{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:43:49Z","timestamp":1767199429259,"version":"build-2238731810"},"reference-count":18,"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":13706,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,9]]},"abstract":"Abstract<\/jats:title>\n Mostowski [11] shows that if a structure has a decidable theory, then its weak direct power has one as well; his proof however never produces decision procedures which are elementary recursive.<\/jats:p>\n Some very general results are obtained here about the nature of the weak direct power of a structure, which in most cases lead to elementary recursive decision procedures for weak direct powers of structures which themselves have elementary recursive procedures.<\/jats:p>\n \n In particular, it is shown that \u3008\n N<\/jats:italic>\n *, +\u3009, the weak direct power of \u3008\n N<\/jats:italic>\n , +\u3009, can be decided in space\n <\/jats:p>\n \n \n \n <\/jats:disp-formula>\n <\/jats:p>\n \n for some constant\n c<\/jats:italic>\n . As corollaries, the same upper bound is obtained for the theory of the structure \u3008\n N<\/jats:italic>\n +<\/jats:sup>\n , \u00b7\u3009 of positive integers under multiplication, and for the theory of finite abelian groups. Fischer and Rabin [7] have shown that the theory of \u3008\n N<\/jats:italic>\n ,* +\u3009\n requires time<\/jats:italic>\n <\/jats:p>\n \n \n \n <\/jats:disp-formula>\n <\/jats:p>\n even on nondeterministic Turing machines.<\/jats:p>","DOI":"10.2307\/2272034","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:42:22Z","timestamp":1146937342000},"page":"561-573","source":"Crossref","is-referenced-by-count":2,"title":["On the complexity of the theories of weak direct powers"],"prefix":"10.1017","volume":"41","author":[{"given":"Charles","family":"Rackoff","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200051136_bib017","first-page":"1","volume-title":"5th ACM Syposium on Theory of Computing","author":"Stockmeyer","year":"1973"},{"key":"S0022481200051136_bib002","volume-title":"Computer and logic group memorandum No. 16","author":"Cooper","year":"1972"},{"key":"S0022481200051136_bib018","doi-asserted-by":"publisher","DOI":"10.4064\/fm-41-2-203-271"},{"key":"S0022481200051136_bib008","first-page":"598","volume-title":"Algebra","author":"MacLane","year":"1968"},{"key":"S0022481200051136_bib016","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1963-0158822-2"},{"key":"S0022481200051136_bib004","doi-asserted-by":"publisher","DOI":"10.1070\/RM1965v020n04ABEH001188"},{"key":"S0022481200051136_bib010","doi-asserted-by":"publisher","DOI":"10.1109\/SWAT.1972.29"},{"key":"S0022481200051136_bib006","doi-asserted-by":"publisher","DOI":"10.1137\/0204006"},{"key":"S0022481200051136_bib001","first-page":"24n30","volume-title":"1964 International Congress for Logic, Methodology, and Philosophy of Science","author":"Cobham","year":"1964"},{"key":"S0022481200051136_bib015","unstructured":"Rackoff, C. 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