Define a Turing machine with doubly$-$infinite tape to be a single$-$tape deterministic Turing machine whose

tape extends infinitely both to the right and the left. Initially, the tape is blank everywhere except where

the input is written and the tape head is initially at the leftmost symbol of the input. Computation is

defined as usual except the tape head never encounters the left end of the tape. Prove that a language is

Turing$-$recognizable if and only if a Turing machine with double$-$infinite tape recognizes it$.$