IP = PSPACE , zero knowledge

A small modification of the previous protocol gives an interactive proof for any language in PSPACE, and hence PSPACE ⊆ IP. In our homework we will prove that IP ⊆ PSPACE, and here we only show the the more interesting direction, namely showing that PSPACE ⊆ IP. We will now work with the PSPACE-complete language TQBF, which (recall) consists of true quantified boolean formulas of the form: ∀x1∃x2 · · ·Qnxn φ(x1, . . . , xn), where φ is a 3CNF formula. We begin by arithmetizing φ as we did in the case of #P; recall, if φ has m clauses this results in a degree-3m polynomial Φ such that, for x1, . . . , xn ∈ {0, 1}, we have Φ(x1, . . . , xn) = 1 if φ(x1, . . . , xn) is true, and Φ(x1, . . . , xn) = 0 if φ(x1, . . . , xn) is false. We next must arithmetize the quantifiers. Let Φ be an arithmetization of φ as above. The arithmetization of an expression of the form ∀xn φ(x1, . . . , xn) is ∏