SEMANTICS, SPECIFICATION LOGIC, AND HOARE LOGIC OF EXACT REAL COMPUTATION

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 Z is not an element of p 7 -> 2p is an element of 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.