I do this via probability (don't click off!) and polynomials. Now instead of explaining where the definitions come from I'll define them and show what cool properties they have.

A and B = A*B

A or B = A B-A*B

A xor B = A or B - A and B

not A = 1-A

A nand B = not A and B

With these definitions you get the following properties. Firstly, it provides a map between logical statements and polynomials. However this is about all it does. In order to make this definition useful we will have to do one more thing. Say our logical statement, with these definitions maps L(A, B, C) to A²C 2A-B 3B³ (I have made this up). We can define a 'reduced' polynomial, which just 'reduces' all powers greater than one to a power of one. This reduced polynomial is how we will map logical statements to polynomials.

I will use the following terminology. A logical expression is to be defined as an expression combining the functions defined above in some way to produce some output {0, 1} when inputs {0, 1} are given for each input. E.g. L(a, b, c) = (a or (a and (b or c))) xor b. And a logical system as a set of these expressions which you can think of as a set of logic gates with some amount of inputs and some amount of outputs. Putting a subscript in either of these is the number of inputs and superscript the number of outputs. So the set of all logical expressions L_n=L^1_n is a subset of the set of logical systems with m outputs L^m_n.

Defining things like this gets you the following properties:

• There is a map between the set of all logical expressions (L_n for all n) and all degree one polynomials of n variables. Not only this, but the map between L_n and the polynomials of n variables is a bijection.

• The caveat to the previous point is that we are saying two logical systems are considered equivalent if they produce the same outputs for the same inputs. (for example a and b = (a and a) and (b and b))

• When the inputs of some L is in the range [0,1] the rules of probability are followed exactly. i.e. imagining the inputs to be probabilities of said input being 0 or 1(where 0.9 is 95% of the input being 1) will produce the probability of the output of that system to be 0 or 1.

*Creating logic gates this way and running simple programs through them (for example a binary adder) produces pretty interesting results for non integer inputs.

In cases of independent inputs (I.e. your system will never simplify to something like A and A). The unreduced polynomials can also be used. The unreduced polynomials are very interesting as well and come with their own properties

• The biggest being that you can 'invert' them. The inversion is in the form of infinite solutions to an equation but one can write for any system L an inversion L^-1 (which will have infinitely many solutions).

• One can also find a 'eigen input' of any set of outputs logical system system by choosing the inversions of the unreduced polynomials such that it gives the output with every input being the same number. This is only possible over C. (Not R).

I am now experimenting with the logical 'gates' themselves in two ways: 1. with group theory and 2. with a bit of functional analysis and extending the domain of iteration of the gates themselves, such as finding the formula for a nth iteration of each gate and seeing what half an application of an and gate would be. (Thinking of a half iteration in the sense, that iterating a half iteration twice is the same as applying the and gate to two variables). I have only done this for some of the gates. (And I don't think there is any closed formula for the others). Interesting the inversions of the gates mentioned above are equivalent to the -1 iterations of the gates.

They have many more properties - I have a 15 page set of notes all about them which I will share soon.

Sorry this isn't as well written or explained as the last post - im writing it quickly.