Subcubic Certificates for CFL Reachability

TL;DR

This paper introduces small, efficiently checkable certificates for CFL reachability, demonstrating that the problem resides in subcubic nondeterministic and co-nondeterministic time, and explores implications for related problems and complexity hypotheses.

Contribution

It provides the first succinct certificates for CFL reachability and related problems, enabling subcubic verification and establishing complexity bounds linked to SAT.

Findings

Certificates are of size O(n^2) for both existence and non-existence of paths.

Verification of certificates can be performed in subcubic matrix multiplication time.

CFL reachability is in nondeterministic and co-nondeterministic subcubic complexity classes.

Abstract

Many problems in interprocedural program analysis can be modeled as the context-free language (CFL) reachability problem on graphs and can be solved in cubic time. Despite years of efforts, there are no known truly sub-cubic algorithms for this problem. We study the related certification task: given an instance of CFL reachability, are there small and efficiently checkable certificates for the existence and for the non-existence of a path? We show that, in both scenarios, there exist succinct certificates ($O(n^2)$ in the size of the problem) and these certificates can be checked in subcubic (matrix multiplication) time. The certificates are based on grammar-based compression of paths (for positive instances) and on invariants represented as matrix constraints (for negative instances). Thus, CFL reachability lies in nondeterministic and co-nondeterministic subcubic time. A natural question is whether faster algorithms for CFL reachability will lead to faster algorithms for combinatorial problems such as Boolean satisfiability (SAT). As a consequence of our certification results, we show that there cannot be a fine-grained reduction from SAT to CFL reachability for a conditional lower bound stronger than $n^\omega$, unless the nondeterministic strong exponential time hypothesis (NSETH) fails. Our results extend to related subcubic equivalent problems: pushdown reachability and two-way nondeterministic pushdown automata (2NPDA) language recognition. For example, we describe succinct certificates for pushdown non-reachability (inductive invariants) and observe that they can be checked in matrix multiplication time. We also extract a new hardest 2NPDA language, capturing the "hard core" of all these problems.