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Peter Chini Delivery until at 12h Exercise 4.1 (Immerman and Szelepcs´enyi) In the lecture we have shown that PATH is inNL

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WS 2015/2016 18.11.2015 Exercises to the lecture

Complexity Theory Sheet 4 Prof. Dr. Roland Meyer

M.Sc. Peter Chini Delivery until 26.11.2015 at 12h

Exercise 4.1 (Immerman and Szelepcs´enyi)

In the lecture we have shown that PATH is inNL. Use this to prove the theorem of Immerman and Szelepcs´enyi:

Fors:N→Nwiths(n)≥log n, we have:

NSPACE(s(n)) = co-NSPACE(s(n)).

Exercise 4.2 (Universal Turing Machine Part II)

Lett2 be time constructible andt21 =o(t2). Show that we have a strict inclusion:

DTIME(t1(n))(DTIME(t2(n)).

Hint: We have already shown thatDSPACE(s1)(DSPACE(s2) is a strict inclusion under suitable assumptions. Recall the proof of this theorem and use the same idea to prove the above result.

Exercise 4.3 (Hierarchies and Padding)

Show the following statements, using the hierarchy and transfer results from the lecture:

a) P(EXP, b) NL(PSPACE,

c) IfNL=P then we also have: PSPACE=EXP.

Delivery until 26.11.2015 at 12h into the box next to 34-401.4

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