Targeting Completeness: Automated Complexity Analysis of Integer Programs

TL;DR

This paper introduces a modular approach for automated complexity analysis of integer programs by leveraging the decidability of periodic rational solvable loops, enhancing analysis scope through program transformations, and implementing it in the KoAT tool.

Contribution

It presents a novel modular method for analyzing the complexity of general integer programs using prs-loops and introduces transformation techniques to expand applicability.

Findings

The approach computes local bounds for prs-loops within integer programs.

Local bounds are effectively lifted to global bounds for entire programs.

Implementation in KoAT demonstrates practical effectiveness.

Abstract

There exist several approaches to infer runtime or resource bounds for integer programs automatically. In this paper, we study the subclass of periodic rational solvable loops (prs-loops), where questions regarding the runtime and the size of variable values are decidable and where we can therefore obtain techniques that are complete for such subclasses. We show how to use these results for the complexity analysis of arbitrary general integer programs. To this end, we present a modular approach which computes local runtime and size bounds for subprograms which correspond to prs-loops. These local bounds are then lifted to global runtime and size bounds for the whole integer program. Furthermore, we introduce several techniques to transform larger programs into prs-loops to increase the scope of the approach. The power of the procedure is shown by our implementation in the complexity analysis tool KoAT.