C.40:5.20 - Choose a basis by the useful variants it can produce
A designer uses three on/off lamps to make indicator patterns. For the present family, the two outer lamps must agree. Two available representations have the same three editable input bits (a,b,c). I displays (a,b,c) directly; S displays (a,b,a), sharing the outer control and leaving c inactive. Both currently store 000 and display 000. The designer wants different acceptable patterns from one inexpensive change, rather than one already specified target.
Use :4.11 to compare the mappings at this starting input. The allowed operation flips exactly one input bit in a fresh copy. Enumerate all three choices; constructing and inspecting each display costs the same in this stipulated case. A trial output that violates outer equality is observable in this offline examination but cannot be returned as an acceptable indicator.
Changed input from 000 | I: output and admissibility | S: output and admissibility |
|---|---|---|
100 | 100, inadmissible | 101, admissible and new |
010 | 010, admissible and new | 010, admissible and new |
001 | 001, inadmissible | 000, admissible but unchanged |
Each mapping produces three distinct trial outputs, so counting distinct trial outputs cannot choose between them. I supplies one new acceptable pattern; S supplies two. S’s unchanged third output is acceptable but adds no new pattern. The joint comparison therefore favors retaining S as the mapping for these one-step variations, with its input and changing operation. The designer can also retain 101 and 010 as usable results; keeping those displays alone would not preserve the shared generating operation.
Now the next use permits asymmetric indicators and specifically requires 100. Under I, flipping a alone produces it. Under S, the first and third output positions are equal for every input, so no sequence of its permitted input flips can produce 100. The earlier trial is still a correct observation of the old use, but it no longer supports preferring S for this one. Return to the mapping: select I for the direct change, or add and examine an independent outer control if preserving S’s shared operation also matters. More attempts under unchanged S cannot repair this limitation.
This finite case compares three specified changes completely. It neither estimates untried tasks nor proves that a sample of good immediate variants supports a longer adaptive search.