Results 1 
9 of
9
Complete Axioms for Categorical Fixedpoint Operators
 In Proceedings of 15th Annual Symposium on Logic in Computer Science
, 2000
"... We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the fre ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
(Show Context)
We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixedpoint operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !continuous functions between !complete pointed partial orders. This possesses a leastfixedpoint oper...
The maximality of the typed lambda calculus and of cartesian closed categories
 Publ. Inst. Math. (N.S
"... From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
(Show Context)
From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.
Linear realizability and full completeness for typed lambda calculi
 Annals of Pure and Applied Logic
, 2005
"... We present the model construction technique called Linear Realizability. It consists in building a category of Partial Equivalence Relations over a Linear Combinatory Algebra. We illustrate how it can be used to provide models, which are fully complete for various typed λcalculi. In particular, we ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
We present the model construction technique called Linear Realizability. It consists in building a category of Partial Equivalence Relations over a Linear Combinatory Algebra. We illustrate how it can be used to provide models, which are fully complete for various typed λcalculi. In particular, we focus on special Linear Combinatory Algebras of partial involutions, and we present PER models over them which are fully complete, inter alia, w.r.t. the following languages and theories: the fragment of System F consisting of MLtypes, the maximal theory on the simply typed λcalculus with finitely many ground constants, and the maximal theory on an infinitary version of this latter calculus. Key words: Typed lambdacalculi, MLpolymorphic types, linear logic, hyperdoctrines, PER models, Geometry of Interaction, (axiomatic) full completeness
Weak Typed Böhm Theorem on IMLL
 Annals of Pure and Applied Logic
, 2007
"... In the Böhm theorem workshop on Crete island, Zoran Petric called Statman’s “Typical Ambiguity theorem ” typed Böhm theorem. Moreover, he gave a new proof of the theorem based on settheoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
In the Böhm theorem workshop on Crete island, Zoran Petric called Statman’s “Typical Ambiguity theorem ” typed Böhm theorem. Moreover, he gave a new proof of the theorem based on settheoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment of intuitionistic linear logic without the multiplicative unit 1 (for short IMLL) weak typed Böhm theorem holds. The system IMLL exactly corresponds to the linear lambda calculus without exponentials, additives and logical constants. The system IMLL also exactly corresponds to the free symmetric monoidal closed category without the unit object. As far as we know, our separation result is the first one with regard to these systems in a purely syntactical manner. 1
An arithmetical proof of the strong normalization for the λcalculus with recursive equations on types
, 2009
"... ..."
An arithmetical proof of the strong normalization for the λcalculus with recursive equations on types
, 2007
"... ..."
and
"... We study a weakening of the notion of logical relations, called prelogical relations, that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of ..."
Abstract
 Add to MetaCart
We study a weakening of the notion of logical relations, called prelogical relations, that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of logical relations to hold only for pairs of functions that are expressible by the same lambda term. Prelogical relations are the minimal weakening of logical relations that gives composability for extensional structures and simultaneously the most liberal definition that gives the Basic Lemma. Prelogical predicates (i.e., unary prelogical relations) coincide with sets that are invariant under Kripke logical relations with varying arity as introduced by Jung and Tiuryn, and prelogical relations are the closure under projection and intersection of logical relations. These conceptually independent characterizations of prelogical relations suggest that the concept is rather intrinsic and robust. The use of prelogical relations gives an improved version of Mitchell’s representation independence theorem which characterizes observational equivalence for all signatures rather than just for firstorder signatures. Prelogical relations can be used in place of logical relations to give an account of data refinement where the fact that prelogical relations compose explains why stepwise refinement is sound.