Final answer:
The question is about proving the decidability of whether a Turing machine loops without reaching the first blank symbol past the input. It can be shown decidable by constructing a modified finite state machine that halts when it either detects a loop or tries to move beyond the input, confirming the problem is decidable.
Step-by-step explanation:
The question asks us to demonstrate that it is decidable to find out if a given Turing machine M would loop endlessly without ever coming across the first blank symbol B beyond the input w on its tape when given an input w. This query relates to the theory of computation and decidability in the context of theoretical computer science.
Examine a modified Turing machine, M', which simulates M but includes an extra restriction, to demonstrate this. M' functions in the same way as M, but it stops whenever M tries to move its head past the final w symbol. The issue then becomes determining whether M' enters a loop before reaching the boundary established by the end of the input, given that the tape of M is infinite but M' only needs to take into account a finite amount of it (up to w). This is equivalent to detecting a cycle in a finite state machine. As finite state machines are decidable, we can construct a decider for M' that halts and accepts if a loop is detected, and halts and rejects if M' tries to move beyond w. This decider proves that the problem, in question, is decidable.