A Turing machine with doubly infinite tape (TMDIT) is similar to an ordinary Turing machine except that its tape is infinite to the left as well as to the right. The tape is initially filled with blanks except for the portion that contains the input. Computation is defined as usual except that the head never encounters an end to the tape as it moves leftward. Show that the class of languages recognized by TDMITs is the same as the class of Turningrecognizable languages

Answer and Explanation:

A TM with doubly infinite tape can simulate an ordinary TM. It marks the left-hand end of the input to detect and prevent the head from moving off of that end. To simulate the doubly infinite tape TM by an ordinary TM, we show how to simulate it with a 2-tape TM, which was already shown to be equivalent in power to an ordinary

TM. The first tape of the 2-tape TM is written with the input string, and the second tape is blank. We cut the tape of the doubly infinite tape TM into two parts, at the starting cell of the input string. The portion with the input string and all the blank spaces to its right appears on the first tape of the 2-tape TM. The portion to the left of the input string appears on the second tape, in reverse order.

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