Collapsing Exact Arithmetic Hierarchies

  • Nikhil Balaji
  • Samir Datta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)


We provide a uniform framework for proving the collapse of the hierarchy \({\sf NC}^1(\mathcal{C})\) for an exact arithmetic class \(\mathcal{C}\) of polynomial degree. These hierarchies collapse all the way down to the third level of the AC 0-hierarchy, \({\sf AC^0_3}(\mathcal{C})\). Our main collapsing exhibits are the classes

$$\mathcal{C} \in \{{\sf C}_={{\sf NC}^1}, {\sf C}_={\sf L}, {\sf C}_={\sf SAC^1}, {\sf C}_={\sf P}\}.$$

NC 1(C = L) and NC 1(C = P) are already known to collapse [1,19,20].

We reiterate that our contribution is a framework that works for all these hierarchies. Our proof generalizes a proof from [9] where it is used to prove the collapse of the AC 0(C = NC 1) hierarchy. It is essentially based on a polynomial degree characterization of each of the base classes.


Polynomial Degree Arithmetic Circuit Vandermonde Matrix Polynomial Size Boolean Circuit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Nikhil Balaji
    • 1
  • Samir Datta
    • 1
  1. 1.Chennai Mathematical InstituteIndia

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