Results 1 
3 of
3
Notions of Computation and Monads
, 1991
"... The i.calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with Iterms. However, if one goes further and uses bnconversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with ..."
Abstract

Cited by 734 (15 self)
 Add to MetaCart
The i.calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with Iterms. However, if one goes further and uses bnconversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from calues to values) that may jeopardise the applicability of theoretical results, In this paper we introduce calculi. based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs for a wide range of notions of computation.
Interpolation in Grothendieck Institutions
 THEORETICAL COMPUTER SCIENCE
, 2003
"... It is well known that interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which ..."
Abstract

Cited by 25 (3 self)
 Add to MetaCart
It is well known that interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which have recently emerged as an important mathematical structure underlying heterogenous multilogic specification. Our main result can be used in the applications in several different ways. It can be used to establish interpolation properties for multilogic Grothendieck institutions, but also to lift interpolation properties from unsorted logics to their many sorted variants. The importance of the latter resides in the fact that, unlike other structural properties of logics, many sorted interpolation is a nontrivial generalisation of unsorted interpolation. The concepts, results, and the applications discussed in this paper are illustrated with several examples from conventional logic and algebraic specification theory.
2008) Classical and quantum structures
"... In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to Frobenius algebras with some additional properties. They express the distinguishing capabilities of classical data: in contrast with quantum data, classical data can be copied and deleted. The algebraic approach thus shifts the paradigm of ”quantization ” of a classical theory to ”classicization ” of a quantum theory. Remarkably, the simple SDM framework suffices not only for this conceptual shift, but even allows us to distinguish the deterministic classical operations (i.e. functions) from the nondeterministic classical operations (i.e. relations), and the probabilistic classical operations (stochastic maps). Moreover, a combination of some basic categorical constructions (due to Kleisli, resp. Grothendieck) with the categorical presentations of quantum states, provides a resource sensitive account of various quantumclassical interactions: of classical control of quantum data, of classical data arising from quantum measurements, as well as of the classical data processing inbetween controls and measurements. A salient feature here is the graphical calculus for categorical quantum mechanics, which allows a purely diagrammatic representation of classicalquantum interaction. 1