A re-construction of the fundamentals of programming as a small mathematical theory (PRISM) based on elementary set theory. Highlights: $\bullet$ Zero axioms. No properties are assumed, all are proved (from standard set theory). $\bullet$ A single concept covers specifications and programs. $\bullet$ Its definition only involves one relation and one set. $\bullet$ Everything proceeds from three operations: choice, composition and restriction. $\bullet$ These techniques suffice to derive the axioms of classic papers on the "laws of programming" as consequences and prove them mechanically. $\bullet$ The ordinary subset operator suffices to define both the notion of program correctness and the concepts of specialization and refinement. $\bullet$ From this basis, the theory deduces dozens of theorems characterizing important properties of programs and programming. $\bullet$ 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 [arXiv:1507.00723]

I’m a fan of the formal specification language TLA+. With TLA+, you can build models of programs or systems, which helps to reason about their behavior. TLA+ is particularly useful for reason…

Invariants are properties that hold true during the evolution of a system, helping to ensure correct behavior in programming. They can simplify reasoning about code, whether it’s for small algorithms or larger systems. By clearly defining invariants, programmers can create robust code and manage complex systems effectively.

The text discusses the future of computing science over the next fifty years, emphasizing the importance of simplicity and elegance in design to prevent complexity. It highlights the close connection between program design and proof design, suggesting that advancements in program design can impact general mathematics. The author encourages embracing the opportunity to simplify processes and design systems that rely on formal mathematics.

Leslie Lamport wrote several papers on verifying and specifying concurrent systems using TLA. He discovered algorithms through formal derivation and emphasized mechanical verification of concurrent algorithms. His work influenced the development of the TLAPS proof system.