Distinctions and states

Distinctions. Everything in this guide is built from one material, the distinction: two alternatives of which exactly one holds. A distinction is one bit, and its two alternatives are written 0 and 1. Nothing else is assumed. There is no space, no order and no number beyond the two values.

Variables. A variable names a distinction. The guide writes the distinctions that loops bring (lesson 1.4) as capital letters, such as A and B, the contact between loops as c, helper variables called witnesses (lesson 1.6) as w, and any other variables as x, y and z. The letters are only names, and renaming them changes nothing.

Joint states. A joint state, or just a state, gives every variable a value. A state is written as its values in a fixed order, so with the order A, B, c the string 110 means A = 1, B = 1 and c = 0. Three variables have eight joint states:

000  001  010  011  100  101  110  111

Each new variable doubles the count, so n variables have 2^n joint states.

The cube of states. It helps to draw the eight states as the corners of a cube, with one direction for each variable: A runs from left to right, B from front to back, and c from bottom to top.

          011 ─────── 111
         ╱ │         ╱ │
      001 ─────── 101  │
       │  010 ─────│─ 110
       │ ╱         │ ╱
      000 ─────── 100

Two states are neighbours, joined by an edge of the cube, when they differ in exactly one variable, as 110 and 111 do. They are opposite when they differ in all three, as 000 and 111 do. Chapters 1 to 3 read many patterns off this cube. It is a picture of the states, not a space in which the variables sit.