On Explicit Substitutions and Names

machines and sharing.Our presentation of a explicit substitution cal-culus has been driven by the correspondence withthe semantics. Of the calculi discussed above, theoriginal -calculus of Abadi et al. [1] is the onlyone with a closer correspondence to a categoricalsemantics. We see our calculus as a direct improve-ment of theirs. Moreover, in earlier work [14] Rittershowed that every reduction strategy that is usedin abstract machines terminates, thus strong nor-malisation is not necessary for their design. He alsoestablished con uence for terms without metavari-ables, which is all one needs for the design of ab-stract machines. Thus, we are not interested in con-uence for metavariables. Designing an abstractmachine that is easy to prove correct with respectto a well-established semantics was our main goal:for that the calculus in this paper seems the appro-priate one.8 Conclusions and Further WorkWe examined choices for designing calculi with ex-plicit substitutions. We presented our own versionof a calculus of explicit substitutions, for the simplytyped -calculus, for which we showed the equiv-alence between its version arising from semantical(equations-in-context) considerations and syntactic(reduction on raw term) ones. (This equivalence iscrucial in establishing the correspondence betweenthe semantics of the calculus and its implementa-tions.) We discussed its typed and untyped vari-ants and the names and de Bruijn avours of thecalculus. Also we proved all the necessary, stan-dard, properties of our calculus. The proofs arealso standard.This calculus contains what we take to be theessential points of our approach of using categor-ical type theory to inform the implementation ofabstract machines. Ritter's PhD thesis is perhapsa more impressive example of the same approach,dealing with the Calculus of Constructions. Butthe point of the paper is to show how \inevitable"this calculus is, given our original goals. This is tobe contrasted with the multitude of other explicitsubstitution calculi. Also it was necessary to clar-ify the case of the simply typed-lambda-calculus,to modify it appropriately, to deal with the linearlambda-calculus. Linearity introduces several newchallenges that we are tackling at the moment.References[1] Martin Abadi, Luca Cardelli, Pierre-Louis Curien, andJean-Jaques Levy. Explicit substitutions. Journal of Func-tional Programming, 1(4):375{416, 1991.[2] Tom Axford and Mike Joy. Aladin: An abstract machinefor integrating functional and procedural programming.Journal of Programming Languages, 4:63{76, 1996.[3] R. Bloo and K.H. Rose. Preesrvation of strong normalisa-tion in named lambda calculi with explicit substitution andgarbage collection. In Proc. CSN'95|Computer Sciencein the Netherlands, pages 62{72, 1995.[4] P.-L. Curien, Th. Hardin, and J.-J.Levy. Con uence prop-erties of weak and strong calculi of explicit substitutions.Journal of the ACM, 43:362{397, March 1996.[5] Pierre-Louis Curien. An abstract framework for environ-ment machines (Note). Theoretical Computer Science,82(2):389{402, 1991.[6] H. B. Curry and R. Feys. Combinatory logic, volume 1.North Holland, 1958.[7] N.G. De Bruijn. Lambda-calculus notation with name-less dummies, a tool for automatic formula manipulation.Indag. Math, 34:381{392, 1972.[8] Herman Geuvers. Logics and Type Systems. PhD thesis,University of Nijmegen, 1993.[9] Therese Hardin. Con uence results for the pure strong cat-egorical logic CCL. -calculi as subsystems of CCL. The-oretical Computer Science, 65:291{342, 1989.[10] Pierre Lescanne. From to : a journey through cal-culi of explicit substitutions. In 21st ACM Symposiumon Principles of Programming Languages, pages 60{69,Portland, Oregon, 1994.[11] Per Martin-Lof. Intuitionistic Type Theory. Bibliopolis,Napoli, 1984.[12] P.-A. Mellies. Typed -calculi with explicit substitu-tion may not terminate. In M. Dezani-Ciancaglini andG. Plotkin, editors, Proceedings of the Second Conferenceon Typed Lambda Calculi and Applications, pages 328{334. Lecture Notes in Computer Science No. 902, Berlin,Heidelberg, New York, 1995.[13] Eike Ritter. Categorical abstract machines for higher-order lambda calculi. Theoretical Computer Science,136(1):125{162, 1994.[14] Eike Ritter. Normalization for typed lambda calculi withexplicit substitution. In Proceedings of the 1993 AnnualConference of the European Association for ComputerScience Logic, pages 295{304. Lecture Notes in ComputerScience No. 832, Berlin, Heidelberg, New York, 1994.[15] Thomas Streicher. Correctness and Completeness of aCategorical Semantics of the Calculus of Constructions.PhD thesis, Universitat Passau, June 1989.[16] A. Tasistro. Formulation of martin-lof's theory types withexplicit substitutions. Technical report, Chalmers Univer-sity, Dept. of Computer Science, May 1993.12