w∈Σ∗, the tape contents left of the head

Universität Koblenz-Landau FB 4 Informatik

Prof. Dr. Viorica Sofronie-Stokkermans^∗1 18.10.2017

M.Ed. Dennis Peuter^∗2

Exercises for Advances in Theoretical Computer Science Exercise Sheet 1

Due at 23.10.2017, 10:00 s.t.

Exercise 1.1

Get acquainted with the following denitions of Turing machines and related concepts:

A Turing machine (TM) Mis a tupleM= (K,Σ, δ, s) with

• K a nite set of states, h6∈K,

• Σan alphabet, L, R6∈Σand #∈Σ,

• δ :K×Σ→(K∪ {h})×(Σ∪ {L, R}) a transition function, and

• s∈K an initial state.

The transition δ(q, a) = (q⁰, x) describes that if a TM is in state q ∈ K and the symbol a∈Σis read, the TM changes its state to q⁰ ∈K∪ {h} and

• moves the head one step to the left, ix=L

• moves the head one step to the right, i x=R

• does not move the head but prints the symbol b∈Σon the tape, ix=b∈Σ A conguration C of a TM M= (K,Σ, δ, s) is a stringC=q, wau, with

• q ∈K∪ {h}, the current state,

• w∈Σ^∗, the tape contents left of the head,

• a∈Σ, the tape content under the head (the current symbol),

• u∈Σ^∗(Σ− {#})∪ {ε}, the tape contents right of the head,

The initial conguration C0 ofMis dened as C0=s,#w#with input w∈Σ^∗.

C₂ =w₂a₂u₂ is a successor conguration ofC₁ =w₁a₁u₁, written as C₁ `_M C₂, i there is a transitionδ(q1, a1) = (q2, b) and:

Case 1: b∈Σ. Thenw₁ =w₂, u₁=u₂, a₂ =b.

Case 2: b=L. Then for w₂ and a₂ : w₁ = w₂a₂. For u₂ : If a₁ = # and u₁ = ε, then u₂=ε, otherwiseu₂=a₁u₁ .

Case 3: b=R. Then for w₂ =w₁a₁. Fora₂ and u₂ : If u₁ =ε, thenu₂=εand a₂= #, otherwise u₁ =a₂u₂ .

C₀ `^∗_MC_n is called computation, i for allC_i with0≤i < n,C_i+1 is a successor congu- ration of Ci.

Exercise 1.2

a) Dene a Turing machine M_a that accepts all wordsw∈ {|}^∗ with an even length, i.e.

M_aholds, i w has even length, otherwiseM_a does not terminate.

b) Dene a Turing machineM_d that decides if a wordw∈ {|}^∗ has an even length.

s,#w#`^∗_M

d h,#Y#if w has even length s,#w#`^∗_M

d h,#N#if w has odd length

c) Dene a Turing machineM_i that adds one|to an input wordw∈ {|}^∗. s,#w#`^∗_M

i h,#w|#.

You can decide to give the formal denition of the Turing machines or to draw it in the ow chart notation.

∗¹ B 225 sofronie@uni-koblenz.de https://userpages.uni-koblenz.de/~sofronie/

∗² B 223 dpeuter@uni-koblenz.de https://userpages.uni-koblenz.de/~dpeuter/

If you want to submit solutions, please do so until 23.10.2017, 10:00 s.t. via the cardboard box in the shelf in room B 222 or via e-mail (with Homework ACTCS in the subject) to dpeuter@uni-koblenz.de.