Monads, Partial Evaluations, and Rewriting

Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of “evaluating an expression partially”: for example, “2+3” can be obtained as a partial evaluation of “2+2+1”. This construction can be given for any monad, and it is linked to the famous bar construction [Saunders Mac Lane, Categories for the Working Mathematician, Springer, 2000, VII.6], of which it gives an operational interpretation: the bar construction is a simplicial set, and its 1-cells are partial evaluations.

We study the properties of partial evaluations for general monads. We prove that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which we give an interpretation in terms of substitution of terms.

For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables, and partial evaluation relation is known as second-order stochastic dominance.

In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.