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Magdeburg","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100012320","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Acta Informatica"],"published-print":{"date-parts":[[2022,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The well-known pumping lemma for regular languages states that, for any regular language <jats:italic>L<\/jats:italic>, there is a constant <jats:italic>p<\/jats:italic> (depending on <jats:italic>L<\/jats:italic>) such that the following holds: If <jats:inline-formula><jats:alternatives><jats:tex-math>$$w\\in L$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>w<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>L<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vert w\\vert \\ge p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mi>w<\/mml:mi>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mi>p<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then there are words <jats:inline-formula><jats:alternatives><jats:tex-math>$$x\\in V^{*}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>V<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mrow\/>\n                        <mml:mo>\u2217<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$y\\in V^+$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>V<\/mml:mi>\n                      <mml:mo>+<\/mml:mo>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and <jats:inline-formula><jats:alternatives><jats:tex-math>$$z\\in V^{*}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>z<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>V<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mrow\/>\n                        <mml:mo>\u2217<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$w=xyz$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>w<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mi>z<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$xy^tz\\in L$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>y<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mi>z<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>L<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:tex-math>$$t\\ge 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The minimal pumping constant <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\,\\mathrm{mpc}\\,}}(L)}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mi>mpc<\/mml:mi>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of <jats:italic>L<\/jats:italic> is the minimal number <jats:italic>p<\/jats:italic> for which the conditions of the pumping lemma are satisfied. We investigate the behaviour of <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\,\\mathrm{mpc}\\,}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mspace\/>\n                    <mml:mi>mpc<\/mml:mi>\n                    <mml:mspace\/>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with respect to operations, i.\u00a0e., for an <jats:italic>n<\/jats:italic>-ary regularity preserving operation <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\circ $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mo>\u2218<\/mml:mo>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we study the set <jats:inline-formula><jats:alternatives><jats:tex-math>$${g_{\\circ }^{{{\\,\\mathrm{mpc}\\,}}}(k_1,k_2,\\ldots ,k_n)}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mi>g<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>\u2218<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mspace\/>\n                        <mml:mi>mpc<\/mml:mi>\n                        <mml:mspace\/>\n                      <\/mml:mrow>\n                    <\/mml:msubsup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mo>\u2026<\/mml:mo>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of all numbers <jats:italic>k<\/jats:italic> such that there are regular languages <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_1,L_2,\\ldots ,L_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2026<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\,\\mathrm{mpc}\\,}}(L_i)=k_i}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mi>mpc<\/mml:mi>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>L<\/mml:mi>\n                        <mml:mi>i<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>k<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\le i\\le n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>i<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\,\\mathrm{mpc}\\,}}(\\circ (L_1,L_2,\\ldots ,L_n)=~k}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mi>mpc<\/mml:mi>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mo>\u2218<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>L<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>L<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mo>\u2026<\/mml:mo>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>L<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mspace\/>\n                    <mml:mi>k<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. With respect to Kleene closure, complement, reversal, prefix and suffix-closure, circular shift, union, intersection, set-subtraction, symmetric difference,and concatenation, we determine <jats:inline-formula><jats:alternatives><jats:tex-math>$${g_{\\circ }^{{{\\,\\mathrm{mpc}\\,}}}(k_1,k_2,\\ldots ,k_n)}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mi>g<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>\u2218<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mspace\/>\n                        <mml:mi>mpc<\/mml:mi>\n                        <mml:mspace\/>\n                      <\/mml:mrow>\n                    <\/mml:msubsup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mo>\u2026<\/mml:mo>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vert xy\\vert \\le p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mi>y<\/mml:mi>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>p<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has to hold.<\/jats:p>","DOI":"10.1007\/s00236-022-00431-3","type":"journal-article","created":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T05:02:58Z","timestamp":1658293378000},"page":"337-355","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Operational complexity and pumping lemmas"],"prefix":"10.1007","volume":"59","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4735-4696","authenticated-orcid":false,"given":"J\u00fcrgen","family":"Dassow","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Isma\u00ebl","family":"Jecker","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,7,20]]},"reference":[{"key":"431_CR1","first-page":"55","volume":"21","author":"J Dassow","year":"2016","unstructured":"Dassow, J.: On the number of accepting states of finite automata. 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