Combining effects: sum and tensor

  • Authors:

  • Martin Hyland;Gordon Plotkin;John Power

  • Affiliations:

  • Department of Mathematics, University of Cambridge, Cambridge, UK;Laboratory for the Foundations of Computer Science, School of Informatics, University of Edinburgh, Edinburgh, Scotland, UK;Laboratory for the Foundations of Computer Science, School of Informatics, University of Edinburgh, Edinburgh, Scotland, UK

  • Venue:
  • Theoretical Computer Science - Clifford lectures and the mathematical foundations of programming semantics
  • Year:
  • 2006

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Abstract

We seek a unified account of modularity for computational effects. We begin by reformulating Moggi's monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi's exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi's side-effects monad transformer. Finally, we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.