pythoncategoryequations
Project description
# pythoncategoryequations
Category is way to represent and generate directed networks by using sinks, sources and connections from sources to sinks. With the tools provided here you can create and simplify category like equations for the given connector operator. On the equations the underlying + and  operations are basic set operations called union and discard. The multiplication operator * connects sources to sinks. The equation system also has a Identity I term and zerO like termination term O. For futher details go https://en.wikipedia.org/wiki/Category_(mathematics)#Definition
## Usage
Here our connector operation is print function called debug which prints an arrow between two objects:
>>> debug('a', 'b') a > b>>> debug('b', 'a') b > a>>> debug('a', 'a') a > a
Get I and O singletons and class C, which use previously defined debug function.
>>> I, O, C = from_operator(debug) >>> I == I True >>> O == I False >>> C(1) C(1)
The items do have differing sinks and sources:
>>> I.sinks {I} >>> I.sources {I}>>> O.sinks set() >>> O.sources set()>>> C(1).sinks {1} >>> C(1).sources {1}
You can write additions also with this notation
>>> C(1,2) == C(1) + C(2) True
The multiplication connects sources to sinks like this:
>>> (C(1,2) * C(3,4)).evaluate() 1 > 3 1 > 4 2 > 3 2 > 4>>> (C(3,4) * C(1,2)).sinks {3, 4}>>> (C(3,4) * C(1,2)).sources {1, 2}
Or
>>> C(1) * C(2, I) == C(1) + C(1) * C(2) True>>> (C(1) * C(2, I)).evaluate() 1 > 2>>> (C(1) * C(2, I)).sinks {1}>>> (C(1) * C(2, I)).sources {1, 2}
And writing C(1,2) instead of C(1) + C(2) works with multiplication too:
>>> C(1,2) * C(3,4) == (C(1) + C(2)) * (C(3) + C(4)) True
The order inside C(…) does not matter:
>>> (C(1,2) * C(3,4)) == (C(2,1) * C(4,3)) True
On the other hand you can not swap the terms like:
>>> (C(1,2) * C(3,4)) == (C(3,4) * C(1,2)) False
Because:
>>> (C(3,4) * C(1,2)).evaluate() 3 > 1 3 > 2 4 > 1 4 > 2
The discard operation works like this:
>>> (C(3,4) * C(1,2)  C(4) * C(1)).evaluate() 3 > 1 3 > 2 4 > 2
But
>>> (C(3,4) * C(1,2)  C(4) * C(1)) == C(3) * C(1,2) + C(4) * C(2) False
Because sinks and sources differ:
>>> (C(3,4) * C(1,2)  C(4) * C(1)).sinks {3} >>> (C(3) * C(1,2) + C(4) * C(2)).sinks {3, 4}
The right form would have been:
>>> (C(3,4) * C(1,2)  C(4) * C(1)) == C(3) * C(1,2) + C(4) * C(2)  C(4) * O  O * C(1) True
The identity I and zero O work together like usual:
>>> I * I == I True >>> O * I * O == O True
Identity I works as a tool for equation simplifying. For example:
>>> C(1,2) * C(3,4) * C(5) + C(1,2) * C(5) == C(1,2) * ( C(3,4) + I ) * C(5) True
Because:
>>> (C(1,2) * C(3,4) * C(5) + C(1,2) * C(5)).evaluate() 1 > 3 1 > 4 1 > 5 2 > 3 2 > 4 2 > 5 3 > 5 4 > 5
and
>>> (C(1,2) * ( C(3,4) + I ) * C(5)).evaluate() 1 > 3 1 > 4 1 > 5 2 > 3 2 > 4 2 > 5 3 > 5 4 > 5
If two terms have the same middle part you can simplify equations via terminating loose sinks or sources with O: For example:
>>> (C(1) * C(2) * C(4) + C(3) * C(4)).evaluate() 1 > 2 2 > 4 3 > 4>>> (C(1) * C(2) * C(4) + O * C(3) * C(4)).evaluate() 1 > 2 2 > 4 3 > 4>>> (C(1) * ( C(2) + O * C(3) ) * C(4)).evaluate() 1 > 2 2 > 4 3 > 4>>> C(1) * C(2) * C(4) + O * C(3) * C(4) == C(1) * ( C(2) + O * C(3) ) * C(4) True
Note that the comparison wont work without the O term because the sinks differ:
>>> C(1) * C(2) * C(4) + C(3) * C(4) == C(1) * ( C(2) + O * C(3) ) * C(4) False
## Equation solving and minimizations
The module contains also (quite unefficient) simplify method, which can be used to expression minimization:
>>> I, O, C = from_operator(debug) >>> m = EquationMap(I, O, C) >>> a = C(1) + C(2) >>> simplify(a, 300, m) (C(1, 2), [C(1) + C(2), C(1, 2)])>>> b = C(1) * C(3) + C(2) * C(3) >>> simplified, path = simplify(b, 100, m) >>> simplified C(1, 2) * C(3) >>> for p in path: ... print(p) C(1) * C(3) + C(2) * C(3) (C(1) * I + C(2) * I) * C(3) (C(1) + C(2) * I) * C(3) (C(1) + C(2)) * C(3) C(1, 2) * C(3)
For proofs use the get_route:
>>> I, O, C = from_operator(debug) >>> m = EquationMap(I, O, C) >>> a = C(1) * C(3) + C(2) * C(3) >>> b = C(1, 2) * C(3) >>> shortest, path = get_route(a,b, 100, m) >>> for p in path: ... print(p) C(1) * C(3) + C(2) * C(3) C(1) * C(3) + C(2) * I * C(3) (C(1) * I + C(2) * I) * C(3) (C(1) + C(2) * I) * C(3) (C(1) + C(2)) * C(3) C(1, 2) * C(3)
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