I’m revisiting this project because it needs more explanation. I’m going to write the explanation in a few stages, beginning with dual rail logic.

Usually when we think about Boolean circuits and binary values, we associate a wire in the circuit with a single binary value - a high voltage represents 1 and a low voltage represents 0. AND and OR gates operate on these single-wire representations and produce an output value that is represented on single-wire output. A NOT gate turns a high voltage to a low voltage and vice versa.

But this isn’t the only way of representing binary values using voltage levels. Another way is to use two wires, and say that a high voltage on one wire represents 1 and a high voltage on the other wire represents 0. This is called ‘dual rail logic’ or the ‘double line trick’. In this scheme there are no NOT gates - to invert a value you simply cross the wires. A dual rail NAND gate can be made from conventional AND and OR gates as shown in the diagram below.

In this diagram the Boolean value a is represented using the wires a₀ and a₁ and similarly for b and for the output (a NAND b). Because inversion is just wire crossing, then by De Morgan’s law dual rail AND, OR, NAND and NOR gates are all essentially the same, and differ only in which inputs and outputs are inverted by crossing the 0 and 1 wires.

Dual rail logic is occasionally used in electronics, but it really comes into it’s own when thinking about how to represent and implement logical operations in other physical systems, where each Boolean value can be represented by the passage of a physical object along one path or another. In this project, the object is water.

The next part of the explanation will talk about one-shot logic, and about how siphons can be used to make the building blocks of a one-shot dual rail logic system.