Results 1 
5 of
5
New Foundations for Fixpoint Computations
, 1990
"... This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of MartinLof's `iteration type' [11]. The type system enforces a separation of comput ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of MartinLof's `iteration type' [11]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a categorytheoretic version of the `logical relations' method. 1 Computation types It is well known that primitive recursion at higher types can be given a categorical characterisation in terms of Lawvere's concept of natural number object [6]. We show that a similar characterisation can be given for general recursion via fixpoint operators of higher types, in terms of a new conceptthat of a fixpoint object in ...
On Fixpoint Objects and Gluing Constructions
, 1997
"... This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a letcategory possessing a fixpoint object. Functional completeness for such categories is developed. We also prove that categories with fixpoint operators do not necessarily have a fixpoint object. In the second part, we extend Freyd's gluing construction for cartesian closed categories to cartesian closed letcategories, and observe that this extension does not obviously apply to categories possessing fixpoint objects. We solve this problem by giving a new gluing construction for a limited class of categories with fixpoint objects; this is the main result of the paper. We use this categorytheoretic construction to prove a typetheoretic conservative extension result. A version of this pap...
New Foundations for Fixpoint Computations: FIXHyperdoctrines and the FIXLogic
"... This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [Mog 89] and contains a version of Martin Löf’s ‘iteration type ’ [MarL 83]. The type system enforces a separation of co ..."
Abstract
 Add to MetaCart
This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [Mog 89] and contains a version of Martin Löf’s ‘iteration type ’ [MarL 83]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a categorytheoretic version of the ‘logical relations ’ method [Plo 85]. 2 1
New Foundations for Fixpoint Computations
"... This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of MartinLof’s ‘iteration type ’ [ll]. The type system enforces a separation of compu ..."
Abstract
 Add to MetaCart
This paper introduces a new higherorder typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of MartinLof’s ‘iteration type ’ [ll]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a categorytheoretic version of the ‘logical relations ’ method. 1 Computation types It is well known that primitive recursion at higher types can be given a categorical characterisation in terms of Lawvere’s concept of natural number object [6]. We show that a similar characterisation can be given for general recursion via fixpoint operators of higher types, in terms of a new conceptthat of a fixpoint object in a suitably