this paper, we study the relationship between the call-by-name and call-by-value paradigms for Parigot's -calculus. The -calculus is an extension of the simply-typed lambda calculus with certain sequential control operators. We show that, in the presence of product and disjunction types, the call-by-name and call-by-value -calculi are isomorphic to each other, in the sense that there exist syntactic translations between them that preserve the operational semantics and that are mutually inverse up to isomorphism of types. These translations take the form of a

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ion. Let \Gamma; x : oe ` M : ø . [[x : oe:M ]] = : (oe * ø ) * 0: (x : oe:M ) = by C-App : (x:C ø (k : ø * 0:k M )) = : (x:C ø ([[M ]])) = j oe*ø (x:C([[M ]])) which is the required expression since x:C oe ([[M ]]) is the abstraction of [[M ]] wrt. x in V. Application. Let \Gamma ` M : oe * ø and \Gamma ` N : oe. [[M N ]] = by definition of [[:]] : ø * 0: (M N ) = by App : ((m:m N ) M ) = by Conv :(m: (m N )) M = by Beta-V :(k:k M ) (m: (m N )) = by Ass :(k:k M ) (m:(n:( (m n))) N ) = by Beta-V :(k:k M ) (m:(l:l N ) (n:( (m n)))) = by definition of app. app([[M ]]; [[N ]]) Complete axiomatisations of control operators 17 A operator. Let \Gamma ` M : 0. [[A oe (M )]] = by definition : oe * 0: (A oe (M )) = by A-Abs :A 0 (M ) = by A 0 -Id :M = by Ident :(x : 0:x) M = by Beta-V :(k:k M )(x : 0:x) = A oe ([[M ]]) C operator. If \Gamma ` M : (oe * 0) * 0 then [[C oe (M )]] = : oe * 0: C(M ) = by C-Nat :C 0 (k : 0 * 0:M (x : oe:k ( x))) = by 0-Endo and Ident :C 0...