Document Zbl 0845.68045

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The degree of the polynomial is related to the accurately of the approximation. Techniques to handle symmetric functions are derived. This yields lower bounds for circuits of And and Or gates with unbounded fan-in and one majority gate.<br class="zbmathjax-paragraph">The method is powerful enough to derive the known $\text{AC}^0$ exponential-size lower bound for computing the parity function (du to Furst, Saxe, and Sipser) and to separate the complexity classes PP and PSPACE relative to a random oracle.<br class="zbmathjax-paragraph">A connection to voting puzzles establishes the power of the voting concept.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7541">M.Hühne (Dortmund)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A617046">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A617046">Cited in <strong>54</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q15" title="MSC2020">68Q15</a> </td> <td class="space"> Complexity classes (hierarchies, relations among complexity classes, etc.) </td> </tr><tr> <td class="mono"> 05E35 </td> <td class="space"> Orthogonal polynomials (combinatorics) (MSC2000) </td> </tr><tr> <td class="mono"> 94C10 </td> <td class="space"> Switching theory, application of Boolean algebra; Boolean functions (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A42C15" title="MSC2020">42C15</a> </td> <td class="space"> General harmonic expansions, frames </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ABoolean+circuits">Boolean circuits</a>; <a href="/?q=ut%3ABoolean+functions">Boolean functions</a>; <a href="/?q=ut%3Avoting+puzzles">voting puzzles</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0845.68045" data-ciurl="/ci/00617046" data-biburl="/bibtex/00617046.bib" data-amsurl="/amsrefs/00617046.bib" data-xmlurl="/xml/00617046.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00617046.pdf" title="Zbl 0845.68045 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF01215346" aria-label="DOI for “The expressive power of voting polynomials”" title="10.1007/BF01215346">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">R. Beigel: Relativized counting classes: Relations among thresholds, parity, and mods.Journal of Computer and System Sciences 42(1) February 1991, 76-96. · <a href="/0717.68031" class="nowrap">Zbl 0717.68031</a> · <a href="https://doi.org/10.1016/0022-0000(91)90040-C" class="nowrap">doi:10.1016/0022-0000(91)90040-C</a></td> </tr><tr> <td>[2]</td> <td class="space">R. Beigel, N. Reingold, andD. Spielman: The perceptron strikes back.Proceedings of the 6th Annual Conference on Structure in Complexity Theory (1991), 286-291.</td> </tr><tr> <td>[3]</td> <td class="space">C. H. Bennett, andJ. Gill: Relative to a random oracleA, P A ?NP A ?co-NP A with probability 1.SIAM Journal on Computing 10(1) February 1981, 96-113. · <a href="/0454.68030" class="nowrap">Zbl 0454.68030</a> · <a href="https://doi.org/10.1137/0210008" class="nowrap">doi:10.1137/0210008</a></td> </tr><tr> <td>[4]</td> <td class="space">J. Bruck: Harmonic analysis of polynomial threshold functions.SIAM Journal on Discrete Mathematics 3(2) May 1990, 168-177. · <a href="/0695.94022" class="nowrap">Zbl 0695.94022</a> · <a href="https://doi.org/10.1137/0403015" class="nowrap">doi:10.1137/0403015</a></td> </tr><tr> <td>[5]</td> <td class="space">J. Bruck, andR. Smolensky: Polynomial threshold functions, AC0 functions and spectral norms. InProceedings of the 31st Annual Symposium on Foundations of Computer Science (1990), 632-641.</td> </tr><tr> <td>[6]</td> <td class="space">V. Chv?tal:Linear Programming. W. H. Freeman and Company, (1983).</td> </tr><tr> <td>[7]</td> <td class="space">M. Furst, J. Saxe, andM. Sipser: Parity, circuits and the polynomial time hierarchy.Mathematical Systems Theory 17 (1984), 13-27. · <a href="/0534.94008" class="nowrap">Zbl 0534.94008</a> · <a href="https://doi.org/10.1007/BF01744431" class="nowrap">doi:10.1007/BF01744431</a></td> </tr><tr> <td>[8]</td> <td class="space">C. Gotsman: On boolean functions, polynomials, and algebraic threshold functions. Unpublished manuscript.</td> </tr><tr> <td>[9]</td> <td class="space">N. Linial, Y. Mansour andN. Nisan: Constant depth circuits, fourier transform, and learnability. InProceedings of the 30th Annual Symposium on Foundations of Computer Science (1989), 574-579.</td> </tr><tr> <td>[10]</td> <td class="space">M. L. Minsky, andS. Papert:Perceptrons. MIT Press, Cambridge, MA, (1988), Expanded Edition. The first edition appeared in 1968.</td> </tr><tr> <td>[11]</td> <td class="space">R. Paturi, andM. E. Saks: On threshold circuits for parity. InProceedings of the 31st Annual Symposium on Foundations of Computer Science (1990), 397-404.</td> </tr><tr> <td>[12]</td> <td class="space">A. A. Razborov: Lower bounds for the size of circuits of bounded depth with basis {?,?}.Math. Notes of the Academy of Sciences of the USSR 41(4) September 1987, 333-338. · <a href="/0632.94030" class="nowrap">Zbl 0632.94030</a> · <a href="https://doi.org/10.1007/BF01137685" class="nowrap">doi:10.1007/BF01137685</a></td> </tr><tr> <td>[13]</td> <td class="space">A. Schrijver:Theory of Linear and Integer Programming. John Wiley and Sons., (1986). · <a href="/0665.90063" class="nowrap">Zbl 0665.90063</a></td> </tr><tr> <td>[14]</td> <td class="space">R. Smolensky: Algebraic methods in the theory of lower bounds for boolean circuit complexity. InProceedings of the 19th Annual ACM Symposium on Theory of Computing (1987), 77-82.</td> </tr><tr> <td>[15]</td> <td class="space">E. Stiemke: ?ber positive L?sungen homogener linearer Gleichungen.Mathematische Annalen 76 (1915), 340-342. · <a href="/45.0168.04" class="nowrap">JFM 45.0168.04</a> · <a href="https://doi.org/10.1007/BF01458147" class="nowrap">doi:10.1007/BF01458147</a></td> </tr><tr> <td>[16]</td> <td class="space">J. Tarui: Randomized polynomials, threshold circuits, and the polynomial hierarchy. Manuscript, August 1990. · <a href="/0764.94026" class="nowrap">Zbl 0764.94026</a></td> </tr><tr> <td>[17]</td> <td class="space">L. G. Valiant, andV. V. Vazirani: NP is as easy as detecting unique solutions. InProceedings of the 17th Annual ACM Symposium on Theory of Computing (1985).</td> </tr><tr> <td>[18]</td> <td class="space">A. Yao: Separating the polynomial-time hierarchy by oracles. InProceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science (1985), 1-10.</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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