Closed forms for numerical loops
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).
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Jan 02, 2019
- Source ID
- 10.1145/3290368
Entities
People
- Jason Breck
- John Cyphert
- Thomas Reps
- Zachary Kincaid
Organizations
- Defense Advanced Research Projects Agency
- Office of Naval Research
- Princeton University
- University of Wisconsin–Madison
- Wisconsin Alumni Research Foundation