# How math deals with state or "algebra of sequential logic"

Boolean algebra is well-studied and you could easily find a lot of introductory materials on different operators, properties, how eveything is nicely related to the algebra of sets, and so on.
Coming from the EE background, I also know that you can construct any circuit using NAND/NOR gates, because of their nice functional completeness properties. The same goes for stateful circuits, as you could build a simple stateful RS-Latch with 2 NAND/NOR gates, by applying them to each other.
This is where it gets interesting - does that mean that the state arises from the recursion? i.e.

 q = nand(r, nand(s, q))


Unfortunately, I couldn't find anything like "algebra of sequential logic" or any notion on how mathematics seems to deal with state (or, better say: time?!). Most likely, my search keywords are compeltely off.
Could you help me out? I would like to know how mathematicians deal with sequential circuits - any introductory book would be great.

• My recommended search terms are Moore Mealy FSM. Aug 8, 2017 at 9:06
• algebraic automaton theory Aug 8, 2017 at 9:07
• @AxelKemper not the state machines, but the state itself. Aug 8, 2017 at 9:33
• The computers in the world run on the sequential switching theory. Montgomery intuited it, Caldwell and his PhD student Huffman developed. That put Americans on the Moon, because Apollo navigation computer used that very logic in its ICs. Sep 26, 2020 at 5:58
• And you're asking right questions. I want to work with you. Please, find me, and get connected. Sep 26, 2020 at 6:02

The math involved is that of Semigroups and its application to "sequential circuits" is named "algebraic theory of machines".