In theoretical computer science, correctness of an algorithm is asserted when it is said that the algorithm is correct with respect to a
specification. Functional correctness refers to the
input-output behaviour of the algorithm (i.e., for each input it produces the
correct output). See also program verification.
A distinction is made between total correctness, where it is additionally
required that the algorithm terminates, and partial correctness, which
simply requires that if an answer is returned it will be correct. Since there is no general solution to the halting problem, a total correctness assertion may lie much deeper.
For example if we are successively searching though integers 1, 2, 3, ... to see if we can find an example of some phenomenon — say an odd perfect number — it is quite easy to write a partially correct program (use integer factorization to check n as perfect or not). But to say this program is totally correct would be to assert something currently not known in number theory.
A proof would have to be a mathematical proof, assuming both the algorithm and specification are given formally. In particular it is not expected to be a correctness assertion for a given program implementing the algorithm on a given machine. That would involve such considerations as limitations on memory.
A deep result in proof theory, the Curry-Howard correspondence, states that a proof of functional correctness in constructive logic corresponds to a certain program in the lambda calculus. Converting a proof in this way is called program extraction.
See also