## Implicit Programming and the Logic of Constructible Duality (1998)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Patterson98implicitprogramming,

author = {Anna L. Patterson},

title = {Implicit Programming and the Logic of Constructible Duality},

year = {1998}

}

### OpenURL

### Abstract

We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a self-dual logic-- constructible duality. We develop a self-dual model by considering an interval of worlds in an intuitionistic Kripke model. The duality arises through how we judge truth and falsity. Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards. We develop a self-dual algebra such that every point in the algebra is representable by some formula in the logic. This algebra arises as an instantiation of a Heyting algebra into several categorical constructions. In particular, we show that this algebra is an instantiation of the Chu construction applied to a Heyting algebra, the second Dialectica construction applied to a Heyting algebra, and as an algebra for the study of recursion and corecursion. Thus the algebra provides a common base for these constructions, and suggests itself as an important part of any constructive logical treatment of duality. Implicit programming is suggested as a new paradigm for computing with constructible duality as its formal system. We show that all the operators that have computable least fixed points are definable explicitly and all operators with computable optimal fixed points are definable implicitly within constructible duality. Implicit programming adds a novel definitional mechanism that allows functions to be defined implicitly. This new programming feature is especially useful for programming with co-recursively defined data-types such as circular lists.