Keep definition, change category  A practical approach to state-based system calculi
Faced with the need to quantify software (un)reliability in the presence of faults, the semantics of state-based systems is urged to evolve towards quantified (e.g. probabilistic) nondeterminism. When one is approaching such semantics from a categorical perspective, this inevitably calls for some technical elaboration, in a monadic setting. This paper proposes that such an evolution be undertaken without sacrificing the simplicity of the original (qualitative) definitions, by keeping quantification implicit rather than explicit. The approach is a monad lifting strategy whereby, under some conditions, definitions can be preserved provided the semantics moves to another category. The technique is illustrated by showing how to introduce probabilism in an existing software component calculus, by moving to a suitable category of matrices and using linear algebra in the reasoning. The paper also addresses the problem of preserving monadic strength in the move from original to target (Kleisli) categories, a topic which bears relationship to recent studies in categorial physics.
linear algebra mealy automata kleisli category relation algebra software calculi