A Symmetric LambdaCalculus Corresponding to the NegationFree Bilateral Natural Deduction
Abstract
Filinski constructed a symmetric lambdacalculus consisting of expressions and continuations which are symmetric, and functions which have duality. In his calculus, functions can be encoded to expressions and continuations using primitive operators. That is, the duality of functions is not derived in the calculus but adopted as a principle of the calculus. In this paper, we propose a simple symmetric lambdacalculus corresponding to the negationfree natural deduction based bilateralism in prooftheoretic semantics. In our calculus, continuation types are represented as not negations of formulae but formulae with negative polarity. Function types are represented as the implication and butnot connectives in intuitionistic and paraconsistent logics, respectively. Our calculus is not only simple but also powerful as it includes a callvalue calculus corresponding to the callbyvalue dual calculus invented by Wadler. We show that mutual transformations between expressions and continuations are definable in our calculus to justify the duality of functions. We also show that every typable function has dual types. Thus, the duality of function is derived from bilateralism.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 DOI:
 10.48550/arXiv.2101.12607
 arXiv:
 arXiv:2101.12607
 Bibcode:
 2021arXiv210112607A
 Keywords:

 Computer Science  Logic in Computer Science