Abstract
A double negation translation (DNT) embeds classical logic into intuitionistic logic. Such translations correspond to continuation passing style (CPS) transformations in programming languages via the Curry-Howard isomorphism. A selective CPS transformation uses a type and effect system to selectively translate only nontrivial expressions possibly with computational effects into CPS functions. In this paper, we review the conventional call-by-value (CBV) CPS transformation and its corresponding DNT, and provide a logical account of a CBV selective CPS transformation by defining a selective DNT via the Curry-Howard isomorphism. By using an annotated proof system derived from the corresponding type and effect system, our selective DNT translates classical proofs into equivalent intuitionistic proofs, which are smaller than those obtained by the usual DNTs. We believe that our work can serve as a reference point for further study on the Curry-Howard isomorphism between CPS transformations and DNTs.
Highlights
The Curry-Howard isomorphism [1,2] states that formulas and proofs in mathematical logic correspond to types and programs in programming languages
We study the logical meaning of a selective continuation passing style (CPS) transformation [16] and present a selective double negation translation (DNT) for classical propositional logic (CPL) via the Curry-Howard isomorphism
We review a CBV selective CPS transformation based on a type and effect system [16] and propose a corresponding selective DNT based on an annotated proof system, with its correctness proof showing that provability is preserved under the translation
Summary
The Curry-Howard isomorphism [1,2] states that formulas and proofs in mathematical logic correspond to types and programs in programming languages. The CPS transformations from λcont into λ→ [3,4], which eliminate control operators, correspond to a DNT from CPL with Peirce’s law or DNE into IPL. The selective CPS transformation only translates nontrivial expressions into CPS functions and keeps trivial expressions intact To do so, it uses a type and effect system to keep track of the uses of control operators. We review the Curry-Howard isomorphism between the standard call-by-value (CBV) CPS transformation with control operators and the corresponding DNT from CPL into IPL. We review a CBV selective CPS transformation based on a type and effect system [16] and propose a corresponding selective DNT based on an annotated proof system, with its correctness proof showing that provability is preserved under the translation.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
