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Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes

Authors Maurice Jansen, Rahul Santhanam



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LIPIcs.STACS.2012.519.pdf
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Maurice Jansen
Rahul Santhanam

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Maurice Jansen and Rahul Santhanam. Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 519-530, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)
https://doi.org/10.4230/LIPIcs.STACS.2012.519

Abstract

We associate to each Boolean language complexity class C the algebraic class a.C consisting of families of polynomials {f_n} for which the evaluation problem over the integers is in C. We prove the following lower bound and randomness-to-hardness results: 1. If polynomial identity testing (PIT) is in NSUBEXP then a.NEXP does not have poly size constant-free arithmetic circuits. 2. a.NEXP^RP does not have poly size constant-free arithmetic circuits. 3. For every fixed k, a.MA does not have arithmetic circuits of size n^k. Items 1 and 2 strengthen two results due to (Kabanets and Impagliazzo, 2004). The third item improves a lower bound due to (Santhanam, 2009). We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case. Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT is in i.o-NTIME[2^{n^{o(1)}}]/n^{o(1)} if and only if there exists a family of multilinear polynomials in a.NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree.
Keywords
  • Computational Complexity
  • Circuit Lower Bounds
  • Polynomial Identity Testing
  • Derandomization

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