Positive and Negative Proofs for Circuits and Branching Programs

• Published in: Descriptional Complexity of Formal Systems pp. 270 - 281
• Main Authors: Dorzweiler, Olga, Flamm, Thomas, Krebs, Andreas, Ludwig, Michael
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Arithmetic Circuit; Complexity Class; Dual Graph; Input Word; Turing Machine
• ISBN: 9783319097039; 3319097032
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-09704-6_24

We extend the # operator in a natural way and derive a new type of counting complexity. While #\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal C$\end{document} classes (where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal C$\end{document} is some circuit-based class like NC1) only count proofs for acceptance of some input in circuits, one can also count proofs for rejection. The here proposed Zap-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal C$\end{document} complexity classes implement this idea. We show that Zap-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal C$\end{document} lies between #\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal C$\end{document} and Gap-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal C$\end{document}. In particular we consider Zap-NC1 and polynomial size branching programs of bounded and unbounded width. We find connections to planar branching programs since the duality of positive and negative proofs can be found again in the duality of graphs and their co-graphs. This links to possible applications of our contribution, like closure properties of complexity classes.