Call-by-value is dual to call-by-name, extended

Abstract

We extend Wadler's work that showed duality between call-by-value and call-by-name by giving mutual translations between the λμ-calculus and the dual calculus. We extend the λμ-calculus and the dual calculus through two stages. We first add a fixed-point operator and an iteration operator to the call-byname and call-by-value systems respectively. Secondly, we add recursive types, T, and ⊥ types to these systems. The extended duality between call-by-name with recursion and call-by-value with iteration has been suggested by Kakutani. He followed Selinger's category-theoretic approach. We completely follow Wadler's syntactic approach. We give mutual translations between our extended λμ-calculus and dual calculus by extending Wadler's translations, and also show that our translations form an equational correspondence, which was defined by Sabry and Felleisen. By composing our translations with duality on the dual calculus, we obtain a duality on our extended λμ-calculus. Wadler's duality on the λμ-calculus was an involution, and our duality on our extended λμ-calculus is also an involution. © Springer-Verlag Berlin Heidelberg 2007.