Peter Chini Delivery until at 12h Exercise 1.1 (COPY can be decided in quadratic time) Let Σ ={a, b

WS 2015/2016 28.10.2015 Exercises to the lecture

Complexity Theory Sheet 1 Prof. Dr. Roland Meyer

M.Sc. Peter Chini Delivery until 04.11.2015 at 12h

Exercise 1.1 (COPY can be decided in quadratic time)

Let Σ ={a, b,#} be an alphabet. Recall the definition of the language COPY from the lecture:

COPY ={w.#.w

w∈ {a, b}^∗}.

Show that COPY is inDTIME(n²).

Hint: Construct a deterministic Turing Machine that decides COPY in quadratic time.

Exercise 1.2 (Crossing sequences of Turing Machines)

LetM be a Turing Machine and x=x1.x2,y=y1.y2 words over an alphabet Σ so that CS(x,|x₁|) =CS(y,|y₁|).

Prove that x₁.x₂ ∈L(M) if and only if x₁.y₂∈L(M).

Exercise 1.3 (Θ, Ω andO-Notation)

Letg:N→Nbe a function. Recall the following definitions from the exercise class:

Θ(g(n)) =

f :N→N

there exist c₁, c₂ >0 and n₀∈Nsuch that 0≤c1g(n)≤f(n)≤c2g(n) for all n≥n0

,

O(g(n)) =

f :N→N

there exist c >0 and n0∈Nsuch that 0≤f(n)≤cg(n) for alln≥n₀

,

Ω(g(n)) =

f :N→N

there exist c >0 andn0 ∈Nsuch that 0≤cg(n)≤f(n) for alln≥n₀

.

Show that the following equality of sets holds:

Θ(g(n)) =O(g(n))∩Ω(g(n)).

Delivery until 04.11.2015 at 12h into the box next to 34-401.4