Chipping Away at P vs NP: How Far Are We from Proving Circuit Size Lower Bounds?

Allender, E.

    Many people are pessimistic about seeing a resolution to the P vs NP question any time soon. This pessimism extends also to questions about other important complexity classes, including two classes that will be the focus of this talk: TC^0 and NC^1. TC^0 captures the complexity of several important computational problems, such as multiplication, division, and sorting; it consists of all problems computable by constant-depth, polynomial-size families of circuits of MAJORITY gates. TC^0_d is the subclass of TC^0 solvable with circuits of depth d. Although TC^0 seems to be a small subclass of P, it is still open if NP = TC^0_3. NC^1 is the class of problems expressible by Boolean formulae of polynomial size. NC^1 contains TC^0, and captures the complexity of evaluating a Boolean formula. Any proof that NP is not equal to TC^0 will have to overcome the obstacles identified by Razborov and Rudich in their paper on 'Natural Proofs'. That is, a 'natural' proof that NP is not equal to TC^0 yields a proof that no pseudorandom function generator is computable in TC^0. This is problematic, since some popular cryptographic conjectures imply that such generators do exist. This leads to pessimism about the even more difficult task of separating NC^1 from TC^0. Some limited lower bounds are within the grasp of current techniques, however. For example, several problems in P are known to require formulae of quadratic size - but this seems to be of little use in trying to prove superpolynomial formula size. Along similar lines, it is known that, for every d, there is a constant c>1 such that the formula evaluation problem (one of the standard complete problems for NC^1) requires TC^0_d circuits of size at least n^c. It might not seem too outrageous to hope to obtain a slightly stronger lower bound, showing that there is a c>1 such that this same set requires uniform TC^0 circuits of size n^c (regardless of the depth d). We show that this would be sufficient to prove that TC^0 is properly contained in NC^1.
Cite as: Allender, E. (2008). Chipping Away at P vs NP: How Far Are We from Proving Circuit Size Lower Bounds?. In Proc. Fourteenth Computing: The Australasian Theory Symposium (CATS 2008), Wollongong, NSW, Australia. CRPIT, 77. Harland, J. and Manyem, P., Eds. ACS. 3.
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