{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T03:40:41Z","timestamp":1777347641403,"version":"3.51.4"},"reference-count":30,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5671,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate the possibility to characterize (multi)functions that are<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline1\"\/>-definable with small<jats:italic>i<\/jats:italic>(<jats:italic>i<\/jats:italic>= 1, 2, 3) in fragments of bounded arithmetic<jats:italic>T<\/jats:italic><jats:sub>2<\/jats:sub>in terms of natural search problems defined over polynomial-time structures. We obtain the following results:<\/jats:p><jats:p>(1) A reformulation of known characterizations of (multi)functions that are<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline2\"\/>and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline3\"\/>-definable in the theories<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline4\"\/>and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline5\"\/>.<\/jats:p><jats:p>(2) New characterizations of (multi)functions that are<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline3\"\/>and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline6\"\/>-definable in the theory<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline7\"\/>.<\/jats:p><jats:p>(3) A new non-conservation result: the theory<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline8\"\/>is not<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline9\"\/>-conservative over the theory<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline10\"\/>.<\/jats:p><jats:p>To prove that the theory<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline8\"\/>is not<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline9\"\/>-conservative over the theory<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline10\"\/>, we present two examples of a<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200014766_inline11\"\/>-principle separating the two theories:<\/jats:p><jats:p>(a) the weak pigeonhole principle WPHP(<jats:italic>a<\/jats:italic><jats:sup>2<\/jats:sup>,<jats:italic>f, g<\/jats:italic>) formalizing that no function<jats:italic>f<\/jats:italic>is a bijection between<jats:italic>a<\/jats:italic><jats:sup>2<\/jats:sup>and<jats:italic>a<\/jats:italic>with the inverse<jats:italic>g<\/jats:italic>,<\/jats:p><jats:p>(b) the iteration principle Iter(<jats:italic>a, R, f<\/jats:italic>) formalizing that no function<jats:italic>f<\/jats:italic>defined on a strict partial order ({0,\u2026, a},<jats:italic>R<\/jats:italic>) can have increasing iterates.<\/jats:p>","DOI":"10.2307\/2586729","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:28Z","timestamp":1146938488000},"page":"1095-1115","source":"Crossref","is-referenced-by-count":17,"title":["Witnessing functions in bounded arithmetic and search 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