On TC0 Lower Bounds for the Permanent
- Cite this paper as:
- Kinne J. (2012) On TC0 Lower Bounds for the Permanent. In: Gudmundsson J., Mestre J., Viglas T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg
Depth O(1), poly − log(n) bits of non-uniformity, and size s(n) such that for all constants c, s(c)(n) < 2n. The size s must satisfy another technical condition that is true of functions normally dealt with (such as compositions of polynomials, logarithms, and exponentials).
Depth o(loglogn), poly − log(n) bits of non-uniformity, and size nO(1).
Depth O(1), no(1) bits of non-uniformity, and size nO(1).
Our proof yields a new “either or” hardness result. One instantiation is that either NP does not have polynomial-size constant-depth threshold circuits that use no(1) bits of non-uniformity, or the permanent does not have polynomial-size general circuits.
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