Cover's test of rationality revisited: Computability aspects of hypothesis testing

In this paper we discuss computability aspects of hypothesis testing. We describe two main results. First we determine the type of sets that admit a weak decision procedure. Surprisingly some non-computable sets admit a computable weak decision procedure. This strengthens results of Cover and Putnam. We then apply the notion of weak decision procedure to the testing of the physical Church-Turing thesis. While our first theorem states that there are non-computable sets that admit weak decision procedures, we are able to show that no weak decision procedure can help us to decide that a physical device is capable of computing non Turing computable functions or that a physical constant encodes the bits of a non-computable real. This has strong implications on the validity of physical theories entailing the failure of the physical Church-Turing thesis.