A type system for Continuation Calculus

TL;DR

This paper extends Continuation Calculus with a type system, enabling canonical data type definitions and iteration schemes, bridging call-by-value and call-by-name evaluation through double negation translation.

Contribution

It introduces a type system for CC with Scott-encoded data types and two iteration schemes, connecting call-by-value and call-by-name semantics.

Findings

Defined data types via Scott encoding in CC

Developed call-by-value and call-by-name iteration schemes

Established a double negation translation between evaluation strategies

Abstract

Continuation Calculus (CC), introduced by Geron and Geuvers, is a simple foundational model for functional computation. It is closely related to lambda calculus and term rewriting, but it has no variable binding and no pattern matching. It is Turing complete and evaluation is deterministic. Notions like "call-by-value" and "call-by-name" computation are available by choosing appropriate function definitions: e.g. there is a call-by-value and a call-by-name addition function. In the present paper we extend CC with types, to be able to define data types in a canonical way, and functions over these data types, defined by iteration. Data type definitions follow the so-called "Scott encoding" of data, as opposed to the more familiar "Church encoding". The iteration scheme comes in two flavors: a call-by-value and a call-by-name iteration scheme. The call-by-value variant is a double negation variant of call-by-name iteration. The double negation translation allows to move between call-by-name and call-by-value.