{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T05:16:22Z","timestamp":1649049382456},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7041,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,12]]},"abstract":"<jats:p>A proof-theoretic characterization of the primitive recursive functions is the \u03a3<jats:sub>1<\/jats:sub>-definable functions in <jats:italic>I<\/jats:italic>\u03a3<jats:sub>1<\/jats:sub> as is shown in Mints [4], Parsons [5], and [8].<\/jats:p><jats:p>Then what is a proof-theoretic characterization of Grzegorzyk's hierarchy? First we discuss a related previous work. In Clote and Takeuti [2], we introduced a theory TAC that corresponds to the computational complexity class AC. TAC has a very weak form of induction. We assign a rank to a proof in TAC in the following way. The rank of a proof <jats:italic>P<\/jats:italic> in TAC is the nesting number of inductions used in <jats:italic>P<\/jats:italic>. Then TAC<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup> is defined to be the subtheory of TAC whose proof has a rank \u2264 <jats:italic>i<\/jats:italic>. We proved that TAC<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup> corresponds to the class AC<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup>.<\/jats:p><jats:p>In this paper we introduce a theory <jats:italic>Iep<\/jats:italic>\u03a3<jats:sub>1<\/jats:sub> which is equivalent to <jats:italic>I<\/jats:italic>\u03a3<jats:sub>1<\/jats:sub>. Then we define the rank of a proof in <jats:italic>Iep<\/jats:italic>\u03a3<jats:sub>1<\/jats:sub> as the nesting number of inductions in the proof and prove that the proofs with rank \u2264 <jats:italic>i<\/jats:italic> correspond to Grzegorcyk's hierarchy <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019277_inline1\" \/> for <jats:italic>i<\/jats:italic> &gt; 0.<\/jats:p><jats:p>We also prove that the system that has proofs with rank 0 is actually equivalent to <jats:italic>I<\/jats:italic> \u0394<jats:sub>0<\/jats:sub>. These facts are interesting since it is proved in [10] that the theory isomorphic to TAC<jats:sup>\u2218<\/jats:sup> by <jats:italic>RSUV<\/jats:italic> isomorphism is a conservative extension of <jats:italic>I<\/jats:italic> \u0394<jats:sub><jats:italic>o<\/jats:italic><\/jats:sub>. Therefore there is some analogy between the class AC and the primitive recursive functions.<\/jats:p>","DOI":"10.2307\/2275705","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:53:29Z","timestamp":1146941609000},"page":"1274-1284","source":"Crossref","is-referenced-by-count":4,"title":["Grzegorcyk's hierarchy and <i>Iep<\/i>\u03a3<sub>1<\/sub>"],"prefix":"10.1017","volume":"59","author":[{"given":"Gaisi","family":"Takeuti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019277_ref009","first-page":"364","volume-title":"Proof theory and computational complexity","author":"Takeuti","year":"1993"},{"key":"S0022481200019277_ref010","unstructured":"Takeuti G. , RSUV isomorphisms for TAC i , TNC i and TLS (to appear)."},{"key":"S0022481200019277_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22156-3"},{"key":"S0022481200019277_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01117472"},{"key":"S0022481200019277_ref001","volume-title":"Bibliopolis","author":"Buss","year":"1986"},{"key":"S0022481200019277_ref002","unstructured":"Clote P. and Takeuti G. , First order bounded arithmetic and small boolean circuit complexity classes (to appear)."},{"key":"S0022481200019277_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70771-7"},{"key":"S0022481200019277_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/2269958"},{"key":"S0022481200019277_ref008","volume-title":"Proof Theory","author":"Takeuti","year":"1975"},{"key":"S0022481200019277_ref007","volume-title":"Subrecursion: functions and hierachies","author":"Rose","year":"1984"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200019277","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T20:41:23Z","timestamp":1557866483000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019277\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12]]},"references-count":10,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1994,12]]}},"alternative-id":["S0022481200019277"],"URL":"https:\/\/doi.org\/10.2307\/2275705","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,12]]}}}