# Mathematical description for boolean logic with “hysteresis” through logic-state dependence.

I am having trouble mathematically representing boolean logic gates that have logical hysteresis as they have "state". In asynchronous circuits for example, I have something similar to a AND gate called a Muller C-element that is very much like a logical AND, but the logic is different depending on the initial condition. Starting from 0,0, the truth table is identical to the AND:

A | B | S
0 | 0 | 0
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1


This results in the boolean equation $S = A \land B$; however, starting from 1,1, the truth table is:

A | B | S
0 | 0 | 0
0 | 1 | 1
1 | 0 | 1
1 | 1 | 1


The fundamental issue that I am having is that this logic gate has "state". Even if I describe the gate using pure boolean logic (I can build a static c-element with 3x ANDs and a 3-input OR), I still have the issue of state when describing the function.

Is there an method in boolean mathematics to define pure boolean logic that has state dependence?

• The natural approach would be to describe the pair of output and new state as a function as input and old state. – Henning Makholm Feb 4 '17 at 17:46
• @HenningMakholm I was considering that, but I am trying to get something general form because what I actually want to describe is very large. Due to the interesting control structures that can be made with devices like this, I was hoping that someone knew of a standard method. I'm sorry to say that boolean logic is something that I know from the practical, but not theoretical aspect. – b degnan Feb 4 '17 at 19:27
• Hmm, "pair of output and new state as a function as input and old state" sounds pretty general to me. – Henning Makholm Feb 4 '17 at 19:30
• If you want it to be even more general you would talk of representing the output as a function of the entire history of all of the inputs ... but then you would have a hard time figuring out whether a given such representation specifies something that has a practical realization. – Henning Makholm Feb 4 '17 at 19:32
• @HenningMakholm My fundamental issue is that the actual thing I want to describe is very large. I will let this question sit here for a few days, and hopefully someone has looked at an esoteric boolean math problem like this. If not, I encourage you to turn your first comment into an answer as that does seem most promising. – b degnan Feb 4 '17 at 19:43