Article

States and exceptions considered as dual effects

01/2010;
Source: arXiv

ABSTRACT 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.

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    ABSTRACT: Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin's mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960's, and how the renewed interest in them in a computer science setting might develop in future.
    Electr. Notes Theor. Comput. Sci. 01/2007; 172:437-458.

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