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Turing Machine
A Turing machine consists of a finite-state control unit, an infinite tape divided into cells to hold symbols, and a read/write head. At each step, the control unit reads the symbol under the head, writes a symbol, and moves the head left or right. It can enter a new state depending on the symbol and its current state. Turing machines can recognize, generate, evaluate, and decide languages by looping through state transitions and tape operations until reaching an accepting or rejecting state. They provide a model for general computation that is theoretically more powerful than finite state machines.
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Turing Machine
- 2. Conceptually aTuring machine, like finite automata A Turing machine consists of a finite-state control unit and a tape that is infinite in both directions and divided into cells, each of which can hold one symbol: At each step, the control unit performs two actions in a way dependent on: The two actions will be: 1. either – move the read/write head one cell to the left or right; 2. put the control unit into a new state
- 3. Turing machine willbe your ultimate model for computer. In Turing machine output is very important. We know that every program has an output but this is not exactly true.
- 4. 1.READ X 2.IF X=1 THEN END 3.IF X=2 THEN DIVIDE X BY 0 4.IF X>2 THEN GOTO STATEMENT 4 We shall see in a moment that the same terminology is applied to Turing machine.
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- 6. ............ Read-Write head No boundaries-- infinite length The head moves Left or Right
- 7. ............ Read-Write head The headat each transition (time step): 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right
- 8. ............ Example: Time 0 ............ Time 1 1.Reads 2. Writes a a cb a b k c a k 3. Moves Left
- 9. ............ Time 1 a bk c ............ Time 2 a k cf 1. Reads 2. Writes b f 3. Moves Right
- 10. ............ ◊ ◊◊ ◊ Blank symbol head ◊a b ca Head starts at the leftmost position of the input string Input string
- 11. 1q 2qLba ,→ ReadWrite Move Left 1q 2qRba ,→ Move Right
- 12. Definition. ATuring machine is a 7-tuple (Q, ∑, Γ, ∂, q0, qaccept, qreject), where: • Q is a set of states ∀ ∑ is a set of symbols (the alphabet) ∀ Γ is a set of symbols that can be written in tape, ∈ Γ and ∑ ⊆ Γ • q0 ∈ Q is the initial state
- 13. • qaccept isthe accepting state • qreject is the rejecting state, qreject ≠ qaccept ∀ ∂ is a collection of transitions defined by the function: ∂: (Q − {qaccept, qreject }) × Γ Q × Γ × {, }
- 14. The machine haltsin a state if there is no transition to follow
- 15. Halting Example 1: ............◊ ◊ ◊ ◊◊a b ca 1q 1q No transition from HALT!!! 1q
- 16. 1q 2q Allowed 1q2q Not Allowed •Accepting states have no outgoing transitions •The machine halts and accepts
- 17. Accept Input If machinehalts in an accept state Reject Input If machine halts in a non-accept state or If machine enters an infinite loop string string
- 18. Accepts the language:*a 0q Raa ,→ L,◊→◊ 1q Input alphabet },{ ba=Σ
- 19. ◊ ◊ ◊◊aaTime 0 0q a 0q Raa ,→ L,◊→◊ 1q
- 20. ◊ ◊ ◊◊aaTime 1 0q a 0q Raa ,→ L,◊→◊ 1q
- 21. ◊ ◊ ◊◊aaTime 2 0q a 0q Raa ,→ L,◊→◊ 1q
- 22. ◊ ◊ ◊◊aaTime 3 0q a 0q Raa ,→ L,◊→◊ 1q
- 23. ◊ ◊ ◊◊aaTime 4 1q a 0q Raa ,→ L,◊→◊ 1q Halt & Accept
- 24. Rejection Example 0q Raa ,→ L,◊→◊ 1q ◊◊ ◊ ◊baTime 0 0q a
- 25. 0q Raa ,→ L,◊→◊ 1q ◊ ◊◊ ◊baTime 1 0q a No possible Transition Halt & Reject
- 26. Accepts the language:*a 0q but for input alphabet }{a=Σ A simpler machine for same language
- 27. ◊ ◊ ◊◊aaTime 0 0q a 0q Halt & Accept Not necessary to scan input
- 32. Turing machineas a language recognizer. Turing machine as a language generator. Turing machine as a language evaluator. Turing machine as a language decider.
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