A Calculus for Modular Loop Acceleration

Abstract

Loop acceleration can be used to prove safety, reachability, runtime bounds, and (non-)termination of programs operating on integers. To this end, a variety of acceleration techniques has been proposed. However, all of them are monolithic: Either they accelerate a loop successfully or they fail completely. In contrast, we present a calculus that allows for combining acceleration techniques in a modular way and we show how to integrate many existing acceleration techniques into our calculus. Moreover, we propose two novel acceleration techniques that can be incorporated into our calculus seamlessly. An empirical evaluation demonstrates the applicability of our approach.⋆ This work has been funded by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).

Proving Non-Termination and Lower Runtime Bounds with LoAT (System Description)

“…LoAT is based on loop acceleration [5,6,[17][18][19]25], which replaces loops by nondeterministic code: The resulting program non-deterministically chooses a value n, representing the number of loop iterations in the original program. To be sound, suitable constraints on n are synthesized to ensure that the original loop allows for at least n iterations, given the current valuation of the program variables.…”

“…7). Contributions The main new feature of the novel version of LoAT presented in this paper is the integration of the loop acceleration calculus from [18,19], which combines different loop acceleration techniques in a modular way, into LoAT's framework to analyze full integer programs (instead of just single-path loops as in [18,19]). To control the application of the calculus, we use a new technique based on unsat cores.…”

“…1. The original implementation of the loop acceleration calculus in [18,19] repeatedly tried to apply acceleration techniques to yet unprocessed inequations in the loop guard until all inequations were processed successfully, or no more progress was possible. In the worst case, such a strategy requires Θ(m 2 ) SMT queries to accelerate a loop whose guard consists of m inequations.…”

“…2. While the original implementation of the loop acceleration calculus in [18,19] only considered single-path loops, we now integrated the calculus and our novel SMT-based approach to control its application into the framework of LoAT (see Sect. 3) to infer lower complexity bounds and non-termination proofs for full programs.…”

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We present the new version of the Loop Acceleration Tool (LoAT), a powerful tool for proving non-termination and worst-case lower bounds for programs operating on integers. It is based on the novel calculus from [18,19] for loop acceleration, i.e., transforming loops into nondeterministic straight-line code, and for finding non-terminating configurations. To implement it efficiently, LoAT uses a new approach based on SMT solving and unsat cores.

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