1. Semantics, Specification Logic, and Hoare Logic of Exact Real Computation
- Author
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Sewon Park, Franz Brauße, Pieter Collins, SunYoung Kim, Michal Konečný, Gyesik Lee, Norbert Müller, Eike Neumann, Norbert Preining, and Martin Ziegler
- Subjects
mathematics - numerical analysis ,computer science - logic in computer science ,03b70, 65y99, 68p, 68n, 68q ,f.3.1 ,g.1.0 ,i.1.2 ,Logic ,BC1-199 ,Electronic computers. Computer science ,QA75.5-76.95 - Abstract
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.
- Published
- 2024
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