Corollary 4.18 Exp Np L(ir; Ir)

{ 15 { codewords. If any of the strings supplied by P k fails a multilinearity test, then V accepts P 3?k 's claim about whether or not w 2 L. It remains to show how V can test equality of two functions that are both 1=4-close to multilinear polynomials in K n X 1 ; : : :; X n ]. This can also be done using a well-known fact about polynomials. Schwartz's Lemma Sc80]: Two distinct polynomials in K n X 1 ; : : :; X n ], each of total degree at most n, must diier on at least (1 ? n=jK n j) jK n j n of the jK n j n points in their domain. This lemma clearly implies that, if P k supplies a function h k that is 1=4-close to a multi-linear polynomial in K n X 1 ; : : : ; X n ], and h 1 6 = h 2 , then the referee V can, with probability 1 ? 2 ?poly(n) , nd a point x such that h 1 (x) 6 = h 2 (x) by sampling polynomially many points in the domain, independently and uniformly at random. Once again, the polynomial exponent in the error probability can be made as large as desired at the cost of increasing the (still polynomial) number of sample points. EXP 2 \ EXP 2. Question 4.19 What is the exact complexity of L(ir; ir)? In particular, can the upper bound be improved to EXP NP ? Finally, we include two general open questions about polynomially deenable game systems. Question 4.20 Reif Re84, Re79] deenes a richer classiication of information structure than we (or traditional game theorists) do and develops an associated complexity hierarchy. Can our model, in which mixed strategies are essential, be combined with his? Question 4.21 Perfect-information games in which the depth of the game tree is polynomial retain the same language-recognition power when we replace one of the all-powerful players by a \random player." This is true in both the bounded-error model GS89] and the unbounded-error model Pa85]. Does a similar result about random players hold for games of imperfect information or imperfect recall? References AH92] R. Proof: Throughout, we will use the fact, explained in the proof of Theorem 4.11, that a multiprover interactive proof can be modelled as a player with imperfect recall. Let …