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While our procedure runs in non-elementary time in general, we show that it yields nearly optimal elementary complexity results for expressive counting extensions of Presburger arithmetic, such as the <jats:italic>threshold counting<\/jats:italic> quantifier <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\exists ^{\\ge c} y\\, \\mathrm {\\Phi }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                <mml:mrow>\n                  <mml:msup>\n                    <mml:mo>\u2203<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>\u2265<\/mml:mo>\n                      <mml:mi>c<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                  <mml:mi>y<\/mml:mi>\n                  <mml:mspace\/>\n                  <mml:mi>\u03a6<\/mml:mi>\n                <\/mml:mrow>\n              <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that requires that the number of different\u00a0<jats:italic>y<\/jats:italic> satisfying <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathrm {\\Phi }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                <mml:mi>\u03a6<\/mml:mi>\n              <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be at least <jats:inline-formula><jats:alternatives><jats:tex-math>$$c\\in \\mathbb {N}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                <mml:mrow>\n                  <mml:mi>c<\/mml:mi>\n                  <mml:mo>\u2208<\/mml:mo>\n                  <mml:mi>N<\/mml:mi>\n                <\/mml:mrow>\n              <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:italic>c<\/jats:italic> can succinctly be defined by a Presburger formula. Our results are cast in terms of what we call the <jats:italic>monadically-guarded fragment<\/jats:italic> of Presburger arithmetic with unary counting quantifiers, for which we develop a <jats:sc>2ExpSpace<\/jats:sc> decision procedure.<\/jats:p>","DOI":"10.1007\/978-3-030-99253-8_12","type":"book-chapter","created":{"date-parts":[[2022,3,28]],"date-time":"2022-03-28T20:02:48Z","timestamp":1648497768000},"page":"225-243","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantifier elimination for counting extensions of Presburger arithmetic"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9055-918X","authenticated-orcid":false,"given":"Dmitry","family":"Chistikov","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5452-936X","authenticated-orcid":false,"given":"Christoph","family":"Haase","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1104-7299","authenticated-orcid":false,"given":"Alessio","family":"Mansutti","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,29]]},"reference":[{"key":"12_CR1","doi-asserted-by":"crossref","unstructured":"Apelt, H.: Axiomatische Untersuchungen \u00fcber einige mit der Presburgerschen Arithmetik verwandte Systeme. 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