Subcubic certificates for CFL reachability

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 reachability) and on invariants represented as matrix inequalities (for non-reachability). 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 ω , unless the nondeterministic strong exponential time hypothesis (NSETH) fails. In a nutshell, reductions from SAT are unlikely to explain the cubic bottleneck for CFL reachability. Our results extend to related subcubic equivalent problems: pushdown reachability and 2NPDA recognition; as well as to all-pairs CFL reachability. 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.

One Algorithm to Evaluate Them All: Unified Linear Algebra Based Approach to Evaluate Both Regular and Context-Free Path Queries

“…Another powerful assumption relative to which many conditional lower bounds are proved is the strong exponential time hypothesis (SETH) introduced by Impagliazzo and Paturi (2001). Recently Chistikov et al (2021) showed that there cannot be a finegrained reduction from SAT to CFL-reachability for a conditional lower bound stronger than n ω , unless the nondeterministic strong exponential time hypothesis (NSETH) fails. Strong assumptions like 3SUM Conjecture by Gajentaan and Overmars (1995), multiphase problem by Patrascu (2010) are variants of OMv conjecture or can be strengthened through it (Henzinger et al, 2015).…”

The Fine-Grained Complexity of CFL Reachability

“…Prior work [Heintze and McAllester 1997] has also explained the sub-cubic barrier by showing that several data-flow reachability problems are 2NPDA-hard, a complexity class that does not admit sub-cubic algorithm for problems lying in this class. The related problem of certifying whether an instance of CFL reachability has a small and efficiently checkable certificate was studied in [Chistikov et al 2022]. Their main result shows that succint certificates of size…”

Automata Equipped with Auxiliary Data Structures and Regular Realizability Problems

“…RR problems have independently been studied under the name regular intersection emptiness problems [21,22]. A restricted version of RR problem (for context-free filters only) is a well-known CFL-reachability problem, which is related to problems in interprocedural program analysis [3,4,6,10,11,23].…”