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          "alternateName": "Informatique et recherche op\u00e9rationnelle, Universit\u00e9 de Montr\u00e9al, C.P. 6128, H3C 3J7, Succ. Centre-Ville Montr\u00e9al, Qu\u00e9bec, Canada", 
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            "Informatique et recherche op\u00e9rationnelle, Universit\u00e9 de Montr\u00e9al, C.P. 6128, H3C 3J7, Succ. Centre-Ville Montr\u00e9al, Qu\u00e9bec, Canada"
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          "alternateName": "Theoretische Informatik, Julius-Maximilians-Universit\u00e4t W\u00fcrzburg, Am Hubland, D-97074, W\u00fcrzburg, Germany", 
          "id": "http://www.grid.ac/institutes/grid.8379.5", 
          "name": [
            "Theoretische Informatik, Julius-Maximilians-Universit\u00e4t W\u00fcrzburg, Am Hubland, D-97074, W\u00fcrzburg, Germany"
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        "familyName": "Wagner", 
        "givenName": "Klaus W.", 
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    "datePublished": "2007-10-05", 
    "datePublishedReg": "2007-10-05", 
    "description": "Abstract.The problem of testing membership in the subset of the natural numbers produced at the output gate of a {\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bigcup, \\bigcap, ^-, +, \\times$$\\end{document}} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bigcup, \\bigcap, +, \\times$$\\end{document}} is shown NEXPTIME-complete, the cases {\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bigcup, \\bigcap, ^-, \\times$$\\end{document}\u00a0}, {\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bigcup, \\bigcap, \\times$$\\end{document}}, {\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bigcup, \\bigcap, +$$\\end{document}} are shown PSPACE-complete, the case {\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bigcup, +$$\\end{document}\u00a0} is shown NP-complete, the case {\u2229,\u00a0+} is shown C=L-complete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for union-intersection-concatenation circuits, and expressing each integer, drawn from a set given as input, as powers of relatively prime integers of one\u2019s choosing. Our results extend in nontrivial ways past work by Stockmeyer and Meyer (1973), Wagner (1984) and Yang (2000).", 
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        "name": "computational complexity", 
        "publisher": "Springer Nature", 
        "type": "Periodical"
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        "issueNumber": "3", 
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      "circuit", 
      "combinational circuits", 
      "NPs", 
      "output gate", 
      "gate", 
      "wide range", 
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    "name": "The Complexity of Membership Problems for Circuits Over Sets of Natural Numbers", 
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