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Isomorphism testing of read-once functions and polynomials

Authors Raghavendra Rao B .V., Jayalal Sarma M. N.



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Raghavendra Rao B .V.
Jayalal Sarma M. N.

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Raghavendra Rao B .V. and Jayalal Sarma M. N.. Isomorphism testing of read-once functions and polynomials. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 115-126, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)
https://doi.org/10.4230/LIPIcs.FSTTCS.2011.115

Abstract

In this paper, we study the isomorphism testing problem of formulas in the Boolean and arithmetic settings. We show that isomorphism testing of Boolean formulas in which a variable is read at most once (known as read-once formulas) is complete for log-space. In contrast, we observe that the problem becomes polynomial time equivalent to the graph isomorphism problem, when the input formulas can be represented as OR of two or more monotone read-once formulas. This classifies the complexity of the problem in terms of the number of reads, as read-3 formula isomorphism problem is hard for \co\NP. We address the polynomial isomorphism problem, a special case of polynomial equivalence problem which in turn is important from a cryptographic perspective[Patarin EUROCRYPT'96, and Kayal SODA'11]. As our main result, we propose a deterministic polynomial time canonization scheme for polynomials computed by constant-free read-once arithmetic formulas. In contrast, we show that when the arithmetic formula is allowed to read a variable twice, this problem is as hard as the graph isomorphism problem.
Keywords
  • Isomorphism Problems
  • Computational Complexity
  • Read-once formulas
  • Read-once Polynomials

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