CS 365: MODELS OF COMPUTATION. ASSIGNMENT 10
DUE ON MARCH 30, 2017
Problem 10: Deciding the undecidable
Submit a solution to the following problem. Acknowledge all collaborators with whom you
discussed it and all external sources that you may have used in the completion of the assignment.
See the website for the full homework policy.
We discussed in class models of randomized computation where Turing machines can flip coins
with a bias different than 12 . These can be formalized as follows:
Definition 1. For a real number ρ in the range 0 < ρ < 1, a ρ-probabilistic Turing machine is
a nondeterministic Turing machine M with two transition functions δ0 and δ1 such that at every
step of the computation, M follows the appropriate transition in δ1 with probability ρ and the
transition in δ0 with probability 1 − ρ. BPP(ρ) is the class of languages that can be decided with
bounded error by polynomial-time ρ-probabilistic Turing machines.
The standard definitions we introduced in class correspond to the ones above when ρ = 12 . We
saw in class that whenever ρ is “polynomial-time computable”, then BPP = BPP(ρ) . For this
problem, you will show that this seemingly artificial condition is necessary.
Theorem 1. For every language L ⊆ {1}∗ , there exists a real number 0 < ρ < 1 such that
L ∈ BPP(ρ) .
Recall that in Exercise 2 of Assignment 6, you showed that there exist languages over the unary
alphabet {1} that are undecidable. Therefore, the above theorem shows that for some values of ρ,
BPP(ρ) contains even undecidable languages.
Hint. You can think of the value of ρ = 0.ρ1 ρ2 ρ3 . . . as an advice string. You can also use the
following lemma without proof.
Lemma 1. For every 0 < ρ < 1 with binary expansion ρ = 0.ρ1 ρ2 ρ3 . . ., there is a ρ-probabilistic
Turing machine that computes ρn for any n ≥ 1 in time poly(n) with bounded error at most .01.1
For bonus marks, complete the proof of the lemma.
1I.e., 99% of the time, the Turing machine correctly returns ρ ; with probability up to 1%, it returns the wrong
n
value 1 − ρn .
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Extra exercises
Complete the following exercises, but do not submit them for marking.
Exercise 1. The language L ⊆ {0, 1}∗ is in the class coRP if its complement L = {0, 1}∗ \ L is in
RP. Show that ZPP ⊆ RP ∩ coRP.
Exercise 2. Show that the converse inclusion also holds: RP ∩ coRP ⊆ ZPP.
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Bonus problem 10: For all the marks
You may submit a solution to the following problem for bonus marks.
Solve one of the open problems regarding BPP that we saw in class and get an automatic 100%
on the class:
(1) Show that BPP ⊆ NP.
(2) Show that there exists a BPP-complete language.
(3) Show that BPTIME(n100 ) 6= BPTIME(n).
Bonus marks will also be awarded for any partial progress or interesting observations about these
(or other related) problems.
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