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49
Calculating invariants as coreflexive bisimulations
, 2008
"... Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as ..."
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Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants ’ proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as “first class citizens” in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds ’ relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulation and invariants. In this process, we provide an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and their instantiation to the classical ParkMilner definition popular in process algebra.
Coinductive Interpreters for Process Calculi
 In Sixth International Symposium on Functional and Logic Programming, volume 2441 of LNCS
, 2002
"... This paper suggests functional programming languages with coinductive types as suitable devices for prototyping process calculi. The proposed approach is independent of any particular process calculus and makes explicit the dierent ingredients present in the design of any such calculi. In partic ..."
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This paper suggests functional programming languages with coinductive types as suitable devices for prototyping process calculi. The proposed approach is independent of any particular process calculus and makes explicit the dierent ingredients present in the design of any such calculi. In particular structural aspects of the underlying behaviour model become clearly separated from the interaction structure which de nes the synchronisation discipline. The approach is illustrated by the detailed development in Charity of an interpreter for a family of process languages.
Covariant Types
 Theoretical Computer Science
, 1997
"... The covariant type system is an impredicative system that is rich enough to represent some polymorphism on inductive types, such as lists and trees, and yet is simple enough to have a settheoretic semantics. Its chief novelty is to replace function types by transformation types, which denote parame ..."
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The covariant type system is an impredicative system that is rich enough to represent some polymorphism on inductive types, such as lists and trees, and yet is simple enough to have a settheoretic semantics. Its chief novelty is to replace function types by transformation types, which denote parametric functions. Their free type variables are all in positive positions, and so can be modelled by covariant functors. Similarly, terms denote natural transformations. There is a translation from the covariant type system to system F which preserves nontrivial reductions. It follows that covariant reduction is strongly normalising and confluent. This work suggests a new approach to the semantics of system F, and new ways of basing type systems on the categorical notions of functor and natural transformation. Keywords covariant types, polymorphism, parametricity, transformation types. 1 Introduction The pros and cons of typing programs are already well known. In brief, static typecheckin...
A.,Some problems and results in synthetic functional analysis
 in Category Theoretic Methods in Geometry, Proceedings Aarhus 1983, Aarhus Various Publication Series No
, 1983
"... This somewhat tentative note aims at making a status about “functional analysis ” in certain ringed toposes E, R, in particular, duality theory for Rmodules in E. This is an area where knowledge still seems fragmentary, but where some of the known facts indicate the importance of such theory. The f ..."
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This somewhat tentative note aims at making a status about “functional analysis ” in certain ringed toposes E, R, in particular, duality theory for Rmodules in E. This is an area where knowledge still seems fragmentary, but where some of the known facts indicate the importance of such theory. The first §, however, does not deal with specific toposes, but is of very general nature, making contact with the general theory of monads on closed categories, [3][6]. Most of the results and problems in this note were presented at the workshop. Closely related viewpoints were contained in Lawvere’s workshopcontribution on Intensive and Extensive Quantities, which have also influenced the presentation given in the following pages. 1 Restricted double dualization monads Let R be a ring in a topos E and let RMod denote the Ecategory of Rmodules (left Rmodules, say). The Evalued homfunctor for it is denoted Hom R(−, −). If X ∈ E and V ∈ RMod, there is a natural structure of Rmodule on V X = ΠXV, and for any U ∈ RMod Hom R(U, V X) ∼ = (Hom R(U, V)) X, naturally in X, U, and V. This expresses that V X is a cotensor of V with X (cf. e.g. [2]). Equivalently, it expresses that, for fixed V, we have a Estrong left adjoint V (−) : E → (RMod) op to the functor Hom R (−, V). Thus we
A Semantic Formulation of ⊤⊤lifting and Logical Predicates for Computational Metalanguage
 In Proc. CSL 2005. LNCS 3634
, 2005
"... Abstract. A semantic formulation of Lindley and Stark’s ⊤⊤lifting is given. We first illustrate our semantic formulation of the ⊤⊤lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤⊤lifting as the lif ..."
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Abstract. A semantic formulation of Lindley and Stark’s ⊤⊤lifting is given. We first illustrate our semantic formulation of the ⊤⊤lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤⊤lifting as the lifting of strong monads across bifibrations with lifted symmetric monoidal closed structures. 1
Matrices, Monads and the Fast Fourier Transform
 Proceedings of the
, 1993
"... This paper presents a formal semantics for vectors and matrices, suitable for static typechecking. This is not available in apl, which produces runtime type errors, or in the usual functional languages, where matrices are typically implemented by lists of lists. Here, a matrix is a vector of vecto ..."
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This paper presents a formal semantics for vectors and matrices, suitable for static typechecking. This is not available in apl, which produces runtime type errors, or in the usual functional languages, where matrices are typically implemented by lists of lists. Here, a matrix is a vector of vectors. Vectors are distinguished from lists by requiring that vector computations determine the length of the result from that of the argument, without reference to values. This leads to a twolevel semantics, with values above and shapes below. Each operation must then specify its action on shapes as well as its action on values. Vectors and matrices inherit much of their structure from lists. In particular, the monadic structure given by singleton lists and the flattening of lists of lists extends in this way. Some new constructions, such as transposition of matrices, have no list counterpart. The power of this calculus for vector and matrix algebra is sufficient to represent the discrete Fou...
Some Calculus With Extensive Quantities: Wave Equation
, 2003
"... We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation. 1. ..."
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We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation. 1.
MONAD COMPOSITIONS I: GENERAL CONSTRUCTIONS AND RECURSIVE DISTRIBUTIVE LAWS
"... ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursivelydefined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad ..."
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ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursivelydefined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad approximations for compositions which fail to be a monad. 1.
COMMUTATIVE MONADS AS A THEORY OF DISTRIBUTIONS
"... Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects cons ..."
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Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability.
Reasoning about meaning in natural language with compact closed categories and frobenius algebras
 Logic and Algebraic Structures in Quantum Computing and Information, Association for Symbolic Logic Lecture Notes in Logic
, 2013
"... Compact closed categories have found applications in modeling quantum information protocols by AbramskyCoecke. They also provide semantics for Lambek’s pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning ..."
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Compact closed categories have found applications in modeling quantum information protocols by AbramskyCoecke. They also provide semantics for Lambek’s pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Specifically, in previous work CoeckeClarkSadrzadeh used the product category of pregroups with vector spaces and provided a distributional model of meaning for sentences. We recast this theory in terms of strongly monoidal functors and advance it via Frobenius algebras over vector spaces. The former are used to formalize topological quantum field theories by Atiyah and BaezDolan, and the latter are used to model classical data in quantum protocols by CoeckePavlovicVicary. The Frobenius algebras enable us to work in a single space in which meanings of words, phrases, and sentences of any structure live. Hence we can compare meanings of different language constructs and enhance the applicability of the theory. We report on experimental results on a number of language tasks and verify the theoretical predictions. 1