Forgetting a variable

What forgetting does. The knot test looks at what remains when a variable is ignored, so it needs an exact way of ignoring one. Forgetting a variable keeps a state of the other variables whenever at least one value of the forgotten variable completes it to an allowed state. It does not set the variable to 0 or to 1; it lets the variable take whichever value suits. What remains is the system's projection onto the other variables, also called its shadow on them.

• Forget c in the atom. Each of the four states of A and B is completed by c = 1, so the shadow on A and B allows all four: the pair is free.
• Forget B in the common hub. The state A = 1, c = 0 has no completion, because ¬A ∨ c rules it out whatever B is. So the shadow on A and c still says that A forces c.

The two ideas fit together: a set of variables is free exactly when forgetting every other variable leaves it unconstrained.

Witnesses. Rules sometimes need helper variables. A witness is a variable added to report on something, such as whether two bits differ or whether two loops are bound. Witnesses are forgotten before any verdict, which is called quantifying them away. They let short rules state long relations. Parity on four variables needs only two three-place rules and one witness w: w = x1 ⊕ x2 and w = x3 ⊕ x4. Forgetting w leaves exactly the states with x1 ⊕ x2 = x3 ⊕ x4, which is the parity x1 ⊕ x2 ⊕ x3 ⊕ x4 = 0. Lesson 3.2 uses the same device to bind any number of loops with three-place rules.