INFR 11102 : Computational Complexity 15 / 02 / 2018 Lecture 10 : Circuit models

Proof. One direction is easy, namely TQBF ∈ PSpace. Once again, to achieve a spaceefficient algorithm, we use recursion. If the leading quantifier is ∃x, then we recursively check the two cases of setting x to 0 and 1, and return true if one of them is true. Similarly, if the leading quantifier is ∀x, then we recursively check the two cases of setting x to 0 and 1, and return true if both of them are true. At any point of the recursion, we will only need polynomial space. The recursion depth is n, and therefore this is a polynomial space algorithm. For the other direction, let M be a TM with space bound p(n) and x be an input. Recall that M accepts x if and only if there is an accepting path from q0 to qacc in the configuration graph GM,x, whose number of vertices is 2 for some constant c. Next we express this property by a TQBF φ. The basic idea is the same as Savitch’s theorem. To encode that q1 can reach q2 in 2 steps, denoted q1 →2l q2, we go through all possible middle points q′. Namely we ask whether ∃q(q1 →2l−1 q′) ∧ (q′ →2l−1 q2). Now, notice that if we recursively expand the → inside, we would end up with an exponential size formula. The trick, is to rewrite (q1 →2l−1 q′) ∧ (q′ →2l−1 q2) as ∀x, y ((x = q1 and y = q′) ∨ (x = q′ and y = q2)) ⇒ (x →2l−1 y).