Formally Verified Interval Arithmetic and Its Application to Program Verification

• Published in: 2024 IEEE/ACM 12th International Conference on Formal Methods in Software Engineering (FormaliSE) pp. 111 - 121
• Main Authors: Brucker, Achim D., Cameron-Burke, Teddy, Stell, Amy
• Format: Conference Proceeding
• Language: English
• Published: ACM 14.04.2024
• Subjects: Cognition; Extended Interval Analysis; Formalising Mathematics; Hardware; Isabelle/HOL; Mathematical analysis; Measurement errors; Program Verification; Programming; Semantics; Software

Interval arithmetic is a well known mathematical technique to analyse or mitigate rounding or measurement errors. Thus, it is promising to integrate interval analysis into program verification environments. Such an integration is not only useful for the verification of numerical algorithms: the need to ensure that computations stay within certain bounds is common. For example, to show that arithmetic computations stay within the hardware bounds of a given number representation.We present a formalisation of (extended) interval arithmetic in Isabelle/HOL, including the concept of inclusion isotone (extended) interval analysis. The main result of this part is the formal proof that interval-splitting converges for Lipschitz-continuous interval isotone functions. We also show how interval types can be used for verifying programs in a C-like programming language.CCS CONCEPTS* Theory of computation → Semantics and reasoning; Interactive proof systems; * Software and its engineering → Software notations and tools; * Mathematics of computing → Mathematical analysis; * Computing methodologies → Symbolic and algebraic manipulation.