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A major open problem in proof complexity is to prove superpolynomial lower bounds for AC0[p]-Frege proofs. This system is the analog of AC0 [p], the class of bounded depth circuits with prime modular counting gates. Despite strong lower bounds for this class dating back thirty years ([28, 30]), there are no significant lower bounds for AC0 [p]-Frege. Significant and extensive degree lower bounds have been obtained for a variety of subsystems of AC0[p]-Frege, including Nullstellensatz ([3]), Polynomial Calculus ([9]), and SOS ([14]). However to date there has been no progress on AC0 [p]-Frege lower bounds. In this paper we study constant-depth extensions of the Polynomial Calculus [13]. We show that these extensions are much more powerful than was previously known. Our main result is that small depth (≤ 43) Polynomial Calculus (over a sufficiently large field) can polynomially effectively simulate all of the well-studied semialgebraic proof systems: Cutting Planes, Sherali-Adams, Sum-of-Squares (SOS), and Positivstellensatz Calculus (Dynamic SOS). Additionally, they can also quasi-polynomially effectively simulate AC0[q]-Frege for any prime q independent of the characteristic of the underlying field. They can also effectively simulate TC0-Frege if the depth is allowed to grow proportionally. Thus, proving strong lower bounds for constant-depth extensions of Polynomial Calculus would not only give lower bounds for AC0 [p]-Frege, but also for systems as strong as TC0-Frege. 

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer from a quantitative loss in parameters, and hence do not give nontrivial implications for models where we don't know super-polynomial lower bounds but do know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can construct PRGs that are essentially best possible without in turn improving the lower bounds. More specifically, say that a circuit family has shrinkage exponent Gamma if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of p^{Gamma + o(1)}. Our PRG uses a seed of length s^{1/(Gamma + 1) + o(1)} to fool circuits in the family of size s. By using this generic construction, we get PRGs with polynomially small error for the following classes of circuits of size s and with the following seed lengths: 1. For de Morgan formulas, seed length s^{1/3+o(1)}; 2. For formulas over an arbitrary basis, seed length s^{1/2+o(1)}; 3. For read-once de Morgan formulas, seed length s^{.234...}; 4. For branching programs of size s, seed length s^{1/2+o(1)}. The previous best PRGs known for these classes used seeds of length bigger than n/2 to output n bits, and worked only when the size s=O(n).