[saurik]

A test is not a proof, and you can prove something works without ever even running it. There are also properties of a system which are impossible to test, or are so non-deterministic that you a test will only fail once every million times the code is executed. You really need to just learn to prove stuff.

The most complex piece of code I have ever written, as a relevant story, took me a month to prove to everyone that it was correct. We then sent it off to multiple external auditors, one of which who had designed tooling that would let them do abstract interpretation with recursion, to verify I hadn't made any incorrect assumptions. The auditors were confused, as the code we sent them didn't do anything at all, as it had a stupid typo in a variable name... I had never managed to figure out how to run it ;P. But... they found no other bugs!

In truth, the people whom I have met whom are, by far, the worst at this, are the people who rely on testing, as they seem to have entirely atrophied the ability to verify correctness by reading the code and modeling it in some mathematical way. They certainly have no typos in their code ;P, but because a test isn't a proof, they always make assumptions in the test suite which are never challenged.

[Kambing]

> A test is not a proof

Actually, a test _is_ a proof! Or more specifically, a traditional test case is a narrow, specific proposition. For example, the test `length([1, 2, 3]) = 3` is a proposition about the behavior of the `length` function on a concrete input value. The proof of this proposition is _automatically generated_ by the runtime, i.e., the proof that this proposition holds is the execution of the left-hand side of the equality and observing it is identical to the right-hand side. In this sense, the runtime serves as an automated theorem prover (and is, perhaps, why test cases are the most accessible form of formal reasoning available to a programmer).

What we colloquially consider "proof" are really more abstract propositions (e.g., using first-order logic) that require reasoning beyond simple program execution. While the difference is, in some sense, academic, it is important to observe that testing and proving (in that colloquial sense) are, at their core, the same endeavor.

[JohnKemeny]

That is stretching the definitions of proofs.

    assert random() != 0.

QED

[pclowes]

Interesting, could you show me a formal proof that can't be expressed in logic (ie. code) and then tested?

My thought here is that since proofs are logic and so is code you can't have a proof that can't be represented in code. Now admittedly this might look very different than typical say JUnit unit tests but it would still be a test validating logic. I am not saying every system is easily testable or deterministic but overall, all else equal, the more tested and testable a system is the better it is.

IME things that are very hard to test are often just poorly designed.

[aljgz]

Consider a function that gets an array of integers and a positive number, and returns the sum of the array elements modulo the number. How can we prove using tests, that this always works for all possible values?

Not discounting the value of tests: we throw a bunch of general and corner cases at the function, and they will ring the alarm if in the future any change to the function breaks any of those.

But they don't prove it's correct for all possible inputs.

[pclowes]

I would lean towards types and property testing here using tools like Coq.

[aljgz]

Can you elaborate?

[layer8]

Tests can generally only test particular inputs and/or particular external states and events. A proof abstracts over all possible inputs, states, and events. It proves that the program does what it is supposed to do regardless of any particular input, state, or events. Tests, on the other hand, usually aren't exhaustive, unless it's something like testing a pure function taking a single 32-bit input, in which case you can actually test its correctness for all 2^32 possible inputs (after you convinced yourself that it's truly a pure function that only depends on its input, which is itself a form of proof). But 99.99% of code that you want to be correct isn’t like that.

As an example, take a sorting procedure that sorts an arbitrarily long list of arbitrarily long strings. You can't establish just through testing that it will produce a correctly sorted output for all possible inputs, because the set of possible inputs is unbounded. An algorithmic proof, on the other hand, can establish that.

That is the crucial difference between reasoning about code versus merely testing code.

[pclowes]

I agree with that but I would say that if I required formal verification of that kind I would move the proof based rationale into the type system to provide those checks.

I would add Tests can be probabilistically exhaustive (eg property based testing) and answer questions beyond what proof based reasoning can provide ie. is this sorting of arbitrary strings efficient and fast?

[joseneca]

Proofs are arguably still better than tests at evaluating efficiency, at least for smaller components/algorithms in isolation. While there are cases where constant factors that can't be described well in a proof matter, in most cases, the crucial element of an algorithm's efficiency lies in how the complexity scales, which can be proven in the vast majority of cases. On the other hand, relying solely on benchmarking introduces a lot of noise that can be difficult to sort through.