Proving Non-Termination via Loop Acceleration

Abstract

We present the first approach to prove non-termination of integer programs that is based on loop acceleration. If our technique cannot show non-termination of a loop, it tries to accelerate it instead in order to find paths to other non-terminating loops automatically. The prerequisites for our novel loop acceleration technique generalize a simple yet effective non-termination criterion. Thus, we can use the same program transformations to facilitate both non-termination proving and loop acceleration. In particular, we present a novel invariant inference technique that is tailored to our approach. An extensive evaluation of our fully automated tool LoAT shows that it is competitive with the state of the art.

A Calculus for Modular Loop Acceleration

“…6) acceleration techniques with our calculus. This improves over prior approaches, where acceleration techniques were used independently, and may thus improve acceleration-based verification techniques [6,7,11,[15][16][17]27] in the future. An empirical evaluation (Sec.…”

“…In the future, we plan to implement the missing features mentioned in Sec. 8 and integrate our novel calculus into our own acceleration-based program analyses to prove lower bounds on the runtime complexity [15,16] and non-termination [11] of integer programs. Furthermore, our experiments indicate that integrating specialized techniques for FMATs (cf.…”

“…We prototypically implemented our approach in our open-source Loop Acceleration Tool LoAT [11,15,16]: https://github.com/aprove-developers/LoAT/tree/tacas20 It uses Z3 [29] to check implications and PURRS [1] to compute closed forms.…”

“…We compared our implementation with LoAT's implementation of acceleration via monotonicity (Thm. 3, [11]) and its implementation of acceleration via metering functions (Thm. 4, [16]), which also incorporates the improvements proposed in [15].…”

“…In the last years, loop acceleration techniques have successfully been used to build static analyses for programs operating on integers [4,11,22,[27][28][29]43]. Essentially, such techniques extract a quantifier-free first-order formula ψ from a singlepath loop T , i.e., a loop without branching in its body, such that ψ under-approximates (or is equivalent to) T .…”