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36
Calculating invariants as coreflexive bisimulations
, 2008
"... Abstract. 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 pred ..."
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Abstract. 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.
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...
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
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.
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.
Equational Systems and Free Constructions (Extended Abstract)
"... Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specif ..."
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Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specifically to the theory of substitution in the presence of variable binding and to models of namepassing process calculi. 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.
DISTRIBUTIVE LAWS IN PROGRAMMING STRUCTURES
, 2009
"... Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approac ..."
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Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2naturality and Gray’s tensor product of 2categories. It generalises the existing more specific notions of distributive
UML Model Refactoring as Refinement: A Coalgebraic Perspective
"... Abstract—Although increasingly popular, Model Driven Architecture (MDA) still lacks suitable formal foundations on top of which rigorous methodologies for the description, analysis and transformation of models could be built. This paper aims to contribute in this direction: building on previous work ..."
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Abstract—Although increasingly popular, Model Driven Architecture (MDA) still lacks suitable formal foundations on top of which rigorous methodologies for the description, analysis and transformation of models could be built. This paper aims to contribute in this direction: building on previous work by the authors on coalgebraic refinement for software components and architectures, it discusses refactoring of models within a coalgebraic semantic framework. Architectures are defined through aggregation based on a coalgebraic semantics for (subsets of) UML. On the other hand, such aggregations, no matter how large and complex they are, can always be dealt with as coalgebras themselves. This paves the way to a discipline of models’ transformations which, being invariant under either behavioural equivalence or refinement, are able to formally capture a large number of refactoring patterns. The main ideas underlying this research are presented through a detailed example in the context of refactoring of UML class diagrams. I.