do not have the least elements. We solve them by considering opfibrations and expressing the equations as one recursive equation about opfibrations. We also show the adequacy theorem for λm following the construction of A. Pitts and use it to derive some syntactic properties