>That isn't what I was trying to express. I was saying you would use: [], [5], [12, 3], and then a few long lists.

Yes and I'm saying this is an arbitrary design choice and therefore NOT part of some mathematical theory. What is it that made you choose these as test cases? How does choosing those test cases make your tests better?

>I was just trying to give an example. Obviously failed.

Yeah sorry, I'm saying can you just give a more clear example rather than using sentences to describe it, write out a full example, test cases and all. I may not be able to parse your descriptions but I could more readily understand a complete code example that is made more "testable" under your definition.

>No, I'm saying it takes two parameters, but we let each parameter accept all three of seconds since an epoch, a relative time reference, e.g., "2 days ago" or a text description "march 15, 2019".

Ok I see what you're saying now. The type of each parameter is a tuple of three values.

This doesn't make any sense in terms of test plan size. How are you Choosing N? It seems to me that you're implying a lower N is a more optimal test.

Let's make that example simpler. Let's reduce the cardinality of the types and make it bools so we can measure it. The cardinality of a bool is 2 (true, false). f(t0 bool, tn bool) will therefore have a total cardinality of 4 (2 times 2) meaning 4 possible variations of inputs (we are disregarding possible outputs and only testing expected output which removes the exponential increase in cardinality of the function type). Now let's make this a tuple of three values each: f((t0,t1,t2), (t3,t4,t5)), the t's are all bools. All possible input cases are now 64 in total. (2 times 2 times 2) times (2 times 2 times 2).

Your test space of possible inputs to measure goes from 4 possible tests to 64 possible tests. This is the measure of the total possible tests you can ever run on the function before you have exhausted every possibility.

If you have N conditions why does increasing the number of test cases required to fully test the experiment (which in your example is nearly infinite, but reduced down to 4 and 64 in my example) suddenly increase the N by a multiple of nine? This makes no sense. Also why do the adapter functions have a test size of 1? What is your metric for determining N?

>It's used in what's called design of experiments in statistics to calculate optimal sampling points for continuous variates, but if you look at what scientists actually do to choose what experiments to run, it is not based in probability.

It's based off of statistics which is itself based off of probability. Probability is the mathematical theory and statistics is theory applied to the real world. Both are math but the latter isn't theory in the sense I'm talking about it.

I'm not really talking about applied experimental design here. I'm talking about a theory that will give me the shortest possible path between point A and B in a cartesian plane. I don't need "design" to help me here, calculation and theory will give me the optimal answer.

In your examples, it seems that there is no exact definition of "optimal" and it seems you're making a bunch of arbitrary test choices to try to converge your tests onto this blurry definition of "optimal."

This is what I mean by there is no "theory" behind tests. Even if you have formulas that give you a bunch of other metrics like "test size" it doesn't mean anything unless N is a concrete number derived from concrete measures. If your "test theory" focuses around just reducing an arbitrary N then I'll give you that, but right now I'm not clear about how this number scales up or down with "testable code"