The differential lambda-mu-calculus Lionel Vaux To cite this version: Lionel Vaux. The differential lambda-mu-calculus. Theoretical Computer Science, Elsevier, 2007, 379 (1-2), pp.166-209. 10.1016/j.tcs.2007.02.028. hal-00349304 HAL Id: hal-00349304 https://hal.archives-ouvertes.fr/hal-00349304 Submitted on 28 Dec 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The differential λµ-calculus Lionel Vaux Institut de Math´ematiquesde Luminy
[email protected] 6th December 2006 Abstract We define a differential λµ-calculus which is an extension of both Parigot’s λµ-calculus and Ehrhard- R´egnier’sdifferential λ-calculus. We prove some basic properties of the system: reduction enjoys Church- Rosser and simply typed terms are strongly normalizing. 1 Introduction Thomas Ehrhard and Laurent R´egniershowed in [ER03] how to extend λ-calculus by means of formal derivatives of λ-terms, following the well-known rules of usual differential calculus. This differential λ-calculus involves a strong relationship between linearity in the logical or computational sense (the argument of a function is used exactly once by this function) and linearity in the usual algebraic sense.