PU COS522/MAT578
Advanced Complexity Theory
Spring 2017
Problem Set 1
Out: February 27
Due: March 7 3:00pm
Guidelines:
Please submit your assignment to the TA/lecturer at the beginning of the appropriate lecture.
You are encouraged to work in groups up-to three people, however we ask that you dedicate
enough time to think about each problem by yourself before consulting others. Moreover,
you must write up each problem solution by yourself without assistance. You are asked to
identify your collaborators on problem sets. If you did not work with anyone, please write
“Collaborators: none”.
You may consult the course notes and the text (Arora-Barak). The assignment questions
may be similar to questions on problem sets from past offerings of this course or courses at
other universities. Using any preexisting solutions from these or any other sources is strictly
prohibited.
When this problem set references a theorem/section, we mean the corresponding theorem/section in the online draft of the Arora-Barak book.
Question 1. Let A ⊆ {0, 1}∗ be a language. Define
A<n = {x ∈ A : |x| < n}.
Assume that it is possible to determine in polynomial time if x ∈ A using the results of
queries of the form “y ∈ A<|x| ?”. The queries may be adaptive, meaning that the polynomial
time procedure may choose later queries depending on the results of earlier ones. Show that
A ∈ PSPACE.
Remark: Observe that T QBF satisfies the above property.
Question 2. Prove that MIP ⊆ NEXP.
See Section 8.8 for the definition of MIP and Section 2.6.2 for the definition of NEXP.
Hint: The proof of Theorem 8.3 (dIP = NP) may be useful.
Question 3. Define the language L
L = {(x, m) ∈ (Zm , N) : ∀y ∈ Zm : x 6≡ y 2 (mod m)}.
1-1
1-2
Show that L ∈ IP by directly constructing an interactive protocol for L and proving its
correctness (not using the IP = PSPACE theorem). The protocol should consist of two
communication rounds. In the first round the verifier sends a message to the prover. In the
second round the prover sends a message to the verifier.
Remark: The function f (y) = y 2 (mod m) is a candidate for a one-way-function. Given y
and m, computing f (y) can be done in polynomial time. However, no polynomial time
algorithm for finding y ∈ f −1 (x), for a random x, is known.
Question 4. Prove that SPACE(O(n)) 6= P.
Remark: While this is an interesting result, it doesn’t seem to shed any light on the major
?
open questions like P = PSPACE.
Hint: The proof of Theorem 2.25 (EXP 6= NEXP implies P 6= NP) and Theorem 3.2
(Space Hierarchy Theorem) may be useful here.
Question 5. Demonstrate an oracle A and a language L ∈ NPA such that L is not
polynomial-time reducible to 3SAT , even when the reduction algorithm is given oracle access
to A. That is, show that L 6A
P 3SAT .
Remark: This shows that the Cook-Levin theorem (3SAT is NP-complete) is nonrelativizing!
Hint: The proof of Theorem 3.9 (Baker-Gill-Solovay) may be useful here. Let B be an
oracle that satisfies PB 6= NPB . Define an oracle that can solve both NP membership
queries and B membership queries.