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We present a weakest-precondition-style calculus for reasoning about the expected values (pre-expectations) of emph{mixed-sign unbounded} random variables after execution of a probabilistic program. The semantics of a while-loop is well-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 this context. 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, allows for compositional reasoning, and present an invariant-based approach for reasoning about pre-expectations of loops.
Types in logic programming have focused on conservative approximations of program semantics by regular types, on one hand, and on type systems based on a prescriptive semantics defined for typed programs, on the other. In this paper, we define a new
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Game semantics has provided adequate models for a variety of programming languages, in which types are interpreted as two-player games and programs as strategies. Melli`es (2018) suggested that such categories of games and strategies may be obtained
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