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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. ...

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.

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.

We survey the techniques used to speed up recursive and stack (pushdown) manipulating algorithms on words. in the case of the parallel speedup new algoritrmic tools are presented: the operation “bush” acting on path systems and a new parallel pebble game. We show how these tools can be applied to some dynamic programming problems related to combinatorial algorithms on words and to some language recognition problems. The techniques are illustrated mostly on pushdown automata (2pda’s, for short) which can be treated as limited algorithms acting on words. The history of the discovery of the fast string matching algorithm [15] shows that 2pda’s can be useful in the design of efficient algorithms on words. In this paper we investigate one aspect of 2pda’s (which in our view is the most important): algorithmic techniques for fast simulation of 2pda’s and their generalisations to recursive programs.

It is proved that any bounded context-free language can be recognized by a two-way deterministic automaton with a finite-rotary counter.

It is shown that k + 1 heads are better than k for one-way multihead pushdown (resp. stack) automata if they do not have endmarkers on the input tape and accept by final state with at least one input head at the right end of the input string. In addition, for two-way multihead pushdown (resp. stack) automata, “hardest languages” are described. It is also shown that for two-way multihead pushdown (resp. stack) automata there is a language with the hardest time and space complexity which can be written as for some one-way multihead pushdown (resp. stack) automaton language L, where is in L for some n ⩾ 1}. A representation theorem for recursively enumerable languages is also given.

There is a context-free language $L_0 $ such that every context-free language is an inverse homomorphic image of $L_0 $ or $L_0 - \{ e\} $. Hence the time complexity of recognition of $L_0 $ is the least upper bound for time complexity of recognition of context-free languages. A similar result holds for quasirealtime Turing machine languages. Several languages are given such that deterministic and nondeterministic polynomial time acceptance are equivalent if and only if any one of them is deterministic polynomial time acceptable.

We considered some of the important unsolved problems in the theory of computation concerning the relationship between deterministic and nondeterministic computations, and between tape and time bounded computations. For each such problem we find an equivalent problem concerning two-way deterministic pushdown automaton languages.

It is shown that under mild conditions on a tape function f, the AFL generated by the languages accepted by f-tape-bounded (deterministic) Turing acceptors is generated by a single language i.e., is principal. The same result holds for the AFL generated by the languages accepted by f-tape-bounded Turing acceptors with a (possibly unbounded) pushdown tape. From these results it follows that the AFL generated by writing pushdown-acceptor languages and the AFL generated by the two-way (deterministic) (nonerasing) stack-acceptor languages are principal. A modification of the main argument shows that the AFL generated by the two-way pushdown-acceptor languages is also principal.