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A Hardest Language Recognized by Two-Way Nondeterministic Pushdown Automata.

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... We identify a new hardest 2NPDA language, that is, a fixed 2NPDA language L 0 such that for every 2NPDA language L there is a homomorphism h such that w ∈ L iff h(w) ∈ L 0 . A different hardest language was previously found by Rytter [44] using language-theoretic techniques. However, our proof and reductions strengthen the link between 2NPDA language recognition and CFL reachability, pointing to the hardest instances of the latter. ...
... Remark 22. Rytter [44] showed there is a fixed hardest 2NPDA language L 0 , 2 based on the classic hardest context-free language by Greibach [28]. Theorem 21 identifies a different hardest 2NPDA language. ...
... There are several ways to deal with this issue. One is reminiscent of Rytter's approach [44]: we can decide we are content with keeping a single endmarker in, i.e., we would only like to find an L 0 such that, for all w ∈ Σ + , one has w ∈ L iff h(w$) ∈ L 0 . Here$ is a fresh symbol. ...
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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.
Article
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.
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