Computational effects may often be interpreted in the Kleisli category of a monad or in the coKleisli category of a comonad. The duality between monads and comonads corresponds, in general, to a symmetry between construction and observation, for inst
ance between raising an exception and looking up a state. Thanks to the properties of adjunction one may go one step further: the coKleisli-on-Kleisli category of a monad provides a kind of observation with respect to a given construction, while dually the Kleisli-on-coKleisli category of a comonad provides a kind of construction with respect to a given observation. In the previous examples this gives rise to catching an exception and updating a state. However, the interpretation of computational effects is usually based on a category which is not self-dual, like the category of sets. This leads to a breaking of the monad-comonad duality. For instance, in a distributive category the state effect has much better properties than the exception effect. This remark provides a novel point of view on the usual mechanism for handling exceptions. The aim of this paper is to build an equational semantics for handling exceptions based on the coKleisli-on-Kleisli category of the monad of exceptions. We focus on n-ary functions and conditionals. We propose a programmers language for exceptions and we prove that it has the required behaviour with respect to n-ary functions and conditionals.
This note is about using computational effects for scalability. With this method, the specification gets more and more complex while its semantics gets more and more correct. We show, from two fundamental examples, that it is possible to design a ded
uction system for a specification involving an effect without expliciting this effect.
An algebraic method is used to study the semantics of exceptions in computer languages. The exceptions form a computational effect, in the sense that there is an apparent mismatch between the syntax of exceptions and their intended semantics. We solv
e this apparent contradiction by efining a logic for exceptions with a proof system which is close to their syntax and where their intended semantics can be seen as a model. This requires a robust framework for logics and their morphisms, which is provided by categorical tools relying on adjunctions, fractions and limit sketches.
We define a proof system for exceptions which is close to the syntax for exceptions, in the sense that the exceptions do not appear explicitly in the type of any expression. This proof system is sound with respect to the intended denotational semanti
cs of exceptions. With this inference system we prove several properties of exceptions.
In this short note we study the semantics of two basic computational effects, exceptions and states, from a new point of view. In the handling of exceptions we dissociate the control from the elementary operation which recovers from the exception. In
this way it becomes apparent that there is a duality, in the categorical sense, between exceptions and states.
Deduction systems and graph rewriting systems are compared within a common categorical framework. This leads to an improved deduction method in diagrammatic logics.
In this paper we consider the two major computational effects of states and exceptions, from the point of view of diagrammatic logics. We get a surprising result: there exists a symmetry between these two effects, based on the well-known categorical
duality between products and coproducts. More precisely, the lookup and update operations for states are respectively dual to the throw and catch operations for exceptions. This symmetry is deeply hidden in the programming languages; in order to unveil it, we start from the monoidal equational logic and we add progressively the logical features which are necessary for dealing with either effect. This approach gives rise to a new point of view on states and exceptions, which bypasses the problems due to the non-algebraicity of handling exceptions.
Diagrammatic logics were introduced in 2002, with emphasis on the notions of specifications and models. In this paper we improve the description of the inference process, which is seen as a Yoneda functor on a bicategory of fractions. A diagrammatic
logic is defined from a morphism of limit sketches (called a propagator) which gives rise to an adjunction, which in turn determines a bicategory of fractions. The propagator, the adjunction and the bicategory provide respectively the syntax, the models and the inference process for the logic. Then diagrammatic logics and their morphisms are applied to the semantics of side effects in computer languages.
This paper is a submission to the contest: How to combine logics? at the World Congress and School on Universal Logic III, 2010. We claim that combining things, whatever these things are, is made easier if these things can be seen as the objects of a
category. We define the category of diagrammatic logics, so that categorical constructions can be used for combining diagrammatic logics. As an example, a combination of logics using an opfibration is presented, in order to study computational side-effects due to the evolution of the state during the execution of an imperative program.
