## 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.

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Citation Context ...judged from picking a world in the future and looking backwards, while truth, as is normal is judged forwards, we generate a new model theory for constructible falsity, denoted N . Earlier, Thomason (=-=Thomason 1969-=-) proposed an ordering on worlds with three truth values true, false and undecided for which he shows that N is sound and complete, c.f. similar models by (Gurevich 1977, Routley 1975). Models with mu... |

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Citation Context ... that ? is a projection from the algebra of constructible duality to a dual-Heyting algebra (also called a Brouwerian algebra); thus ? projects the logic of constructible duality to Brouwerian logic (=-=McKinsey and Tarski 1944-=-, Birkhoff 1948). Proposition 132 ! maps H \ThetasH op to H \Thetas1 and ? maps H \ThetasH op to 1 \ThetasHop. Proof: Let A be a point A = ha1; a2i in H \ThetasHop. Then, by definition, !A = A ^ I whe... |

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Citation Context ...d point such that ( lub F XP C(\Phi ) ) v f2, we have f2 v f3. These optimal fixed points were first defined by (Manna and Shamir 1976) and studied by (Manna and Shamir 1977, Gallier 1980, Chen 1996, =-=Lassez and Maher 1985-=-). We present the original formulation, rather than later presentation, as Manna and Shamir also studied the relationship between fixed points and equational definitions of functions. Definition 167 (... |

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Citation Context ...lier, Thomason (Thomason 1969) proposed an ordering on worlds with three truth values true, false and undecided for which he shows that N is sound and complete, c.f. similar models by (Gurevich 1977, =-=Routley 1975-=-). Models with multiple viewpoints are used in relevant logics to avoid the rule of weakening c.f. (Hautam"aki 1982) for a generalization. We give an ordering on worlds having only two truth values tr... |

6 |
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Citation Context ...n's and Zaslavski^i's symmetrizations of intuitionistic logic which kept the disjunction property in the presence of the additional negation. Recent uses for N (Fitting 1991, Pearce and Wansing 1992, =-=Wagner 1991-=-, Wansing 1993) have also been found in the realm of logic programming using explicitly "false" facts. 3.2.2 Models The first model theory for N was developed by Thomason(Thomason 1969), who used a 3-... |