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.

中文翻译：

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CFL可达性的亚立方证书

过程间程序分析中的许多问题可以建模为图形上的上下文无关语言（CFL）可达性问题，并且可以在三次时间内解决。尽管经过多年的努力，仍没有针对此问题的真正次立方算法。我们研究了相关的认证任务：给定CFL可达性的实例，是否存在小型且可有效检查的证书，用于路径的存在和不存在？我们表明，在两种情况下，都存在简洁的证书（问题的大小为$ O（n ^ 2）$），并且可以在次三次（矩阵乘法）时间内检查这些证书。证书基于路径的基于语法的压缩（对于肯定的实例）和表示为矩阵约束的不变式（对于否定的实例）。因此，CFL的可达性在于不确定的和共同不确定的亚立方时间。一个自然的问题是，用于CFL可达性的更快算法是否会导致用于组合问题（例如布尔可满足性（SAT））的更快算法。作为我们的认证结果的结果，我们表明，对于强于$ n ^ \ omega $的条件下界，无法从SAT到CFL的可达性进行细粒度的降低，除非不确定的强指数时间假设（NSETH）失败。我们的研究结果扩展到相关的立方次等价问题：下推可达性和双向不确定下推自动机（2NPDA）语言识别。例如，我们为下推不可及性（归纳不变性）描述了简洁的证书，并观察到可以在矩阵乘法时间内对其进行检查。