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 to be essential to real number computation, we use a Plotkin powerdomain and make 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 $\mathbb{Z}\ni p\mapsto 2^p\in\mathbb{R}$, we arrive at a first-order theory which we prove to be decidable and model-complete. 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 specifications. Various examples demonstrate the practicality and convenience of our language and the extended Hoare logic.

@article{paperbot2751,
  title = {Semantics, Specification Logic, and Hoare Logic of Exact Real Computation},
  author = {Sewon Park and Franz Brauße and Pieter Collins and SunYoung Kim and Michal Konečný and Gyesik Lee and Norbert Müller and Eike Neumann and Norbert Preining and Martin Ziegler},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 20, Issue 2},
  year = {2024},
  doi = {10.46298/lmcs-20(2:17)2024}
}