Proper and Loop-Free Accepters
Proper PDA is a proper subclass of the accepters whose language-recognizing ability is equivalent to that of full class of pushdown accepters.
A pushdown accepter cannot scan beyond the end of the string on its input tape because it can make no move in which a sharp is scanned.
A pushdown accepter cannot move its stack head left of the first square of the storage tape because it can make no move in which a sharp is read from the stack.
In the program of a pushdown accepter M, a write instruction:
q] write (u,q')
is improper if q' is the label of any read instruction. A pushdown accepter is proper if its program contains no improper instruction.
Let M be any pushdown accepter. One can construct a proper pushdown accepter M' such that L(M)=L(M').
A proper pushdown accepter is loop free if its program contains no loop of write instructions. For any deterministic pushdown accepter M, one can construct a loop-free pushdown accepter M' such that L(M)=L(M').