Closed forms for numerical loops

Zachary Kincaid, Jason Breck, John Cyphert, Thomas Reps

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

This paper investigates the problem of reasoning about non-linear behavior of simple numerical loops. Our approach builds on classical techniques for analyzing the behavior of linear dynamical systems. It is well-known that a closed-form representation of the behavior of a linear dynamical system can always be expressed using algebraic numbers, but this approach can create formulas that present an obstacle for automated-reasoning tools. This paper characterizes when linear loops have closed forms in simpler theories that are more amenable to automated reasoning. The algorithms for computing closed forms described in the paper avoid the use of algebraic numbers, and produce closed forms expressed using polynomials and exponentials over rational numbers. We show that the logic for expressing closed forms is decidable, yielding decision procedures for verifying safety and termination of a class of numerical loops over rational numbers. We also show that the procedure for computing closed forms for this class of numerical loops can be used to over-approximate the behavior of arbitrary numerical programs (with unrestricted control flow, non-deterministic assignments, and recursive procedures).

Original languageEnglish (US)
Article number55
JournalProceedings of the ACM on Programming Languages
Volume3
Issue numberPOPL
DOIs
StatePublished - Jan 2019

All Science Journal Classification (ASJC) codes

  • Software
  • Safety, Risk, Reliability and Quality

Keywords

  • Decision procedures
  • Invariant generation
  • Loop summarization

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