A proof theoretical approach to Continuation Passing Style

In this paper we study the relation with continuation passing style(CPS) and classical proofs, proof theoretically. There is two style of CPS translation { Call by name (CBN) and Call by value(CBV). We revealed that in the case of classical proofs, this two reduction strategy may induce di erent result even if the language is purely functional(i.e. proof theoretical). We explain this through the witness of peirce's law as a program with the help of computational property of classical linear logic. In Curry-Howard correspondence, CPS translation on programs is conversion on classical proofs. We show that CBN/CBV CPS translation precisely corresponds to proof conversion from classical logic(LK) into two style of restricted classical logic: LKT/LKQ respectively. LKT/LKQ [5], and its accompanying linear decoration, presented by Danos, Joinet and Schellinx are variation on Gentzen's original system LK. LKT/LKQ enrich two coloring (T and Q) to the formula of LK and is still complete for classical provability. The key idea to show that this proof conversion being CBN/CBV, is intuitionistic decoration which maps LKT/LKQ proofs into equivalent (w.r.t the cut elimination procedure) LJ proofs. Resulting LJ proofs is exactly the one known as the result of CBN/CBV translation. Since LKT/LKQ also has classical linear decoration which is more precise from the point of view of computation, this gives us an idea to construct CBN/CBV term calculus inheriting computational property of linear logic. We present con uent, strongly normalizing LKQ term calculus (CBV calculus on classical proofs) in Gentzen-style sequent as the subsystem of Proof Net of Girard[9].