Primitive Direcursion and Difunctorial Semantics of Typed Object Calculus

Abstract: In the first part of this thesis, we contribute to the semantics of typed object calculus by giving (a) a category-theoretic denotational semantics using partial maps making use of an algebraic compactness assumption, (b) a notion of "wrappers'' by which algebraic datatypes can be represented as object types, and (c) proofs of computational soundness and adequacy of typed object calculus via Plotkin's FPC (with lazy operational semantics), thus making models of FPC suitable also for first-order typed object calculus (with recursive objects supporting method update, but not subtyping). It follows that a valid equation in the model induces operationally congruent terms in the language, so that program algebras can be studied. For (c), we also develop an extended first-order typed object calculus, and prove subject reduction. The second part of the thesis concerns recursion principles on datatypes including the untyped lambda calculus as a special case. Freyd showed that in certain domain theoretic categories, locally continuous functors have minimal invariants, which possess a structure that he termed dialgebra. This gives rise to a category of dialgebras and homomorphisms, where the minimal invariants are initial, inducing a powerful recursion scheme (direcursion) on a complete partial order. We identify a problem that appears when we translate (co)iterative functions to direcursion, and as a solution to this problem we develop a recursion scheme (primitive direcursion). This immediately gives a number of examples of direcursive functions, improving on the situation in the literature where only a few examples have appeared. By means of a case study, this line of work is connected to object calculus models.