Results 21  30
of
35
Equational lifting monads
 Proceedings CTCS '99, Electronic Notes in Computer Science
, 1999
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the (partial) equational properties of partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of the Kleisli categories of equational lifting monads as abstract Kleisli categories with extra structure. This axiomatization is of interest in its own right. 1
Synthetic Domain Theory in Type Theory: Another Logic of Computable Functions
 In Proceedings of TPHOL
, 1996
"... Abstract. We will present a Logic of Computable Functions based on the idea of Synthetic Domain Theory such that all functions are automatically continuous. Its implementation in the Lego proofchecker – the logic is formalized on top of the Extended Calculus of Constructions – has two main advantag ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. We will present a Logic of Computable Functions based on the idea of Synthetic Domain Theory such that all functions are automatically continuous. Its implementation in the Lego proofchecker – the logic is formalized on top of the Extended Calculus of Constructions – has two main advantages. First, one gets machine checked proofs verifying that the chosen logical presentation of Synthetic Domain Theory is correct. Second, it gives rise to a LCFlike theory for verification of functional programs where continuity proofs are obsolete. Because of the powerful type theory even modular programs and specifications can be coded such that one gets a prototype setting for modular software verification and development. 1
Metric Spaces in Synthetic Topology
, 2010
"... We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and me ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree. 1
Sheaf Toposes for Realizability
, 2001
"... We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part o ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott. 1
Partial Hyperdoctrines: Categorical Models for Partial Function Logic and Hoare Logic
, 1993
"... this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important ne ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important new aspect of the resulting partial (first order) hyperdoctrine is a different notion of morphism between the fibres. We also use this partial hyperdoctrine to give a model for Beeson's Partial Function Logic such that (a version of) his axiomatization is complete w.r.t. this model. This shows the usefulness of the notion independent of its intended use as a model for Hoare logic. 1. Introduction
Wellfoundedness in Realizability
, 2005
"... Introduction Let < be a binary relation on a set X. In ZFC, the following three statementsare equivalent: 1) There are no infinite <descending sequences in X: i.e. no sequences( ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Introduction Let < be a binary relation on a set X. In ZFC, the following three statementsare equivalent: 1) There are no infinite <descending sequences in X: i.e. no sequences(
A Survey of Categorical Computation: Fixed Points, . . .
, 1990
"... Machine by Curien [Cur86]. It is based upon a weak categorical combinatory logic, viz. lacking surjective pairing and extensionality, that arose as a direct semantictosyntactic translation of the lambda calculus of tuples. The computational mode was combinator term reduction through rewriting usin ..."
Abstract
 Add to MetaCart
Machine by Curien [Cur86]. It is based upon a weak categorical combinatory logic, viz. lacking surjective pairing and extensionality, that arose as a direct semantictosyntactic translation of the lambda calculus of tuples. The computational mode was combinator term reduction through rewriting using a direct lefttoright parse algorithm, initially making the evaluation strategy inefficiently eager 1 . Application is therefore simply juxtaposition, losing the full expressiveness ofreduction that computes via substitution. Its overly strong bias towards the lambda calculus was another factor that limited its expressiveness. On one hand the CAM demanded the existence of categorical products but on the other it had no coproducts for developing many useful data structures. Nevertheless, the high acceptance and efficiency of the CAMbased ML compiler, CAML, gives significant encouragement towards developing a highlyprogrammable categorical computing paradigm. Some prominent workers in ...
Domains in H
"... We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Int ..."
Abstract
 Add to MetaCart
We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domainlike sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a settheoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...
A Presentation Of The Initial LiftAlgebra
 Journal of Pure and Applied Algebra
, 1997
"... The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the success ..."
Abstract
 Add to MetaCart
The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1 : N ! N, arbitrary operations su : N u ! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set.
Part II Local Realizability Toposes and a Modal Logic for
"... 5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: ..."
Abstract
 Add to MetaCart
5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: