Why can't programs be proven?

Question

Why can\'t a computer program be proven just as a mathematical statement can? A mathematical proof is built up on other proofs, which are built up from yet more proofs and

Answer 1

I read a bit about this, but there are two problems.

First, you can't prove some abstract thing called correctness. You can, if things are set up properly, prove that two formal systems are equivalent. You can prove that a program implements a set of specifications, and it's easiest to do this by constructing the proof and program more or less in parallel. Therefore, the specifications must be sufficiently detailed to provide something to prove against, and therefore the specification is effectively a program. The problem of writing a program to satisfy a purpose becomes the problem of writing a formal detailed specification of a program to satisfy a purpose, and that's not necessarily a step forward.

Second, programs are complicated. So are proofs of correctness. If you can make a mistake writing a program, you sure can make one proving. Dijkstra and Gries told me, essentially, that if I was a perfect mathematician I could be a good programmer. The value here is that proving and programming are two somewhat different thought processes, and at least in my experience I make different sorts of mistakes.

In my experience, proving programs isn't useless. When I am trying to do something I can describe formally, proving the implementation correct eliminates a whole lot of hard-to-find errors, primarily leaving the dumb ones, which I can catch easily in testing. On a project that must produce extremely bug-free code, it can be a useful adjunct. It isn't suitable for every application, and it's certainly no silver bullet.