Invariants,Well-Founded Statements and Real-Time Program Algebra

• Published in: FM 2014: Formal Methods Vol. 8442; pp. 318 - 334
• Main Authors: Hayes, Ian J., Meinicke, Larissa
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2014; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Algebraic Property; Idle Statement; Iteration Operator; Operational Semantic; Sequential Composition
• ISBN: 3319064096; 9783319064093
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-06410-9_23

Program algebras based on Kleene algebra abstract the essential properties of programming languages in the form of algebraic laws. The proof of a refinement law may be expressed in terms of the algebraic properties of programs required for the law to hold, rather than directly in terms of the semantics of a language. This has the advantage that the law is then valid for any programming language that satisfies the axioms of the algebra. In this paper we explore the notion of well-founded statements and their relationship to well-founded relations and iterations. The laws about well-founded statements and relations are combined with invariants to derive a simpler proof of a while-loop introduction law. The algebra is then applied to a real-time programming language. The main difference is that tests within conditions and loops take time to evaluate and during that time the values of program inputs may change. This requires new definitions for conditionals and while loops but the proofs of the introduction laws for these constructs can still make use of the more basic algebraic properties of iterations.