Question
Answer and Explanation
A Turing machine that accepts odd-length strings is a theoretical computing model designed to recognize input strings that contain an odd number of symbols. Let's delve into the details:
Understanding Turing Machines:
A Turing machine consists of:
1. A Tape: An infinitely long tape divided into cells, each capable of storing a single symbol from a finite alphabet.
2. A Head: A device that can read from or write to the current cell and move one cell left or right.
3. A Finite Control Unit: This unit has a set of states, and based on the current state and symbol under the head, it changes the state, writes a new symbol, and moves the head.
Designing a Turing Machine for Odd-Length Strings:
To create a Turing machine that accepts odd-length strings, we'll use a basic approach. Let's consider an alphabet Σ = {a, b}. The language consists of strings like "a", "aba", "bbb", "aabbb", etc. The key is to keep track of whether we've seen an even or odd number of symbols so far. Here’s a high-level description of the Turing Machine:
1. States:
`q0`: The starting state (representing an even number of symbols seen so far).
`q1`: An intermediate state (representing an odd number of symbols seen so far).
`q_accept`: The accepting state.
`q_reject`: The rejecting state (implicitly included if not specified in transitions).
2. Transitions:
If in state `q0` and the head reads any symbol (let’s say 'a' or 'b'), move to the state `q1`, move the head to the right and do not change the symbol.
If in state `q1` and the head reads any symbol ('a' or 'b'), move back to `q0`, move the head to the right and do not change the symbol.
If in state `q0` and the head reads a blank symbol (empty space), go to reject state because there is an even number of symbols.
If in state `q1` and the head reads a blank symbol (empty space), go to accept state because there is an odd number of symbols.
3. Initial Configuration:
The machine starts in state `q0` with the head at the left end of the input string on the tape.
4. Acceptance:
The machine will halt in the accepting state `q_accept` if it reads a blank symbol in state `q1` otherwise will reject (will halt in rejecting state or will keep going forever)
Example:
Let's see how this machine would work for the string "aba":
1. Start: state `q0` , tape: `|a|b|a|`, head at 'a'.
2. Transition: read 'a', move to state `q1`, head moves right: tape:`|a|b|a|`, head at 'b'.
3. Transition: read 'b', move to state `q0`, head moves right: tape:`|a|b|a|`, head at 'a'.
4. Transition: read 'a', move to state `q1`, head moves right: tape:`|a|b|a|`, head at blank.
5. Transition: read blank at `q1`, go to accept state.
The machine accepts since it ends in the accept state after reading the blank symbol in state `q1`
Formal Definition (Simplified):
A Turing machine that accepts odd-length strings can be defined formally as a 7-tuple: (Q, Σ, Γ, δ, q0, q_accept, q_reject), where:
Q = {`q0`, `q1`, `q_accept`, `q_reject`} is the set of states.
Σ = {a, b} is the input alphabet.
Γ = {a, b, blank} is the tape alphabet.
δ is the transition function.
q0 is the initial state.
q_accept is the accepting state.
q_reject is the rejecting state.
Conclusion:
The Turing machine for odd-length strings demonstrates how these theoretical machines can be designed to accept specific language patterns based on input criteria. By using simple states and transition rules, this example provides a fundamental understanding of Turing machines and their operation.