Output cones and semantic support

This overview explains how a circuit's output cone identifies relevant structure, while cancellation can reduce semantic support further. The surrounding circuit size and nominal input list may overstate the function actually being evaluated. Start at the output and follow its inputs backward. The output depends structurally on u and v; those depend on A through D. This identifies the output cone, the part of the graph that can feed the requested output. Now evaluate forward with one one zero one. The first and gives one; the or gives one; their final and gives one. Backtracing identifies relevant structure, while forward evaluation supplies values for a particular assignment. With A fixed to zero, u becomes zero and masks v at the final and. Changing C or D in that context cannot change the answer. This local fact does not prove that C and D are irrelevant to the unrestricted function. An output cone contains the gates that can influence that output. Structural inputs can overestimate semantic support when logic cancels. Neither the surrounding gate count nor the nominal input list alone describes the function being measured. The AND gate combines A and B into the intermediate result S. The next AND gate combines S with NOT C. The final OR gate combines that result with D. An unrelated input E feeds a different output and is outside this cone. At A B C D equal 1100, the selected output is one. Changing E cannot affect it. Each of A, B, C and D has some context that changes the output; the companion enumerates those witnesses. This example is a teaching cone, not an EPFL result row. The retained EPFL AND/INV parity comparison belongs to its selected cones, exact checks and compiled-kernel boundary. Later descriptor-screening studies ask for a different artifact and must not replace that parity conclusion. Return to Circuit One: u equals A and B, v equals C or D, and the output is u and v. Circuit Two in the intervening explanation instead computes A and B and not C, then or D. At A B C D equal one one zero zero, Circuit One gives zero and Circuit Two gives one. The formulas differ; neither answer contradicts the other. The first branch is one, so it no longer masks the second branch. With C and D both zero, v is zero and the output is zero. Changing only D to one makes v one and the output one. This fixed context demonstrates that D can influence the complete function. This case establishes a specific contrast. A structural path can carry influence. Use a fixed context to demonstrate it. 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. In this different circuit, A appears twice in a branch leading to the output. A graph traversal therefore finds A in the output cone. Reachability alone does not yet establish semantic influence. For either value of A, exclusive-or of A with itself is zero. The output reduces to B. Changing A never changes the answer, even though A is structurally reachable in the unreduced graph. An output cone describes graph reachability. Live support describes variables that can affect the function in some context. Use the appropriate notion for the task and do not treat a structural over-approximation as an exact support calculation. Trace the gates that can reach the selected output. Distinguish structural cone membership from the inputs that can change its value.