Honors Theory, Spring 2002, Yap Homework 6

This is due on Wed April 17. All reducibility concepts assume Karp-reducibility.

1. (15 Points) The proof of Cook’s theorem in Chapter 3 reduces any language A ∈ NP to SAT. If M is an NP-machine for A, for any input w, the proof constructs a 3CNF formula F_{w} such that F_{w} is satisfiable iff w ∈ A. We claim that something stronger is true: #(F_{w}) is equal to the number of accepting computations of M on input w. Actually, this claim needs a mild condition needed on M. What is it? Prove this claim under this mild condition. Hint: if you do not see what this condition might be, we suggest the strategy of ignoring it at first, and simply try to prove the claim without the condition. If you are careful, you will see what else is needed.

2. (20 Points) Prove that QBF is PSPACE-complete. Since this proof can be found in the literature, I will enforce some originality in your solution by asking you to use the same framework as my description of Cook’s theorem in Chapter 3. The additional idea you need comes from the key idea in the proof of Savage’s theorem: if C ⊢^{2m} C′ (i.e., there is an 2m-step path from C to C′ then C ⊢^{m} C′′ and C′′ ⊢^{m} C′ for some C′′.

3. (20 Points) This exercise helps you gain some facility with the group theoretic ideas in the NONISO ∈ IP proof. Let V = V_{n} = {1, …, n} and S_{n} be the set of permutations on V_{n}. The trivial permutation is denoted 1_{n} (or simply 1). Write the composition of σ, σ′ ∈ S_{n} in the form of a product σσ′, instead of σ ∘ σ′.

(i) Let 2 ≤ k ≤ n. If \{a_{1}, …, a_{k}\} ⊆ \binom{V}{k}, then (a_{1}, …, a_{k}) ∈ S_{n} denotes the permutation which takes each a_{i} to a_{(i+1 mod k)}, called a cyclic permutation. Two special cases are k = 2 or k = n. Then (a_{1}, …, a_{k}) is transpose or a Hamiltonian permutation, respectively. Two cyclic permutations (a_{1}, …, a_{k}), (b_{1}, …, b_{ℓ}) are disjoint if a_{i} ≠ b_{j} for all i, j. For instance, (132)(45) = (45)(132) is a product of two disjoint cycles. The order of writing disjoint products does not matter. Show that every non-trivial permutation is a product of disjoint permutations.

(a) G_{0}  (a) G_{1} 1 3 4 5  2  1  2 5  3 4

Figure 1: Two labeled digraphs G_{0}, G_{1}.

(ii) Let G_{0} be the digraph shown in Figure 1. Determine iso(G_{0}) and iso(G_{1}). What the sizes of these two sets? (iii) Determine aut(G_{0}) and aut(G_{1}). What the sizes of these two sets?