No Brain, No Pain

1) The entropy of a function is determined by how predictable the output of the function is, based on its inputs.  In other words, I think we can we can take the basic concepts of statistical thermodynamics and apply it to software.

 

2) Every function can be assigned an entropy value between zero and one.  The following is a zero-entropy function (as it doesn't change the input parameters at all):

int func0(int x)
{
     return x;
}

Conversely, the following function has an entropy of one (output is completely independent of the input):

int func1(int x)
{
     return rand();
}

(Actually, this function would have an entropy slightly less than 1.0, since any software random number generator isn't completely random, but... you get the basic gist.  In any case, the point is I'm assigning arbitrary values to define a range of possible entropy values)

3) Any two functions which have identical return values for identical inputs, have the same entropy.

4) The entropy of a function is not equal to the sum of the entropy of each line that comprises the function.

 

Proof:

 

By #2, the following function has zero entropy:

int func0(int x)
{
     return x;
}

Conversely, these functions have positive, non-zero (albeit small), entropy:

int func1(int x)
{
    return x + 1;
}

int func2(int x)
{
     return x - 1;
}

Now, consider the following function:

int func3(int x)
{
     int y = func1(x);
     return func2(y);
}

By #3 the func3() has the same entropy as func0(), since the output of both functions is identical (they both return the original value passed to the function).  Thus the entropy of func3() is zero, even though the lines the comprise the function have positive entropy.

Given all of these, I propose the following statement:

5) The efficiency of a function is the entropy of the function, divided by the sum of entropy of each statement that make up the function.