A weakest pre-expectation semantics for mixed-sign expectations

We present a weakest-precondition-style calculus for reasoning about the expected values (pre-expectations) of mixed-sign unbounded random variables after execution of a probabilistic program. The semantics of a while-loop is defined as the limit of iteratively applying a functional to a zero-element just as in the traditional weakest pre-expectation calculus, even though a standard least fixed point argument is not applicable in our semantics. A striking feature of our semantics is that it is always well-defined, even if the expected values do not exist. We show that the calculus is sound and allows for compositional reasoning. Furthermore, we present an invariant-based approach for reasoning about pre-expectations of loops.