{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6646","name":"The Power of Depth 2 Circuits over Algebras","abstract":"We study the problem of polynomial identity testing (PIT) for depth\r\n$2$ arithmetic circuits over matrix algebra. We show that identity\r\ntesting of depth $3$ ($\\Sigma \\Pi \\Sigma$) arithmetic circuits over a\r\nfield $\\F$ is polynomial time equivalent to identity testing of depth\r\n$2$ ($\\Pi \\Sigma$) arithmetic circuits over\r\n$\\mathsf{U}_2(\\mathbb{F})$, the algebra of upper-triangular $2\\times\r\n2$ matrices with entries from $\\F$. Such a connection is a bit\r\nsurprising since we also show that, as computational models, $\\Pi\r\n\\Sigma$ circuits over $\\mathsf{U}_2(\\mathbb{F})$ are strictly `weaker'\r\nthan $\\Sigma \\Pi \\Sigma$ circuits over $\\mathbb{F}$. The equivalence\r\nfurther implies that PIT of $\\Sigma \\Pi \\Sigma$ circuits reduces to PIT\r\nof width-$2$ commutative \\emph{Algebraic Branching\r\n Programs}(ABP). Further, we give a deterministic polynomial time\r\nidentity testing algorithm for a $\\Pi \\Sigma$ circuit of size $s$ over\r\ncommutative algebras of dimension $O(\\log s\/\\log\\log s)$ over\r\n$\\F$. Over commutative algebras of dimension $\\poly(s)$, we show that\r\nidentity testing of $\\Pi \\Sigma$ circuits is at least as hard as that\r\nof $\\Sigma \\Pi \\Sigma$ circuits over $\\mathbb{F}$.","keywords":["Polynomial identity testing","depth 3 circuits","matrix algebras","local rings"],"author":[{"@type":"Person","name":"Saha, Chandan","givenName":"Chandan","familyName":"Saha"},{"@type":"Person","name":"Saptharishi, Ramprasad","givenName":"Ramprasad","familyName":"Saptharishi"},{"@type":"Person","name":"Saxena, Nitin","givenName":"Nitin","familyName":"Saxena"}],"position":32,"pageStart":371,"pageEnd":382,"dateCreated":"2009-12-14","datePublished":"2009-12-14","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Saha, Chandan","givenName":"Chandan","familyName":"Saha"},{"@type":"Person","name":"Saptharishi, Ramprasad","givenName":"Ramprasad","familyName":"Saptharishi"},{"@type":"Person","name":"Saxena, Nitin","givenName":"Nitin","familyName":"Saxena"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.FSTTCS.2009.2333","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6207","volumeNumber":4,"name":"IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science","dateCreated":"2009-12-07","datePublished":"2009-12-07","editor":[{"@type":"Person","name":"Kannan, Ravi","givenName":"Ravi","familyName":"Kannan"},{"@type":"Person","name":"Narayan Kumar, K.","givenName":"K.","familyName":"Narayan Kumar"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6646","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6207"}}}