Modern Aspects of Complexity Within Formal Languages

Fernau, Henning

doi:10.1007/978-3-030-13435-8_1

Cited by 4 publications

(3 citation statements)

References 81 publications

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“…More broadly, a range of problems in formal languages are now being approached with tools from modern algorithms and complexity [Fernau 2019]. Our work contributes to this ongoing effort.…”

Section: Related Workmentioning

confidence: 99%

Subcubic certificates for CFL reachability

Chistikov

Majumdar

Schepper

2022

Proc. ACM Program. Lang.

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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 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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“…More broadly, a range of problems in formal languages are now being approached with tools from modern algorithms and complexity [Fernau 2019]. Our work contributes to this ongoing effort.…”

Section: Related Workmentioning

confidence: 99%

Subcubic certificates for CFL reachability

Chistikov

Majumdar

Schepper

2022

Proc. ACM Program. Lang.

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“…It is also interesting to observe that most of our problems come from the area of formal languages, in most cases, from automata theory. This area has been a bit neglected from the perspective of parameterized complexity, see Fernau (2019) for discussing further questions in this regard. Also, the class WNL mostly hosts formal language problems, as already exhibited by Guillemot (2011).…”

Section: Inputmentioning

confidence: 99%

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Fernau

Bruchertseifer²

2022

Math. Struct. Comp. Sci.

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The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $\textsf{W}[2]$ -hardness results, we show that these problems belong to $\textsf{A}[2]$ , $\textsf{W}[\textsf{P}],$ and $\textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $\textsf{W}[\textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $\textsf{W}[\textsf{Sync}]$ or are hard or complete for this new class.

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“…In general, the variability of taste priorities is a natural thing. Indeed, the speech itself is an extremely volatile and multifaceted phenomenon (Fernau, 2019). Even the literary norm sometimes allows options.…”

Section: Literature Reviewmentioning

confidence: 99%

The Structure of the Literary Problem in the Formation of the Local Text Substrate

Bayanbaeva¹,

Sinyachkin²,

Dzholdasbekova³

et al. 2020

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The article aims to study the structure of the literary problem in the formation of the local text substrate. The study uses the methodology of studying the language when it changes in time and space. The article explains the basics of the methodological support of the translation complex and the structure of its application in private studies of foreign cultures and communicants. The results of the study showed the possibility of interaction between the subjects of linguistic exchange and the dynamics of the translation and literary component. The novelty of the study is determined by the fact that the work defines methods that can be used not only by local researchers but also by foreign-speaking communicants. The research results can be used in practical activities to bridge the gap between understanding the local text in translation studies and its structuring in the local versions of individual authors.

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Modern Aspects of Complexity Within Formal Languages

Cited by 4 publications

References 81 publications

Subcubic certificates for CFL reachability

Subcubic certificates for CFL reachability

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

The Structure of the Literary Problem in the Formation of the Local Text Substrate

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