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Lehr- und Forschungsgebiet

Mathematische Grundlagen der Informatik RWTH Aachen

Prof. Dr. E. Grädel, D. Fischer, T. Ganzow, B. Puchala

WS 09/10

Complexity Theory and Quantum Computing — Assignment 7 Due: Monday, December 14, 12:00

Exercise 1

Let GEN be the problem from Assignment 4, Exercise 1, (b). Construct an explicitALogspace- algorithm for GEN.

Exercise 2

An ATM M with adress tape uses one of the working tapes to adress cells of the input tape.

If M writes the number i in binary representation to the adress tape, then the head on the input tape goes to cell numberiin one step. This means, that M can read arbitrary bits of the input in logarithmic time. Therefore, it makes sense to define complexity classes Atime(T(n)) forT(n)< n.

(a) Give a precise definition of this machine model.

(b) Prove that PAL ∈ ALogtime = Atime(O(logn)), where PAL is the problem from Assi- gnment 2, Exercise 3.

Exercise 3

Let A, B and C ben×n matrices. The decision problem whether AB = C can obviously be solved in timeO(n3). This trivial bound can be improved, however, no deterministic algorithm is known whichs solves this problem in time less than O(n2.3). Now consider the following randomized algorithm.

input:A,B,C

choose a vector x∈ {−1,1}n at random if A(Bx)6=Cx: reject, else: accept.

Prove that this algorithm solves the problem with O(n2) arithmetical operations and with an error probability less than 12.

Exercise 4

Prove the following fact: If NP⊆BPP, thenRP=NP.

Hint: First, prove that from aPtimealgorithm for SAT one can construct aPtime algorithm which constructs, for a given satisfiable formula, a model of this formula. Now use the fact that from NP⊆BPPin particular it follows that SAT∈BPP.

http://www.logic.rwth-aachen.de/Teaching/KTQC-WS09/

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