Live support and display dimensions

This overview explains which variables can actually affect an output, compared with the variables named by a surrounding interface. Irrelevant display variables can enlarge a layout without adding distinct behavior. Use the rule A or B. To test whether A can influence the answer, hold B fixed at zero and compare A zero with A one. The answers change from zero to one. This pair is a witness that A belongs to the live support of the function. Now hold B at one instead. Changing A no longer changes the answer, because B already makes the or true. That local masking does not erase the earlier witness. Live support asks whether some fixed context exposes an influence, not whether every context does. A variable can also be written in the expression without influencing the function at all. In A exclusive-or A exclusive-or B, the two A terms cancel for both values of A. The result is always B. Textual occurrence and semantic support are different properties. An input is in semantic support if changing it can alter the output in at least one context. The ambient universe may name extra variables, and a written variable list can overestimate support before simplification. The function takes the exclusive-or of two results: A AND B, and C OR D. Its ambient universe includes A through F, although E and F never occur in the rule. Fix C to one. C OR D becomes one for either D, leaving NOT of A AND B. With B equal one, toggling A changes the output. With A equal one, toggling B changes it too. D cannot change the restricted output in any context. It is outside the remaining support. The four A-B assignments produce four distinct cases. Retaining D as a display axis repeats the outputs across eight cells. With B equal zero, toggling A happens to change nothing. The B-equals-one witness still establishes that A is in support. Return to the opening two-input rule F equals A or B. A and B are Boolean inputs. The intervening restricted four-input rule is a separate example. Here set A to zero and B to zero; then change only B to one. At zero zero the output is zero. Hold A at zero and change B to one; the output becomes one. This is a witness that B can influence the unrestricted function. The earlier context with A one masked that influence but did not invalidate this witness. This case establishes a specific contrast. Some fixed context exposes B. One masked context does not remove support. Keep the input, intermediate values, and conclusion together when comparing it with the previous case. The calculated answer belongs to this complete assignment under the stated rule. A restriction fixes an input and produces the remaining function. With B fixed to zero, A or zero becomes A. We have not filled in an unknown answer by guessing. We have explicitly selected one slice of the original function. With B fixed to one, the remaining function is the constant one. A was live in the original function but is not live after this restriction. The support of a restricted function must be determined for that new function. Suppose we display the original function using an extra input C that does not occur in it. Every A B assignment now appears twice, once for each C value, with the same output. The larger ambient layout has more cells, but the rule has gained no dependence on C. Support means that some context can distinguish an input's values. Declare fixed inputs and display axes separately.