{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T14:32:44Z","timestamp":1777645964369,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"1-4","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Fundamenta Informaticae"],"published-print":{"date-parts":[[2011,6]]},"abstract":"<jats:p>\n                    The number of states in a two-way nondeterministic finite automaton (2NFA) needed to represent intersection of languages given by an m-state 2NFA and an n-state 2NFA is shown to be at least m+n and at most m+n+1. For the union operation, the number of states is exactly m+n. The lower bound is established for languages over a one-letter alphabet. The key point of the argument is the following number-theoretic lemma: for all m, n \u2265 2 with m, n \u2260 6 (and with finitely many other exceptions), there exist partitions m = p\n                    <jats:sub>1<\/jats:sub>\n                    +. . .+ p\n                    <jats:sub>k<\/jats:sub>\n                    and n = q\n                    <jats:sub>1<\/jats:sub>\n                    +. . .+q\n                    <jats:sub>l<\/jats:sub>\n                    , where all numbers p\n                    <jats:sub>1<\/jats:sub>\n                    , . . . , p\n                    <jats:sub>k<\/jats:sub>\n                    , q\n                    <jats:sub>1<\/jats:sub>\n                    , . . . , q\n                    <jats:sub>l<\/jats:sub>\n                    \u2265 2 are powers of pairwise distinct primes. For completeness, an analogous statement about partitions of any two numbers m, n \u2209 {4, 6} (with a few more exceptions) into sums of pairwise distinct primes is established as well.\n                  <\/jats:p>","DOI":"10.3233\/fi-2011-540","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T23:44:54Z","timestamp":1575330294000},"page":"231-239","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":8,"title":["State Complexity of Union and Intersection for Two-way Nondeterministic Finite Automata"],"prefix":"10.1177","volume":"110","author":[{"given":"Michal","family":"Kunc","sequence":"first","affiliation":[{"name":"Department of Mathematics, Masaryk University, Brno, Czech Republic. kunc@math.muni.cz"}]},{"given":"Alexander","family":"Okhotin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Turku, Turku, FI\u201320014, Finland. alexander.okhotin@utu.fi"}]}],"member":"179","published-online":{"date-parts":[[2011,1,1]]},"container-title":["Fundamenta Informaticae"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2011-540","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2011-540","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T06:33:55Z","timestamp":1777444435000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/FI-2011-540"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,1]]},"references-count":0,"journal-issue":{"issue":"1-4","published-print":{"date-parts":[[2011,6]]}},"alternative-id":["10.3233\/FI-2011-540"],"URL":"https:\/\/doi.org\/10.3233\/fi-2011-540","relation":{},"ISSN":["0169-2968","1875-8681"],"issn-type":[{"value":"0169-2968","type":"print"},{"value":"1875-8681","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1,1]]}}}