So far, relations just say when they hold or how input and output of functions relate. However, you can also use relations to compute new values by being creative with the variables.

For example, say you want to flip an integer variable between 0 and 1 in an assignment (if it is 0 it becomes 1, if it is 1 it becomes 0). You would normally write i := 1 - i but how to express this assignment as a relation?

The key point is to understand that the i value at the left-hand side is not the same as the i value at the right-hand side. The right-hand side value exists until performing the assignment, the value of the left-hand side exists only after the assignment. For clarity, the left-hand side i is written as i+, and we get i+ := 1 - i. Now this can be expressed as a function between variables i+ and i:

A ::=    (i+ == 0 and i == 1)
      or (i+ == 1 and i == 0)

You can read the first line of the A relation as ‘when i equals 1, i+ must equal `0’. With another relation that represents the value of all current variables like

C ::= i == 0 and j == 3

you select the correct line in A with A and C, yielding relation U that says

U ::= i+ == 1 and i == 0 and j == 3

Almost there, except i+ must become i and the existing i should be removed. This is what variable replacement does. Apply replace(i+, i) on U and you get a new D relation

D ::= i == 1 and j == 3

This relation is just like the C relation, it contains all variables with their existing values. In other words we computed i := 1 - i from state C, resulting in state D.

While this example is quite easy, there is no inherent upper limit to what you can consider to be an ‘assignment’. Basically anything that you can describe as a function between input and output works.

Thus if you construct a relation E that expresses the input - output relation of all edges of an automaton, and you have a relation C expressing the current state, then C and E followed by replace for all variables (assuming that E is complete for all variables), you get a new state C' containing the combined result of taking one of the edges.