Boolean rules and exact decisions

This overview explains how precise Boolean inputs determine a reproducible decision. An explicit rule lets us trace disagreements to a particular input and operation. Let A mean a valid pass and B mean access is enabled. The first branch, P, requires both. With A one and B zero, P is zero because enabled access is missing. With both one, P is one. The branch already has a complete meaning before we combine it with anything else. The second branch accepts either of two alternative conditions, C and D. This is inclusive or: one true condition is enough, and two true conditions still give one. In this teaching rule, these alternatives form a different branch; they are not extra requirements added to P. The final rule takes exclusive-or of the branch results. It accepts exactly one true branch. Neither branch gives zero; either branch alone gives one; both branches together give zero. This is a deliberately chosen rule, not the usual policy that grants access whenever any route succeeds. Give every input a precise true-or-false meaning. A complete assignment then determines one output under the rule. First compute A AND B and C OR D. Our function takes the exclusive-or of those two results. At 1011, A AND B is zero and C OR D is one. Zero exclusive-or one is one. This traces the decision from its inputs. Change only B to one. Both inner branches now return one, so exclusive-or returns zero. Starting again at 1011, changing D to zero leaves the answer at one because C is still true. A changed input does not change the output in every context. A matrix, program, and packed vector must agree on every assignment in their declared universe. Work a second complete case with A 1, B 1, C 1, D 0. Keep this assignment fixed throughout the calculation. Each symbol retains the meaning already introduced; a new value does not create a new logical rule. Both pass requirements hold, so P is one. The first alternative condition holds, so Q is also one even though D is zero. The outer exclusive-or compares the two branch outputs and returns zero. It accepts exactly one true branch, not one or more. This case establishes a specific contrast. Inclusive OR and XOR differ. The outer gate determines the final rule. 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. For one zero one one, first compute P as one and zero, giving zero. Compute Q as one or one, giving one. The final exclusive-or is zero with one, so the decision is one. Keeping the intermediate values visible prevents us from combining the four inputs in the wrong order. Change only B to one. P becomes one while Q stays one. Now both branches hold, so exclusive-or returns zero. This flip is a direct consequence of the outer gate. It is not an error in the inner or operation. Reset to the first request and change D to zero. C is still one, so Q remains one and the final answer is unchanged. In the different context one zero zero zero, changing D to one changes Q and the final answer. An input can matter to the function even when one particular change is masked. Keep input meanings and the rule precise. Every representation must preserve the same assignment-to-output mapping.