CMP.1:5.2 - An event decider would decide halting

A team asks for a procedure that always decides whether an arbitrary deterministic program with unbounded working memory will eventually emit a designated event. The input is a finite program description, its finite initial data and the event to be recognized. The guarantee includes terminating with “no” for a program that never emits it.

Use the halting problem as A: given a program P and input x, decide whether P(x) terminates. Under the ordinary Turing-computable model, no total algorithm decides this for all programs and inputs.

Construct a program Q with x and P’s description included. Q simulates P on x, suppresses the simulated program’s output, and emits the designated event if and when the simulation halts. The description of Q is obtained by placing the supplied data inside this fixed wrapper; constructing it does not run P(x).

If P(x) halts, Q emits. If P(x) does not halt, Q never reaches its emitting step. A supposed total event-decider applied to Q would therefore decide A in both cases. This contradicts the halting result, so the requested universal event-decider is unavailable under these assumptions.

The direction matters: halting was reduced to event decision. The argument constructed a halting decider from the assumed event decider.

Now change the admitted system to a fully represented deterministic finite-state machine with effective transitions and a decidable emitted-event label on each transition. Starting from its initial state, follow transitions while remembering visited states. Return yes upon the event; return no if the machine halts without it or repeats a state before emitting it. At most the number of reachable states can be visited before such repetition or termination. Determinism and complete state make the future repeat as well.

This supplies a usable decision procedure for the changed class. If an environment can add unrepresented inputs or the “state” omits a changing counter, restore those inputs or state before applying this finite-state result. C.29.2 and A.3.3.TR supply that formulation work. The original universal impossibility and this restricted procedure remain compatible.