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On a Theorem of Razborov

Abstract

In an unpublished Russian manuscript, Razborov proved that a matrix family with high rigidity over a finite field would yield a language outside the polynomial hierarchy in communication complexity. We present an alternative proof that strengthens the original result in several ways. In particular, we replace rigidity by the strictly weaker notion of toggle rigidity.

It turns out that Razborov’s astounding result is actually a corollary of a slight generalization of Toda’s First Theorem in communication complexity and that matrix rigidity over a finite field is a lower-bound method for bounded-error modular communication complexity. We also give evidence that Razborov’s strategy is a promising one by presenting a protocol with few alternations for the inner product function mod two and by discussing problems possibly outside the communication complexity version of the polynomial hierarchy.

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Wunderlich, H. On a Theorem of Razborov. comput. complex. 21, 431–477 (2012). https://doi.org/10.1007/s00037-011-0021-5

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Keywords

  • Structural complexity
  • communication complexity
  • Toda’s theorems
  • theorem of Razborov
  • matrix rigidity
  • approximate rank
  • quasi-random graph

Subject classification

  • 68Q10
  • 68Q15
  • 68Q17
  • 68R99