Programming really is simple mathematics

PRISM is a novel framework that offers a fresh perspective on programming. Traditionally, key aspects of programming — such as specification and implementation, formal and applied methods, and static and dynamic verification — are often viewed as distinct and even opposing concepts. PRISM challenges this notion by providing a unified, mathematically rigorous foundation that integrates these elements seamlessly.

Currently, the project comprises a research paper, soon submitted as part of a Festschrift honoring Yuri Gurevich, along with the formal proofs of the paper’s theorems, developed in Isabelle HOL. Moving forward, PRISM will continue to be refined and expanded, with the ultimate goal of enabling the generation of formally verified programming languages.

Abstract

A re-construction of the fundamentals of programming as a small mathematical theory (“PRISM”) based on elementary set theory. Highlights:

• Zero axioms. No properties are assumed, all are proved (from standard set theory).
• A single concept covers specifications and programs.
• Its definition only involves one relation and one set.
• Everything proceeds from three operations: choice, composition and restriction.
• These techniques suffice to derive the axioms of classic papers on the “laws of programming” as consequences and prove them mechanically.
• The ordinary subset operator suffices to define both the notion of program correctness and the concepts of specialization and refinement.
• From this basis, the theory deduces dozens of theorems characterizing important properties of programs and programming.
• All these theorems have been mechanically verified (using Isabelle/HOL); the proofs are available in a public repository.

This paper is a considerable extension and rewrite of an earlier contribution.