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.
Highlights
Context-free reachability is a fundamental problem in interprocedural program analysis, verification of recursive programs, and database theory [Alur et al 2005; Bouajjani et al 1997; Dolev et al 1982; Melski and Reps 2000; Reps et al 1995; Yannakakis 1990]
Our results extend to related subcubic equivalent problems: pushdown reachability and 2NPDA recognition; as well as to all-pairs context-free language (CFL) reachability
By exploiting fine-grained reductions between CFL reachability, pushdown reachability, the emptiness problem for pushdown automata, and the recognition problem for two-way nondeterministic pushdown automata (2NPDA), we show all these problems have subcubic certificate systems
Summary
Context-free reachability is a fundamental problem in interprocedural program analysis, verification of recursive programs, and database theory [Alur et al 2005; Bouajjani et al 1997; Dolev et al 1982; Melski and Reps 2000; Reps et al 1995; Yannakakis 1990]. Our main result shows that CFL reachability has subcubic certificate systems: every positive or negative instance has a quadratic certificate and a checker that runs in O (nω ) time, n = |V |. (1) We propose certificate systems for positive and negative instances of CFL reachability, with certificates of size O (n2) which can be verified in time subcubic in the size of the graph We prove these systems sound and complete (Section 3). (2) We show that our certificate systems extend: to subcubic equivalent problems such as pushdown reachability, the emptiness problem for PDA, and 2NPDA language recognition (Subsections 5 and 6.1); as well as to all-pairs CFL reachability (Subsection 3.3) We show they support verification in randomized quadratic time (with correct certificates never rejected), based on Freivalds’ algorithm for verifying matrix multiplication [Freivalds 1979] (Subsection 3.2).
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