Semantics, Specification Logic, and Hoare Logic of Exact Real Computation

We propose a simple imperative programming language, ERC, that features arbitrary real numbers as primitive data type, exactly. Equipped with a denotational semantics, ERC provides a formal programming language-theoretic foundation to the algorithmic processing of real numbers. In order to capture multi-valuedness, which is well-known essential to real number computation, we use a carefully designed Plotkin powerdomain. This makes our programming language semantics computable and complete: all and only real functions computable in computable analysis can be realized in ERC. The base programming language supports real arithmetic as well as implicit limits; expansions support additional primitive operations (such as a user-defined exponential function). By restricting integers to Presburger arithmetic and real coercion to the `precision' embedding $Z\ni p\mapsto 2^p\in R$, we arrive at a first-order theory which we prove to be decidable, model-complete, and expressive for the base programming language. Based on said logic as specification language for preconditions and postconditions, we extend Hoare logic to a sound (w.r.t. the denotational semantics) and expressive system for deriving correct TOTAL correctness predicates. Various examples demonstrate the practicality and convenience of our language and proof rules.