{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T08:10:20Z","timestamp":1707034220036},"reference-count":5,"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>Let<jats:italic>L<jats:sub>0<\/jats:sub><\/jats:italic>be a language on \u2115 consisting of 0, 1, +, \u2238, \u00b7, \u230a\u00bd<jats:italic>a<\/jats:italic>\u230b, \u2223<jats:italic>a<\/jats:italic>\u2223, #, \u2227(<jats:italic>a,b<\/jats:italic>), \u2228(<jats:italic>a,b<\/jats:italic>), \u00ac(<jats:italic>a<\/jats:italic>), \u2264 (<jats:italic>a,b<\/jats:italic>), and<jats:italic>\u03bcx<\/jats:italic>\u2264 \u2223<jats:italic>s<\/jats:italic>\u2223<jats:italic>t<\/jats:italic>(<jats:italic>x<\/jats:italic>). Here<jats:italic>\u03bcx<\/jats:italic>\u2264 \u2223<jats:italic>s<\/jats:italic>\u2223<jats:italic>t<\/jats:italic>(<jats:italic>x<\/jats:italic>) is the smallest number<jats:italic>x<\/jats:italic>\u2264 \u2223<jats:italic>s<\/jats:italic>\u2223 satisfying<jats:italic>t<\/jats:italic>(<jats:italic>x<\/jats:italic>) &gt; 0 and 0 if there exist no such<jats:italic>x<\/jats:italic>and we stipulate that if<jats:italic>s<\/jats:italic>and<jats:italic>t<\/jats:italic>(<jats:italic>a<\/jats:italic>) are terms in<jats:italic>L<jats:sub>o<\/jats:sub><\/jats:italic>, then<jats:italic>\u03bcx<\/jats:italic>\u2264 \u2223<jats:italic>s<\/jats:italic>\u2223<jats:italic>t<\/jats:italic>(<jats:italic>x<\/jats:italic>) is also a term in<jats:italic>L<jats:sub>o<\/jats:sub><\/jats:italic>. The defining axioms of functions \u2227(<jats:italic>a,b<\/jats:italic>), \u2228(<jats:italic>a,b<\/jats:italic>), \u00ac(<jats:italic>a<\/jats:italic>), \u2264 (<jats:italic>a,b<\/jats:italic>) are as follows:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Let<jats:italic>L<\/jats:italic>a language on \u2115 with only predicate constant = and<jats:italic>L<jats:sub>0<\/jats:sub><\/jats:italic>\u2286<jats:italic>L<\/jats:italic>. Let<jats:italic>f<\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>a<jats:sub>m<\/jats:sub><\/jats:italic>) be a function for \u2115<jats:sup><jats:italic>m<\/jats:italic>+1<\/jats:sup>into \u2115. We say \u201c<jats:italic>f<\/jats:italic>is weakly expressed by terms<jats:italic>t<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>a<jats:sub>m<\/jats:sub><\/jats:italic>),\u2026,<jats:italic>t<jats:sub>r<\/jats:sub><\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>a<jats:sub>m<\/jats:sub><\/jats:italic>) in<jats:italic>L<\/jats:italic>\u201d if for every<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>a<jats:sub>m<\/jats:sub><\/jats:italic>\u2208 \u2115,<jats:italic>f<\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>a<jats:sub>m<\/jats:sub><\/jats:italic>) is equal to one of<jats:italic>t<jats:sub>i<\/jats:sub><\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026,<jats:italic>a<jats:sub>m<\/jats:sub><\/jats:italic>). The critical number of<jats:italic>b<\/jats:italic>in<jats:italic>f<\/jats:italic>with respect to<jats:italic>L<\/jats:italic>is the minimum number<jats:italic>n<\/jats:italic>such that whenever<jats:italic>f<\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic>) is weakly expressed by terms<jats:italic>t<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026,<jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic>),\u2026, the number of occurrences of<jats:italic>b<\/jats:italic>in some<jats:italic>t<jats:sub>i<\/jats:sub><\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic>) is at least<jats:italic>n<\/jats:italic>.<\/jats:p><jats:p>As is defined in [1], a function<jats:italic>f<\/jats:italic>is defined by a limited iteration from<jats:italic>g<\/jats:italic>and<jats:italic>h<\/jats:italic>with respect to<jats:italic>L<\/jats:italic>iff the following holds: Let<jats:italic>\u03c4<\/jats:italic>be defined as<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_eqnU2\" \/><\/jats:disp-formula><\/jats:p><jats:p>with the condition<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_inline1\" \/>; and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_inline2\" \/>is defined by<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_eqnU3\" \/><\/jats:disp-formula><\/jats:p><jats:p>where<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_inline3\" \/>and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_inline4\" \/>are terms in<jats:italic>L<\/jats:italic>. We say \u201c<jats:italic>f<\/jats:italic>is defined by a short limited iteration from<jats:italic>g<\/jats:italic>and<jats:italic>h<\/jats:italic>\u201d if<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_inline2\" \/>is defined by<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_eqnU4\" \/><\/jats:disp-formula><\/jats:p><jats:p>where<jats:italic>\u03c4<\/jats:italic>,<jats:italic>s, t<\/jats:italic>are the same as above satisfying the condition<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_inline5\" \/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001923X_inline6\" \/>.<\/jats:p>","DOI":"10.2307\/2275701","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:53:29Z","timestamp":1146956009000},"page":"1228-1244","source":"Crossref","is-referenced-by-count":1,"title":["The critical number of a variable in a function"],"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":"S002248120001923X_ref005","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1093\/oso\/9780198536901.003.0016","volume-title":"Arithmetic, proof theory and computational complexity","author":"Takeuti","year":"1993"},{"key":"S002248120001923X_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01531025"},{"key":"S002248120001923X_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(92)90068-B"},{"key":"S002248120001923X_ref001","volume-title":"Bounded arithmetic","author":"Buss","year":"1986"},{"key":"S002248120001923X_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3466-1_4"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120001923X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:28:09Z","timestamp":1707031689000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120001923X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1994,12]]}},"alternative-id":["S002248120001923X"],"URL":"https:\/\/doi.org\/10.2307\/2275701","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,12]]}}}