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When the machine starts, it is in state 1, it is positioned at the extreme left end of the tape, and the tape extends indefinitely to the right.
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A particular TM will have a set of instructions it understands. Each instruction consists of a 5-tuple (rhymes with couple), which is a mathematical way of saying that one instruction consists of five values. These values are 1 2 3 4 5 the current state the current symbol being read the symbol with which to replace the current symbol the next state to enter the direction to move the tape (Right, Left, or Stationary) As a first example, suppose a TM includes these three instructions ( means blank): 1 (1, 0, 1, 1, Right ) 2 (1, 1, 0, 1, Right ) 3 (1, , , halt, Stationary ) The first says that if the symbol being read is a 0, replace it with a 1 and move right. The second says that if the symbol being read is a 1, replace it with a 0 and move right. The third says that if the symbol being read is a blank, halt the machine without moving the tape. Assume the tape presented to this TM contains the symbols: 1 1 0 1 0 1 0 0 ... Starting in state 1, and positioned at the extreme left of the tape, the machine reads the symbol 1. Instruction 2 applies to this situation, so the instruction causes the 1 to be replaced by a 0, the machine state to remain 1, and the machine to move 1 cell to the right on the tape. Next the TM reads another 1. Instruction 2 applies again, so the TM changes the second 1 to a 0, and moves right again, remaining in state 1. When the TM reads the symbol 0, instruction 1 applies, so instruction 1 causes the 0 to be replaced by a 1, the machine to stay in state 1, and the machine to move right once again. As the machine advances down the tape, every 1 will be changed to a 0, and every 0 will be changed to a 1. Finally, the machine will read a blank. In that case, instruction 3 will apply, and the machine will halt. This simple TM is a machine for complementing (inverting) the bits of a binary number. The result of the computation will be a tape that contains these symbols: 0 0 1 0 1 0 1 1 ... Complementing the bits of a binary number is a frequently required task, so this is a useful TM. A slightly more complex task is that of complementing and incrementing a binary number. That operation is often used by computers to perform binary subtraction. In fact, in the old days when the only calculating machines available were mechanical adding machines, people performed subtraction the same way in base 10, using the 10 s complement method. To subtract 14 from 17 in base 10, they found the 9 s complement of 14, which is 85 (subtract 1 from 9 to get the 8, and subtract 4 from 9 to get the 5). They incremented 85 by 1, to get 86, or what s called the 10 s complement. Adding 17 and 86 gave 103. Ignoring the carry digit gave the answer of 3! To perform binary subtraction by the 2 s complement method, the subtrahend is complemented and incremented, and then added to the minuend. For instance, to subtract 2 from 5, we can complement and increment 2, and add that to 5 to get 3: 010 101 110 +101 1011 2 (in base 2: 0 fours, 1 two, 0 units) 2 complemented (1s --> 0s; 0s --> 1s) 2 complemented & incremented (adding 001 to 101 --> 110 in base 2) 5 (1 four, 0 twos, 1 unit) 3 (in base 2: 0 fours, 1 two, 1 unit -ignore the carry bit to the left)
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