{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T14:45:40Z","timestamp":1791297940841,"version":"4.1.0"},"reference-count":1,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2016,3,9]],"date-time":"2016-03-09T00:00:00Z","timestamp":1457481600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>Slot and van Emde Boas' weak invariance thesis states that reasonable\nmachines can simulate each other within a polynomially overhead in time. Is\nlambda-calculus a reasonable machine? Is there a way to measure the\ncomputational complexity of a lambda-term? This paper presents the first\ncomplete positive answer to this long-standing problem. Moreover, our answer is\ncompletely machine-independent and based over a standard notion in the theory\nof lambda-calculus: the length of a leftmost-outermost derivation to normal\nform is an invariant cost model. Such a theorem cannot be proved by directly\nrelating lambda-calculus with Turing machines or random access machines,\nbecause of the size explosion problem: there are terms that in a linear number\nof steps produce an exponentially long output. The first step towards the\nsolution is to shift to a notion of evaluation for which the length and the\nsize of the output are linearly related. This is done by adopting the linear\nsubstitution calculus (LSC), a calculus of explicit substitutions modeled after\nlinear logic proof nets and admitting a decomposition of leftmost-outermost\nderivations with the desired property. Thus, the LSC is invariant with respect\nto, say, random access machines. The second step is to show that LSC is\ninvariant with respect to the lambda-calculus. The size explosion problem seems\nto imply that this is not possible: having the same notions of normal form,\nevaluation in the LSC is exponentially longer than in the lambda-calculus. We\nsolve such an impasse by introducing a new form of shared normal form and\nshared reduction, deemed useful. Useful evaluation avoids those steps that only\nunshare the output without contributing to beta-redexes, i.e. the steps that\ncause the blow-up in size. The main technical contribution of the paper is\nindeed the definition of useful reductions and the thorough analysis of their\nproperties.<\/jats:p>","DOI":"10.2168\/lmcs-12(1:4)2016","type":"journal-article","created":{"date-parts":[[2016,11,21]],"date-time":"2016-11-21T13:47:11Z","timestamp":1479736031000},"source":"Crossref","is-referenced-by-count":27,"title":["(Leftmost-Outermost) Beta Reduction is Invariant, Indeed"],"prefix":"10.46298","volume":"Volume 12, Issue 1","author":[{"given":"Beniamino","family":"Accattoli","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9200-070X","authenticated-orcid":false,"given":"Ugo Dal","family":"Lago","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"25203","published-online":{"date-parts":[[2016,3,9]]},"reference":[{"key":"1140:not-found"}],"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/1627\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/1627\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T20:08:22Z","timestamp":1681243702000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/1627"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,9]]},"references-count":1,"URL":"https:\/\/doi.org\/10.2168\/lmcs-12(1:4)2016","relation":{"is-same-as":[{"id-type":"arxiv","id":"1601.01233","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1601.01233","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,3,9]]},"article-number":"1627"}}