Impossibility results for recycling random bits in two-prover proof systems

Recycling random bits (that is, replacing independent random bits by dependent random bits extracted from a pseudo-random bit generator) has been a successful enterprise in many scenarios, including cryptography, NC computations, space bounded computation, RP and BPP algorithms, and interactive proofs. A wide variety of techniques have been introduced for this purpose. Recycling random bits is highly motivated in the context of parallel repetition of two-prover one-round proof systems, where (if it were possible) it would lead to stronger hardness results for approximating NP-hard optimization problems. In this paper we show that the great success enjoyed by general techniques for recycling random bits in other contexts meets its limits when MIP(2,1) proof systems are concerned. That is, there are natural MIP(2,1) proof systems for 3-SAT for which parallel repetition using independent random bits can reduce the error to be polynomially small, but parallel repetition using pseudo-random bits cannot reduce the error below a constant, regardless of the nature of the pseudo-random source.