Multi Head, Multi Tape Turing Machine
- 1. UNIT IV– TURING MACHINE October 11, 2018 1UNIT IV TURING MACHINE
- 2. A multi head Turing machine is a single tape TM having n heads reading symbols on the same tape. In one step all the heads sense the scanned symbols and move or write independently. The heads are numbered 1 through n and move of TM depends upon the state and symbols scanned by each head. In one move the heads may move left , right or remain stationary. This type of TM is as powerful as one tape Turing machine. The multi head Turing Machine is as shown in the following October 11, 2018 2UNIT IV TURING MACHINE
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- 4. The multi tape Turing Machine is a type of Turing machine in which there are more than one input tapes. Each tape is divided into cells and each cells can hold any symbol of finite tape alphabet. The multi tape Turing machine is as shown in the following image. This TM is more powerful than the basic Turing machine. Because finite control reads more than one input tape and more symbols can be scanned at a time. October 11, 2018 UNIT IV TURING MACHINE 4
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- 6. Turing Machine has a great computational capabilities and hence it can be used as a general mathematical model for modern computers. Turing Machine can model even recursive enumerable languages. Thus the advantage of Turing machine is that it can model all the computable functions as well as the languages for which the algorithm is possible. October 11, 2018 UNIT IV TURING MACHINE 6
- 7. Halting Problem An important question of computing science is “Are there problems that cannot be solved?” There are, and probably the most famous of these is the halting problem described by Turing. He was thinking in terms of Turing machines (there were no computers), but it is easy to extend the idea. Can we write a program that will look at any computer program and its input and decide if the program will halt (not run infinitely)? A practical solution might be to run the program and if it halts you have your answer. If after a given amount of time it doesn’t halt, guess that it won’t halt. However, you wouldn’t know if the program would eventually halt.
- 8. Halting Problem The state halting problem we will consider the given configuration of a turing machine. The output of TM can be HALT The machine starting at this configuration will halts after a finite number of states. No HALT The machine starting at this configuration never reaches a halt state, no matter how long it turns. Now question is arises on these two observation. Given any functional matrix , input data tape and initial configuration, then it is possible to determine whether the process will ever halts? This is called HP. Yes Solvable and No Unsolvable. Now we prove why it is unsolvable.
- 9. Halting Problem October 11, 2018 UNIT IV TURING MACHINE 9 Consider TM M1 which decides whether or not any computation by a TM T will ever halt when a description dT of T and tape t of T is given. Then for every input (t,dT) to M1 if T halt for input t, M1 also halts which is called accept halt. Similarly if T does not halt for input t then the M1 will halt which is called reject halt.
- 10. Halting Problem October 11, 2018 UNIT IV TURING MACHINE 10 Now consider another TM M2 which takes an input dT. It first copies dT and duplicates dT on its tape and then this duplicated tape information is given as input to machine M1. but machine M1 is a modified machine with the modification that whenever M1 is supposed to reach an accept halt, M2 never loops forever. It loops if T halts for input t=dT and halts if T does not halt for T=dT.