Murthy, J and Nair, V and Saha, C (2020) Randomized polynomialtime equivalence between determinant and traceIMM equivalence tests. In: 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, 2526 August 2020, Prague; Czech Republic.

PDF
lei_int_pro_inf_lip_170_2020.pdf  Published Version Download (600kB)  Preview 
Abstract
Equivalence testing for a polynomial family gmmâ��N over a field F is the following problem: Given blackbox access to an nvariate polynomial f(x), where n is the number of variables in gm for some m â�� N, check if there exists an A â�� GL(n, F) such that f(x) = gm(Ax). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the determinant (Det) and the family of iterated matrix multiplication polynomials. Two popular variants of the iterated matrix multiplication polynomial are: IMMw,d (the (1, 1) entry of the product of d many w Ã� w symbolic matrices) and TrIMMw,d (the trace of the product of d many w Ã� w symbolic matrices). The families  Det, IMM and TrIMM  are VBPcomplete under pprojections, and so, in this sense, they have the same complexity. But, do they have the same equivalence testing complexity? We show that the answer is â��yesâ�� for Det and TrIMM (modulo the use of randomness). The above result may appear a bit surprising as the complexity of equivalence testing for IMM and that for Det are quite different over Q: a randomized polytime equivalence testing for IMM over Q is known 28, whereas 15 showed that equivalence testing for Det over Q is integer factoring hard (under randomized reductions and assuming GRH). To our knowledge, the complexity of equivalence testing for TrIMM was not known before this work. We show that, despite the syntactic similarity between IMM and TrIMM, equivalence testing for TrIMM and that for Det are randomized polytime Turing reducible to each other over any field of characteristic zero or sufficiently large. The result is obtained by connecting the two problems via another wellstudied problem in computer algebra, namely the full matrix algebra isomorphism problem (FMAI). In particular, we prove the following: 1. Testing equivalence of polynomials to TrIMMw,d, for d â�¥ 3 and w â�¥ 2, is randomized polynomialtime Turing reducible to testing equivalence of polynomials to Detw, the determinant of the w Ã� w matrix of formal variables. (Here, d need not be a constant.) 2. FMAI is randomized polynomialtime Turing reducible to equivalence testing (in fact, to tensor isomorphism testing) for the family of matrix multiplication tensors {TrIMMw,3}wâ��N. These results, in conjunction with the randomized polytime reduction (shown in 15) from determinant equivalence testing to FMAI, imply that the four problems  FMAI, equivalence testing for TrIMM and for Det, and the 3tensor isomorphism problem for the family of matrix multiplication tensors  are randomized polytime equivalent under Turing reductions. Â© Nathalie Bertrand; licensed under Creative Commons License CCBY 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020).
Item Type:  Conference Paper 

Publication:  Leibniz International Proceedings in Informatics, LIPIcs 
Publisher:  Schloss Dagstuhl LeibnizZentrum fur Informatik GmbH, Dagstuhl Publishing 
Additional Information:  The copyright of this article belongs to Schloss Dagstuhl LeibnizZentrum fur Informatik GmbH, Dagstuhl Publishing 
Keywords:  Equivalence classes; Matrix algebra; Polynomial approximation; Set theory; Tensors, Equivalence testing; Equivalence tests; Isomorphism problems; Isomorphism testing; MAtrix multiplication; Randomized reductions; Syntactic similarities; Testing equivalence, Blackbox testing 
Department/Centre:  Division of Electrical Sciences > Computer Science & Automation 
Date Deposited:  23 Sep 2020 10:35 
Last Modified:  23 Sep 2020 10:35 
URI:  http://eprints.iisc.ac.in/id/eprint/66604 
Actions (login required)
View Item 