When production in a factory is divided into many smaller chunks productivity increases. The question is whether the same thing is true in programming as well. If a function is divided into smaller segments (base vectors), and each segment has some specific structure then the complexity of computations will increase. However, the upside is that based on this structure programs turn into a network of base vectors. Many functions may share the same base vector and this enhances the interaction between different functions.
Consider the example of the sort function which gets a set of words and outputs them alphabetically. This is an imprecise construction:
An n-dimensional space should be constructed. Each axis contains all letters in the alphabet (A=1, B=2, etc.). n specifies the length of the longest word in the set. Then words turn into points in that space, for example, the word “ad” is the point (1,4,0,0,…,0).
To generalize this process there should be a base space (for example R^n), called S_intitial or S_i. Then different functions will turn S_i into required spaces specific for that function (for example in this example, points in S_i are associated with letters).
Now that words in the previous example have turned into points in that space, the sort function will be a sequence of base vectors operating on those points. These are roughly the base vectors:
F_1) a function turns the set of inputs into points as described before. In other words, this function turns S_i into the space of letters and outputs a set of points.
F_2) a function assigns a value to each point.
f( [x_1,x_2,…]) = c_1*x_1+ c_2*x_2 +… , where [x_1, x_2, …] is a point associated to a word. and c_i are coefficients. This method ensures that the first letter in words gets the highest weight, and so on.
F_3) this is the max function and outputs the highest value.
F_4) this is the inverse of F_2 and turns the value into a point.
F_5) this is the inverse of F_1 function and turns a point into a word.
imagine in this rough example several functions share F_2. And coefficients in F_2 can be influenced by those functions. This is comparable to the process of design where the properties of one element influence the shape and properties of other elements (see 28 Sep 2024).
After mathematically constructing this idea, I can start writing articles. There are many incomplete sections in things I’ve written so far. Those sections will be worked on in the next versions of the articles.
UPDATE 1:
Spaces similar to F_2 provide a space for the modifications of functions. This space contains all modifications that retain the symmetry of the function and is the wave group theory.