MMP.8:5.3 - Find a strategy, rather than an answer chosen with future knowledge

A program repeatedly receives a bit b and emits a bit a. A requirement asks it to emit the same bit. If receipt precedes emission, the rule a=b works on every round. If emission must precede receipt, each possible future bit has a matching answer, but no deterministic rule using only the earlier history can guarantee a match against every admitted next bit.

To see the failure, hold the earlier history fixed. The rule chooses either 0 or 1. Both next input bits are still admitted, including the opposite one. That continuation refutes the guarantee for this history. Inspecting more successful traces cannot remove it.

A changed requirement may ask for success probability under independent fair input bits. Any earlier choice then matches with probability 1/2 in one round, including a randomized earlier choice independent of the next bit. Success in every one of N such rounds has probability 2^(-N). These probabilistic claims use the new input assumption. They do not establish success against every input stream.

This is a continuing computational interaction. The construction determines how each response may depend on incoming information. The same observation-order construction identifies what a distributed team or controller would need to know before acting.