Monads, partial evaluations, and rewriting
| Autor: | Tobias Fritz, Paolo Perrone |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
FOS: Computer and information sciences
Transitive relation Computer Science - Logic in Computer Science Interpretation (logic) General Computer Science Substitution (logic) Mathematics - Category Theory Monad (functional programming) Partial evaluation Logic in Computer Science (cs.LO) Theoretical Computer Science Algebra Mathematics::Category Theory 18C15 18G30 FOS: Mathematics Simplicial set Universal algebra Category Theory (math.CT) Rewriting Mathematics |
| Zdroj: | MFPS |
| Popis: | 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, of which it gives an operational interpretation: the bar construction induces 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. In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction. Originally written for the ACT Adjoint School 2019. To appear in Proceedings of MFPS 2020 |
| Databáze: | OpenAIRE |
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