Incremental Dead State Detection in Logarithmic Time

TL;DR

This paper introduces guided incremental digraphs (GIDs) for dead state detection, achieving logarithmic time complexity and significant speedups over previous algorithms in formal verification tasks.

Contribution

The paper proposes GIDs and two algorithms that enable dead state detection in logarithmic time, improving efficiency over existing methods for incremental graph analysis.

Findings

Achieved $O(\log m)$ amortized time per edge for dead state detection.

Implemented and compared algorithms showing 110-530x speedups over the state-of-the-art.

Validated improvements across various graph classes and regex benchmark graphs.

Abstract

Identifying live and dead states in an abstract transition system is a recurring problem in formal verification; for example, it arises in our recent work on efficiently deciding regex constraints in SMT. However, state-of-the-art graph algorithms for maintaining reachability information incrementally (that is, as states are visited and before the entire state space is explored) assume that new edges can be added from any state at any time, whereas in many applications, outgoing edges are added from each state as it is explored. To formalize the latter situation, we propose guided incremental digraphs (GIDs), incremental graphs which support labeling closed states (states which will not receive further outgoing edges). Our main result is that dead state detection in GIDs is solvable in $O(\log m)$ amortized time per edge for $m$ edges, improving upon $O(\sqrt{m})$ per edge due to Bender, Fineman, Gilbert, and Tarjan (BFGT) for general incremental directed graphs. We introduce two algorithms for GIDs: one establishing the logarithmic time bound, and a second algorithm to explore a lazy heuristics-based approach. To enable an apples-to-apples experimental comparison, we implemented both algorithms, two simpler baselines, and the state-of-the-art BFGT baseline using a common directed graph interface in Rust. Our evaluation shows $110$-$530$x speedups over BFGT for the largest input graphs over a range of graph classes, random graphs, and graphs arising from regex benchmarks.