Lab 02 - Introduction to Turing Machines

Key concepts

1. How is a Turing Machine (TM) defined?
2. What is a configuration?
3. How is the execution of a TM defined?

A Turing Machine consists of:

• an alphabet $ \Sigma$
• a set of states $ K$
• an initial state $ q_0$
• a transition function $ \delta : K \times \Sigma \rightarrow K \times \Sigma \times \{L,H,R\}$
• a set of final states $ F \subseteq K$

Which of the following components of an assembly language would best correspond to the above? $ K,\Sigma, \delta, q_0, F$

• the processor
• the memory
• registers
• assembly instructions

1.1 What does the following TM do?

$ M=(K,\Sigma,q_0,\delta,F)$ where $ K=\{q_0,q_1,q_2\}$ , $ F=\{q_2\}$ , $ \Sigma=\{0,1,\#\}$ and $ \delta$ is defined as below:

1.2 Write a TM which enters the final state only if the input is a binary encoding of an even natural number.

1.3 Write a TM which verifies if a given word over alphabet $ {A,B}$ contains the sequence ABA.

1.4 Write a TM which adds 5 to a number encoded in binary on the tape.

1.5 Write a TM which checks if a binary number x is strictly larger than y. Hint: use a separator symbol between words.

How would the following algorithm be represented as a Turing Machine:

Algorithm(vector V, integer M) {
   integer s = 0
   for-each x in V
      s += x
   if (s > M)
   then  return 1
   else  return 0
}

Helpful questions:

• how should the tape be organised?
• when should the machine accept?
• how would foreach x in V be implemented?
• how would s += x be implemented?
• how would if (s > M) then … else … be implemented ?

• Write a TM which verifies if a string has the same number of ones and zeroes. Give hints - live (what should the machine do?)
• write a TM which accepts a given regular expression
• write a TM which reverses a given binary string (always accepts)