{"id":168052,"date":"2014-06-01T00:00:00","date_gmt":"2014-06-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/a-super-polynomial-lower-bound-for-regular-arithmetic-formulas\/"},"modified":"2018-10-16T20:06:01","modified_gmt":"2018-10-17T03:06:01","slug":"a-super-polynomial-lower-bound-for-regular-arithmetic-formulas","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/a-super-polynomial-lower-bound-for-regular-arithmetic-formulas\/","title":{"rendered":"A super-polynomial lower bound for regular arithmetic formulas."},"content":{"rendered":"<div class=\"asset-content\">\n<p>We consider arithmetic formulas consisting of alternating layers of addition and multiplication gates such that the fanin of all the gates in any fixed layer is the same. Such a formula  which additionally has the property that its formal\/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a <em>regular formula<\/em>. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (<em>n<\/em><sup>2<\/sup> + 1)-variate polynomial of degree 2<em>n<\/em> in VNP such that any regular formula computing it must be of size at least <em>n<\/em> <sup>(log<\/sup> <em><sup>n<\/em><\/sup><sup>)<\/sup>.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We consider arithmetic formulas consisting of alternating layers of addition and multiplication gates such that the fanin of all the gates in any fixed layer is the same. Such a formula which additionally has the property that its formal\/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-168052","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"ACM - Association for Computing Machinery","msr_edition":"Symposium on Theory of 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