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A Turing machine is both the most primitive computer it is possible to build and the second most powerful. It consists of four simple components: an infinitely long tape, a read/write head, a state register, and a transition table (Fig. 16-2, Table 16-1). The tape contains an infinite series of cells that each hold one symbol from an alphabet. A simple alphabet consists of a blank (0) and a one (1). Each cell on the tape can be written to as often as desired, and of course you can read the symbol that is in a cell. All cells on the tape are assumed to hold a blank unless they have been explicitly set to a different symbol. The head interacts with the tape. It reads the contents of the cell it is positioned over and can write a new value to that cell. It can also move the tape forward and backward, one cell at a time. The state register is like a counter on the head. It keeps track of the current state, which may be a number or a name. A state is like a marker, a piece of memory that keeps track of where you are in the computation.
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Fig. 16-2.
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Turing machine.
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Table 16-1 Transition table State 1 2 2 3 3 4 4 5 5 Read 1 1 0 0 1 1 0 1 0 State 2 2 3 4 3 4 5 5 1 Write 0 1 0 1 1 1 0 1 1 Move R R R L R L L L R
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The transition table contains the rules that drive the Turing machine. It defines a pattern to match, consisting of a state and a cell symbol. It maps this pattern to actions for the machine to perform. These actions consist of a new state for the machine, a symbol to write in the current cell, and a
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direction to move the tape. If there is no pattern matching the current state/ cell, the machine stops and the program is complete. Table 16-1 is an example transition table from Wikipedia. State 1 starts the process and the program execution is given in Table 16-2. The position of the head is indicated by the marked cell.
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Table 16-2 Step No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Turing machine execution State 1 2 2 3 4 5 5 1 2 3 3 4 4 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 Tape cells 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 halt 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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The most powerful computer is a universal Turing machine. In this universal machine, the content of the tape is expanded so that it describes the transition table of a Turing machine as well as the program it runs. The universal Turing machine then simulates this described machine. Any language that has all of the capability of a universal Turing machine can emulate any other computational machine. Such a language is known as a Turing complete language. A real computer language can, in fact, be as simple as the Turing machine. The rudely named language that we shell refer to as BF is one such. It s more an excercise in what is possible than a practical language. It has only eight instructions: > < . , [ ] Increment position Decrement position Add 1 to the value at the current position Subtract 1 from the current value Output the current value as a character Input a character from the keyboard and store it as the current value If the current value is zero, jump to the statement after the next ] If the current value is not zero, jump to the statement after the previous [
BF programs are, essentially, unreadable strings of characters. For example, the Wikipedia sample program that prints Hello World! is:
[> > > > <<<<-] > .> . .. .> .<< . >. .- - - - - -.- - - - - - - -.> .>.
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