Halting Problem - Common Pitfalls

Common Pitfalls

The difficulty in the halting problem lies in the requirement that the decision procedure must work for all programs and inputs. A particular program either halts on a given input or does not halt. Consider one algorithm that always answers "halts" and another that always answers "doesn't halt". For any specific program and input, one of these two algorithms answers correctly, even though nobody may know which one.

There are programs (interpreters) that simulate the execution of whatever source code they are given. Such programs can demonstrate that a program does halt if this is the case: the interpreter itself will eventually halt its simulation, which shows that the original program halted. However, an interpreter will not halt if its input program does not halt, so this approach cannot solve the halting problem as stated. It does not successfully answer "doesn't halt" for programs that do not halt.

The halting problem is theoretically decidable for linear bounded automata (LBAs) or deterministic machines with finite memory. A machine with finite memory has a finite number of states, and thus any deterministic program on it must eventually either halt or repeat a previous state:

...any finite-state machine, if left completely to itself, will fall eventually into a perfectly periodic repetitive pattern. The duration of this repeating pattern cannot exceed the number of internal states of the machine... (italics in original, Minsky 1967, p. 24)

Minsky warns us, however, that machines such as computers with e.g. a million small parts, each with two states, will have at least 21,000,000 possible states:

This is a 1 followed by about three hundred thousand zeroes ... Even if such a machine were to operate at the frequencies of cosmic rays, the aeons of galactic evolution would be as nothing compared to the time of a journey through such a cycle (Minsky 1967 p. 25):

Minsky exhorts the reader to be suspicious—although a machine may be finite, and finite automata "have a number of theoretical limitations":

...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness the state diagram may not carry a great deal of significance. (Minsky p. 25)

It can also be decided automatically whether a nondeterministic machine with finite memory halts on none of, some of, or all of the possible sequences of nondeterministic decisions, by enumerating states after each possible decision.