A constructive proof presenting languages in $\Sigma_2^P$ that cannot be decided by circuit families of size $n^k$

Sunny Daniels

A constructive proof presenting languages in $\Sigma_2^P$ that cannot be decided by circuit families of size $n^k$

Computational Complexity 2014-09-18 v2

Authors: Sunny Daniels

Abstract

As far as I know, at the time that I originally devised this result (1998), this was the first constructive proof that, for any integer $k$ , there is a language in $\Sigma_2^P$ that cannot be simulated by a family of logic circuits of size $n^k$ . However, this result had previously been proved non-constructively: see Cai and Watanabe [CW08] for more information on the history of this problem. This constructive proof is based upon constructing a language $\Gamma$ derived from the satisfiabiility problem, and a language $\Lambda_k$ defined by an alternating Turing machine. We show that the union of $\Gamma$ and $\Lambda_k$ cannot be simulated by circuits of size $n^k$ .

Keywords

constructive proof Sigma_2^P circuit lower bounds logic circuits alternating Turing machine satisfiability problem

Cite

@article{arxiv.1408.6334,
  title  = {A constructive proof presenting languages in $\Sigma_2^P$ that cannot be decided by circuit families of size $n^k$},
  author = {Sunny Daniels},
  journal= {arXiv preprint arXiv:1408.6334},
  year   = {2014}
}

Comments

This is a corrected version of my previous article (of the same name) which attracted the attention of Professor Lance Fortnow at Georgia Institute of Technology ("Sixteen Years in the Making" in his Complexity Theory Blog). The original had a missing closing bracket in a footnote and a reference to the wrong step in the machine for $\Lambda_k$

A constructive proof presenting languages in $\Sigma_2^P$ that cannot be decided by circuit families of size $n^k$

Abstract

Keywords

Cite

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