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

Further, what are the axioms of programming? The very atomic truths of the field?

I've TA'ed a course called Contract Based Programming (course homepage: http://www.daimi.au.dk/KBP2/). Here what I can extrapolate from the course (and other courses I've taken)

You have to formally (mathematically) define the semantics of your language. Let's think of a simple programming language; one that has global variables only, ints, int arrays, arithmetic, if-then-else, while, assignment and do-nothing [you can probably use a subset of any mainstream language as an "implementation" of this].

An execution state would be a list of pairs (variable name, value of variable). Read "{Q1} S1 {Q2}" as "executing statement S1 takes you from execution state Q1 to state Q2".

One axiom would then be "if both {Q1} S1 {Q2} and {Q2} S2 {Q3}, then {Q1} S1; S2 {Q3}". That is, if statement S1 takes you from state Q1 to Q2, and statement S2 takes you from Q2 to Q3, then "S1; S2" (S1 followed by S2) takes you from state Q1 to state Q3.

Another axiom would be "if {Q1 && e != 0} S1 {Q2} and {Q1 && e == 0} S2 {Q2}, then {Q1} if e then S1 else S2 {Q2}".

Now, a bit of refinement: the Qn's in {}'s would actually be statements about states, not states themselves.

Suppose that M(out, A1, A2) is a statement which does a merging of two sorted arrays and stores the result in out, and that all the words I use in the next example were expressed formally (mathematically). Then "{sorted(A1) && sorted(A2)} A := M(A1, A2) {sorted(A) && permutationOf(A, A1 concatened with A2)}" is the claim tha M correctly implements the merge algorithm.

One can try to prove this by using the above axioms (a few others would probably be needed. You're likely to need a loop, for one).

I hope this illustrates a bit of what proving programs correct might look like. And trust me: it takes a lot of work, even for seemingly simple algorithms, to prove them correct. I know, I read a lot of attempts ;-)

[if you read this: your hand-in was fine, it's all the other ones that caused me headaches ;-)]