Abstract In this work, we study reduction monads, which are essentially the same as monads relative to the free functor from sets into multigraphs. Reduction monads account for two aspects of the lambda calculus: on the one hand, in the monadic viewpoint, the lambda calculus is an object equipped with a well-behaved substitution; on the other hand, in the graphical viewpoint, it is an oriented multigraph whose vertices are terms and whose edges witness the reductions between two terms. We study presentations of reduction monads. To this end, we propose a notion of reduction signature. As usual, such a signature plays the role of a virtual presentation, and specifies arities for generating operations—possibly subject to equations—together with arities for generating reduction rules. For each such signature, we define a category of models; any model is, in particular, a reduction monad. If the initial object of this category of models exists, we call it the reduction monad presented (or specified) by the given reduction signature. Our main result identifies a class of reduction signatures which specify a reduction monad in the above sense. We show in the examples that our approach covers several standard variants of the lambda calculus.
Reduction Monads and Their Signatures / Benedikt Ahrens; André Hirschowitz; Ambroise Lafont; Marco Maggesi. - In: PROCEEDINGS OF ACM ON PROGRAMMING LANGUAGES. - ISSN 2475-1421. - ELETTRONICO. - 4:(2020), pp. 1-29. [10.1145/3371099]
Reduction Monads and Their Signatures
Benedikt Ahrens;Marco Maggesi
2020
Abstract
Abstract In this work, we study reduction monads, which are essentially the same as monads relative to the free functor from sets into multigraphs. Reduction monads account for two aspects of the lambda calculus: on the one hand, in the monadic viewpoint, the lambda calculus is an object equipped with a well-behaved substitution; on the other hand, in the graphical viewpoint, it is an oriented multigraph whose vertices are terms and whose edges witness the reductions between two terms. We study presentations of reduction monads. To this end, we propose a notion of reduction signature. As usual, such a signature plays the role of a virtual presentation, and specifies arities for generating operations—possibly subject to equations—together with arities for generating reduction rules. For each such signature, we define a category of models; any model is, in particular, a reduction monad. If the initial object of this category of models exists, we call it the reduction monad presented (or specified) by the given reduction signature. Our main result identifies a class of reduction signatures which specify a reduction monad in the above sense. We show in the examples that our approach covers several standard variants of the lambda calculus.File | Dimensione | Formato | |
---|---|---|---|
3371099.pdf
accesso aperto
Tipologia:
Pdf editoriale (Version of record)
Licenza:
Open Access
Dimensione
456.17 kB
Formato
Adobe PDF
|
456.17 kB | Adobe PDF |
I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.