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Chapter 13: Problem 7

Construct a Turing machine with tape symbols 0, 1, and B that, when given a bit string as input, replaces the first 0 with a 1 and does not change any of the other symbols on the tape.

Short Answer

Expert verified
The Turing machine has states q0, q1, and q_accept. It replaces the first 0 with 1 and stops.

Step by step solution

01

Define the States

Identify the states needed for the Turing machine. Let's define three states: `q0` (initial state), `q1` (state after finding the first 0 and replacing it with 1), and `q_accept` (accepting state).
02

Start in Initial State

Begin in the initial state `q0` and scan the tape from left to right until a 0 is found.
03

Transition on Finding 0

When the machine is in state `q0` and reads a 0, it should replace this 0 with a 1, move the tape head to the right, and transition to state `q1`.
04

Transition to Accepting State

When the machine is in state `q1`, it simply transitions to the accepting state `q_accept`, signifying the end of the computation.
05

Write the Transition Function

Compile the transitions into the format: (current_state, read_symbol) -> (new_state, write_symbol, move_direction). For this Turing machine, the transitions are as follows: - (q0, 0) -> (q1, 1, R)- (q0, 1) -> (q0, 1, R)- (q0, B) -> (q_accept, B, R)- (q1, 0) -> (q1, 0, R)- (q1, 1) -> (q1, 1, R)- (q1, B) -> (q_accept, B, R)
06

Verify the Machine

Ensure the Turing machine correctly handles the tape input and halts in the accepting state (`q_accept`) after replacing the first 0 with 1 and not altering other symbols.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

tape symbols
In the context of a Turing machine, tape symbols are a crucial component. They represent the different characters or symbols that can be found on the tape.
The tape itself is an infinite strip divided into cells, each capable of holding one symbol.
For our specific Turing machine exercise, the tape symbols are `0`, `1`, and `B` (where `B` denotes a blank space).
The Turing machine will read and write these symbols during its operation.
Understanding tape symbols helps elucidate how the machine interprets and manipulates data.
Key points to remember include:
  • The tape head reads and writes symbols one at a time.
  • The machine's instructions depend on current tape symbol and state.
  • Changes to tape symbols are critical in achieving the desired calculation or transformation.
finite states
The Turing machine operates through a finite number of states.
Each state symbolizes a unique configuration or step in the computation process.
Let's break down the finite states defined in our Turing machine exercise:
  • `q0`: The initial state where the machine begins.
  • `q1`: The state entered after locating the first `0` and converting it to `1`.
  • `q_accept`: The accepting state, indicating that the machine has successfully completed its task.

The machine starts in the initial state (`q0`) and based on the tape symbols it reads, transitions between states as defined by the transition function.
This ultimately helps the machine to process input and reach a final state.
transition function
The transition function of a Turing machine maps its current state and tape symbol to a new state, tape symbol, and tape head direction.
Essentially, it acts as a set of rules guiding the machine's operations.
For our exercise, the transition function is structured as follows:
  • In state `q0` reading `0`, transition to state `q1`, write `1`, and move the tape head right.
  • In state `q0` reading `1`, remain in `q0`, write `1`, and move tape head right.
  • In state `q0` reading `B`, transition to `q_accept`, write `B`, and move the tape head right.
  • In state `q1` reading `0`, remain in `q1`, write `0`, and move tape head right.
  • In state `q1` reading `1`, remain in `q1`, write `1`, and move tape head right.
  • In state `q1` reading `B`, transition to `q_accept`, write `B`, and move the tape head right.

These transitions collectively enable the machine to perform its intended function - replacing the first `0` with a `1` and halting correctly.
computation verification
Verification is essential to ensure the Turing machine functions correctly.
After defining states and the transition function, it is crucial to test if the machine behaves as intended.
For our exercise, verification steps include:
  • Starting with various input bit strings.
  • Running the machine step-by-step to observe the transitions.
  • Confirming that the first `0` is replaced by `1` and no other symbols are altered.
  • Checking if the Turing machine halts in the `q_accept` state.

Through these verification steps, we confirm that the machine meets its designed objective. Computation verification helps in identifying and rectifying any potential errors in the transition function or state definitions.

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Most popular questions from this chapter

Construct a finite-state machine for a toll machine that opens a gate after 25 cents, in nickels, dimes, or quarters, has been deposited. No change is given for overpayment, and no credit is given to the next driver when more than 25 cents has been deposited.

Find a phrase-structure grammar for each of these languages. a) the set of all bit strings containing an even number of 0 s and no 1 \(\mathrm{s}\) b) the set of all bit strings made up of a 1 followed by an odd number of 0s c) the set of all bit strings containing an even number of 0s and an even number of 1s d) the set of all strings containing 10 or more 0s and no 1s e) the set of all strings containing more 0s than 1s f ) the set of all strings containing an equal number of 0s and 1s g) the set of all strings containing an unequal number of 0s and 1s

Express each of these sets using a regular expression. a) the set consisting of the strings \(0,11,\) and 010 b) the set of strings of three 0 s followed by two or more 0 \(\mathrm{s}\) c) the set of strings of odd length d) the set of strings that contain exactly one 1 e) the set of strings ending in 1 and not containing 000

Let \(M=\left(S, I, f, s_{0}, F\right)\) be a deterministic finite-state automaton. Show that the language recognized by \(M, L(M),\) is infinite if and only if there is a word \(x\) recognized by \(M\) with \(l(x) \geq|S|\) .

Find a phrase-structure grammar for each of these languages. a) the set consisting of the bit strings \(10,01,\) and 101 b) the set of bit strings that start with 00 and end with one or more 1 \(\mathrm{s}\) c) the set of bit strings consisting of an even number of 1 s followed by a final 0 d) the set of bit strings that have neither two consecutive 0 s nor two consecutive 1 \(\mathrm{s}\)
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