{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T11:46:14Z","timestamp":1773143174028,"version":"3.50.1"},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8777,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:p>This paper is a contribution to the following natural problem in complexity theory:<\/jats:p><jats:p>(*) <jats:italic>Is there a complexity theory for isomorphism types of recursive countable relational structures<\/jats:italic>? I.e. given a recursive relational structure \u211b over the set <jats:bold>N<\/jats:bold> of nonnegative integers, is there a nontrivial lower bound for the time-space complexity of recursive structures isomorphic (resp. recursively isomorphic) to \u211b?<\/jats:p><jats:p>For unary recursive relations <jats:italic>R<\/jats:italic>, the answer is trivially negative: either <jats:italic>R<\/jats:italic> is finite or coinfinite or \u3008<jats:bold>N<\/jats:bold>, <jats:italic>R<\/jats:italic>\u3009 is recursively isomorphic to \u3008<jats:bold>N<\/jats:bold>, {<jats:italic>x<\/jats:italic> \u03f5 <jats:bold>N<\/jats:bold>: <jats:italic>x<\/jats:italic> is even}\u3009.<\/jats:p><jats:p>The general problem for relations with arity 2 (or greater) is open.<\/jats:p><jats:p>Related to this problem, a classical result (going back to S. C. Kleene [4], 1955) states that every recursive ordinal is in fact primitive recursive.<\/jats:p><jats:p>In [3] Patrick Dehornoy, using methods relevant to computer science, improves this result, showing that every recursive ordinal can be represented by a recursive total ordering over <jats:bold>N<\/jats:bold> which has linear deterministic time complexity relative to the binary representation of integers. As he notices, his proof applies to every recursive total order type <jats:italic>\u03b1<\/jats:italic> such that the isomorphism type of <jats:italic>\u03b1<\/jats:italic> is not changed if points are replaced by arbitrary finite nonempty subsets of consecutive points.<\/jats:p><jats:p>In this paper we extend Dehornoy's result to <jats:italic>all<\/jats:italic> recursive total orderings over <jats:bold>N<\/jats:bold> and get minimal complexity for both time and space simultaneously.<\/jats:p>","DOI":"10.2307\/2274966","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:33:48Z","timestamp":1146954828000},"page":"260-276","source":"Crossref","is-referenced-by-count":36,"title":["Every recursive linear ordering has a copy in DTIME-SPACE(<i>n<\/i>,log(<i>n<\/i>))"],"prefix":"10.1017","volume":"55","author":[{"given":"Serge","family":"Grigorieff","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026554_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1981-0624937-1"},{"key":"S0022481200026554_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/2372632"},{"key":"S0022481200026554_ref006","first-page":"572","volume":"46","author":"Remmel","year":"1981","journal-title":"Recursive isomorphism types of recursive Boolean algebras"},{"key":"S0022481200026554_ref011","volume-title":"Computational complexity","author":"Wagner","year":"1986"},{"key":"S0022481200026554_ref001","unstructured":"Cenzer Douglas and Remmel Jeffrey , Polynomial-time complexity of models (to appear)."},{"key":"S0022481200026554_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01746527"},{"key":"S0022481200026554_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90028-9"},{"key":"S0022481200026554_ref008","volume-title":"Linear orderings","author":"Rosenstein","year":"1982"},{"key":"S0022481200026554_ref009","first-page":"179","volume-title":"Proceedings of the IEEE conference on switching, circuit theory and logical design","author":"Stearns","year":"1965"},{"key":"S0022481200026554_ref010","first-page":"563","volume":"49","author":"Watnick","year":"1984","journal-title":"A generalization of Tennenbaum's theorem on effectively finite recursive linear orderings"},{"key":"S0022481200026554_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(86)90130-4"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026554","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:51:52Z","timestamp":1558216312000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026554\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026554"],"URL":"https:\/\/doi.org\/10.2307\/2274966","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}