wheresrhys

I did say, and I quote:

Now, somebody has probably already done this, but I’ll throw in my twopenneth anyway.

with respect to finding analytical function for the logic gates. But, lo and behold, I’m the top result in Google for it, and no-one else seems to have thought of exploring this before.

So maybe it’s not so obvious as I thought.

Now, somebody has probably already done this, but I’ll throw in my twopenneth anyway.

Yesterday morning I woke feeling strangely alert, so decided to do some maths. Namely, finding analytical functions of the real numbers taht agree with the logical operators NOT, AND, OR and XOR on the values of 0 and 1, and here they are:

• NOT(x) = 1-x
• AND(x,y) = xy
• OR(x,y) = 1-(x-1)(y-1)
• XOR(x,y) = (1-xy)(x+y)

Now, the above begs a few questions

1. Are they any use? Well, I think so. They can be combined and recombined to form an polynomial function LP: {1,0}^n —> {1,0} to represent any logical proposition, where n is the number of elementary propositions. So given the truth or falsity of all these propositions the truth or falsity of the compound statement can easily be deduced
2. Are these the simplest analytical functions that agree with the logical operators on 1 and 0? Probably NOT, OR and AND are; they’re all quadratic or less. But XOR is a cubic expression, which is unexpected. I can’t help thinking a hyberbolic parabola – quadratic – with the relevant constants shoudl work. Will have a think. *edit – success!
3. Can they be extended over the reals? Well, yes – they’re analytical! But a good question is ‘What is the real world interpretation of a function like this?’ Extending the factorial function over the reals has proved useful, but would extending logic to things being doubly true, trebly true, negatively true, make any sense? Search me. The functions above could withstand the insertion of a few square/cube roots here and there, thus making the graphs more linear, and maybe they would be more likely to lend themselves to a real world interpretation. But the shapes of the graphs for the non-rooted functions (and probably ones with roots taken) (see below) defy interpretation I think.
4. What do these functions look like?