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Turing Machine

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Turing Machine is a topic from the subject of Theory Of computation

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Turing Machine

  1. 1. Theory of Computation Topic: Definition of Turing Machine
  2. 2. INTRODUCING TURING MACHINES Introduced by Alan Turing in 1936. A simple mathematical model of a computer. Models the computing capability of a computer.
  3. 3. DEFINITION A Turing machine (TM) is a finite-state machine with an infinite tape and a tape head that can read or write one tape cell and move left or right. It normally accepts the input string, or completes its computation, by entering a final or accepting state. Tape is use for input and working storage.
  4. 4. Representation of Turing Machine • Turing Machine is represented by- • M=(Q,Σ, Γ,δ,q0,B,F) , • Where • Q is the finite state of states • Σ a set of τ not including B, is the set of input symbols, • τ is the finite state of allowable tape symbols, • δ is the next move function, a mapping from Q × Γ to • Q × Γ ×{L,R} • Q0 in Q is the start state, • B a symbol of Γ is the blank, • F is the set of final states.
  6. 6. Transition function One move (denoted by |---) in a TM does the following: δ(q , X) = (p ,Y ,R/L) • q is the current state • X is the current tape symbol pointed by tape head • State changes from q to p
  7. 7. Turing machine as language acceptors A Turing machine halts when it no longer has available moves.  If it halts in a final state, it accepts its input, otherwise it rejects its input. For language accepted by M ,we define L(M)={ w ε ∑+ : q0w |– x1qfx2 for some qf ε F , x1 ,x2ε Γ * }
  8. 8. Turing machine as transducers • To use a Turing machine as a transducer, treat the entire nonblank portion of the initial tape as input • Treat the entire nonblank portion of the tape when the machine halts as output. A Turing machine defines a function y = f (x) for strings x, y ε ∑* if q0x |*– qf y • A function index is “Turing computable” if there exists a Turing machine that can perform the above task.
  9. 9. ID of a TMID of a TM • Instantaneous Description or ID :  X1 X2…Xi-1 q Xi Xi+1 …Xn Means:  q is the current state  Tape head is pointing to Xi  X1X2…Xi-1XiXi+1… Xn are the current tape symbols  δ (q , Xi ) = (p ,Y , R ) is same as:δ (q , Xi ) = (p ,Y , R ) is same as: X1 X2…Xi-1 q Xi Xi+1 …Xn|---- X1 X2…Xi-1 YY pp Xi+1…Xn  δδ (q Xi) = (p Y L) same as:(q Xi) = (p Y L) same as: X1 X2…Xi-1 q Xi Xi+1 …Xn|---- X1 X2…ppXi-1YY Xi+1 …Xn
  10. 10. VARIATIONS OF TURING MACHINESVARIATIONS OF TURING MACHINES Multitape Turing Machines Non deterministic Turing machines Multihead Turing Machines Off-line Turing machines Multidimensional Turing machines
  11. 11. MULTITAPE TURING MACHINES A Turing Machine with several tapes  Every Tape’s have their Controlled own R/W Head  For N- tape TM M=(Q,Σ, Γ,δ,q0,B,F) • we define δ : Q X ΓN Q X ΓN X { L , R} N
  12. 12. NON DETERMINISTIC TURINGNON DETERMINISTIC TURING MACHINESMACHINES It is similar to DTM except that for any input symbol and current state it has a number of choices A string is accepted by a NDTM if there is a sequence of moves that leads to a final state The transaction function δ : Q X Γ 2 QXΓ X{L,R}
  13. 13. MULTIHEAD TURING MACHINEMULTIHEAD TURING MACHINE Multihead TM has a number of heads instead of one.  Each head indepently read/ write symbols and move left / right or keep stationery.
  14. 14. OFF- LINE TURING MACHINEOFF- LINE TURING MACHINE An Offline Turing Machine has two tapes 1. One tape is read-only and contains the input 2. The other is read-write and is initially blank.
  15. 15. MULTIDIMENSIONAL TURINGMULTIDIMENSIONAL TURING MACHINEMACHINE A Multidimensional TM has a multidimensional tape. • For example, a two-dimensional Turing machine would read and write on an infinite plane divided into squares, like a checkerboard.  For a two- Dimensional Turing Machine transaction function define as: • δ : Q X Γ Q X Γ X { L , R,U,D}
  16. 16. Applications of TM • Check Decidability If TM cannot solve a problem in countable time  then there could not be any algorithm which could solve that  problem (That is the problem is undecidable).For a decision problem  if its TM halt in countable time for all finite length inputs then we can  say that the problem could be solved by an algorithm in countable  time. • Classify Problem TM helps to classify decidable problems into classes  of Polynomial Hierarchy. Suppose we found that the problem is  decidable. Then out target become how efficiently we can solve it.  The efficiency been calculated in number of steps, extra space used ,  length of the code/size of the FSM. • Design and Implement Algorithm for Practical Machines TM helps to  propagate idea of algorithm in other practical machines. After the  successful check of 1,2 criteria we can use our practical  devices/computers to design and implement algorithm.
  17. 17. Thankyou