In non-trivial settings, (some but not all) structural differences between programs lead to differences in input/output behaviour, even if there is a large domain for which they are behaviourally equivalent.
I think this is a crux (of why we’re talking past each other; I don’t actually know if we have a substantive disagreement). The post was about detecting “smaller than a lookup table would support” implementations, which implied that the input/output functionally-identical-as-tested were actually tested in the broadest possible domain. I fully agree that “tested” and “potential” input/output pairs are not the same sets, but I assert that, in a black-box situation, it CAN be tested in a very broad set of inputs, so the distinction usually won’t matter. That said, nobody has built a pure lookup table anywhere near as complete as it would take to matter (unless the universe or my experience is simulated that way, but I’ll never know).
My narrower but stronger point is that “lookup table vs algorithm” is almost never as important as “what specific algorithm” for any question we want to predict about the black box. Oh, and almost all real-world programs are a mix of algorithm and lookup.
I think this is a crux (of why we’re talking past each other; I don’t actually know if we have a substantive disagreement). The post was about detecting “smaller than a lookup table would support” implementations, which implied that the input/output functionally-identical-as-tested were actually tested in the broadest possible domain. I fully agree that “tested” and “potential” input/output pairs are not the same sets, but I assert that, in a black-box situation, it CAN be tested in a very broad set of inputs, so the distinction usually won’t matter. That said, nobody has built a pure lookup table anywhere near as complete as it would take to matter (unless the universe or my experience is simulated that way, but I’ll never know).
My narrower but stronger point is that “lookup table vs algorithm” is almost never as important as “what specific algorithm” for any question we want to predict about the black box. Oh, and almost all real-world programs are a mix of algorithm and lookup.
Cool, that all sounds fair to me. I don’t think we have any substantive disagreements.