Dyck Language Research Articles

If the Current Clique Algorithms Are Optimal, so Is Valiant's Parser

The CFG recognition problem is as follows: given a context-free grammar $\mathcal{G}$ and a string $w$ of length $n$, decide whether $w$ can be obtained from $\mathcal{G}$. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in $O(n^{\omega})$ time, where $\omega<2.373$ is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic $O(n^3/\log^3{n})$ time complexity. Evidence for the nonexistence of more efficient algorithms was given in a result of Lee (JACM'01), who showed that any algorithm with running time $O(|\mathcal{G}|\cdot n^{3-\varepsilon})$, which can solve a more general parsing problem in the unusual case that $|\mathcal{G}|=\Omega(n^6)$, can be converted into a surprising subcubic time algorithm for Boolean Matrix Multiplication. However, nothing was known for the more relevant case of constant size grammars. In this work, we make the first progress on the constant size grammar case in 40 years, and prove that any improvement on Valiant's algorithm, either in terms of runtime or by avoiding the inefficiencies of the known fast matrix multiplication algorithms, would imply a breakthrough algorithm for the $k$-Clique problem: given a graph of $n$ nodes, decide whether there are $k$ nodes that form a clique. Using similar techniques, we derive similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding, a central problem in computational biology, and Dyck Language Edit Distance, answering an open question of Saha (FOCS'14).

Noncommutative Valiant's Classes

In this article, we explore the noncommutative analogues, VP nc and VNP nc , of Valiant’s algebraic complexity classes and show some striking connections to classical formal language theory. Our main results are the following: — We show that Dyck polynomials (defined from the Dyck languages of formal language theory) are complete for the class VP nc under ≤ abp reductions. To the best of our knowledge, these are the first natural polynomial families shown to be VP nc -complete. Likewise, it turns out that PAL (palindrome polynomials defined from palindromes) are complete for the class VSKEW nc (defined by polynomial-size skew circuits) under ≤ abp reductions. The proof of these results is by suitably adapting the classical Chomsky-Schützenberger theorem showing that Dyck languages are the hardest CFLs. — Assuming that VP nc ≠ VNP nc , we exhibit a strictly infinite hierarchy of p-families, with respect to the projection reducibility, between the complexity classes VP nc and VNP nc (analogous to Ladner’s theorem [Ladner 1975]). — Additionally, inside VP nc , we show that there is a strict hierarchy of p-families (based on the nesting depth of Dyck polynomials) with respect to the ≤ abp reducibility (defined explicitly in this article).

THE CHOMSKY-SCHÜTZENBERGER THEOREM FOR QUANTITATIVE CONTEXT-FREE LANGUAGES

Weighted automata model quantitative aspects of systems like the consumption of resources during executions. Traditionally, the weights are assumed to form the algebraic structure of a semiring, but recently also other weight computations like average have been considered. Here, we investigate quantitative context-free languages over very general weight structures incorporating all semirings, average computations, lattices. In our main result, we derive the Chomsky-Schützenberger Theorem for such quantitative context-free languages, showing that each arises as the image of the intersection of a Dyck language and a recognizable language under a suitable morphism. Moreover, we show that quantitative context-free languages are expressively equivalent to a model of weighted pushdown automata. This generalizes results previously known only for semirings. We also investigate under which conditions quantitative context-free languages assume only finitely many values.

Fast enumeration of words generated by Dyck grammars

The problem of enumerating and denumerating words generated by Dyck grammars arises in the work of compilers for high-level programming languages and a number of other applications. The present paper proposes an algorithm for the fast enumeration and denumeration of words of Dyck languages; the complexity of this algorithm per one symbol of enumerated words is O(log3n log log n) bit operations, provided that the Schonhage-Strassen multiplication and division algorithmis used. The well-knownmethods applied earlier possess complexityO(n) per one symbol of enumerated words. The construction of the proposed algorithm is based on the Ryabko method.

Multidimensional trees and a Chomsky–Schützenberger–Weir representation theorem for simple context-free tree grammars

Weir [34] proved a Chomsky-Schutzenberger-like representation theorem for the string languages of tree-adjoining grammars, where the Dyck language Dn in the Chomsky-Schutzenberger characterization is replaced by the intersection D2n ∩ g(D2n), where g is a certain bijection on the alphabet consisting of 2n pairs of brackets. This paper presents a generalization of this theorem to the string languages that are the yield images of the tree languages generated by simple (i.e., linear and non-deleting) context-free tree grammars. This result is obtained through a natural generalization of the original Chomsky-Schutzenberger theorem to the tree languages of simple context-free tree grammars. We use Rogers’s [24,23] notion of multi-dimensional trees to state this latter theorem in a very general, abstract form.

Numéro 10/2013. Réponses

Among the variety of representation theorems for context-free languages, the Chomsky–Schützenberger theorem is unique in that it consists of Dyck languages, regular languages, and simple operations. In the present paper, we obtain some characterizations of and representation theorems for languages in the Chomsky hierarchy by using insertion systems, strictly locally testable languages, and morphisms in the framework of the Chomsky–Schützenberger theorem. For instance, each context-free language L can be represented in the form L=h(L(γ)∩R), where γ is an insertion system of weight (1,1), R is a strictly 2-testable language, and h is a morphism. In contrast, if R is instead a strictly 1-testable language, the class of languages generated is a proper subclass of context-free languages and incomparable with the class of regular languages.

Fast algorithms for Dyck-CFL-reachability with applications to alias analysis

The context-free language (CFL) reachability problem is a well-known fundamental formulation in program analysis. In practice, many program analyses, especially pointer analyses, adopt a restricted version of CFL-reachability, Dyck-CFL-reachability, and compute on edge-labeled bidirected graphs. Solving the all-pairs Dyck-CFL-reachability on such bidirected graphs is expensive. For a bidirected graph with n nodes and m edges, the traditional dynamic programming style algorithm exhibits a subcubic time complexity for the Dyck language with k kinds of parentheses. When the underlying graphs are restricted to bidirected trees, an algorithm with O(n log n log k) time complexity was proposed recently. This paper studies the Dyck-CFL-reachability problems on bidirected trees and graphs. In particular, it presents two fast algorithms with O(n) and O(n + m log m) time complexities on trees and graphs respectively. We have implemented and evaluated our algorithms on a state-of-the-art alias analysis for Java. Results on standard benchmarks show that our algorithms achieve orders of magnitude speedup and consume less memory.

Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition

This paper makes three main contributions to the theory of communication complexity and stream computation. First, we present new bounds on the information complexity of augmented-index. In contrast to analogous results for index by Jain, Radhakrishnan, and Sen [J. ACM, 56 (2009), article 33], we have to overcome the significant technical challenge that protocols for augmented-index may violate the “rectangle property” due to the inherent input sharing. Second, we use these bounds to resolve an open problem of Magniez, Mathieu, and Nayak [Proceedings of the 42nd Annual ACM Symposium on Theory of Computing, 2010, pp. 261--270] that asked about the multipass complexity of recognizing Dyck languages. This results in a natural separation between the standard multipass model and the multipass model that permits reverse passes. Third, we present the first passive memory checkers that verify the interaction transcripts of priority queues, stacks, and double-ended queues. We obtain tight upper and lower bounds for these problems, thereby addressing an important subclass of the memory checking framework of Blum et al. [Algorithmica, 12 (1994), pp. 225--244].

Morphic characterizations with insertion systems controlled by a context of length one

Among the variety of representation theorems for context-free languages, the Chomsky–Schützenberger theorem is unique in that it consists of Dyck languages, regular languages, and simple operations. In the present paper, we obtain some characterizations of and representation theorems for languages in the Chomsky hierarchy by using insertion systems, strictly locally testable languages, and morphisms in the framework of the Chomsky–Schützenberger theorem. For instance, each context-free language L can be represented in the form L=h(L(γ)∩R), where γ is an insertion system of weight (1,1), R is a strictly 2-testable language, and h is a morphism. In contrast, if R is instead a strictly 1-testable language, the class of languages generated is a proper subclass of context-free languages and incomparable with the class of regular languages.

THE INCLUSION PROBLEM OF CONTEXT-FREE LANGUAGES: SOME TRACTABLE CASES

We study the problem of testing whether a context-free language is included in a fixed set L0, where L0 is the language of words reducing to the empty word in the monoid defined by a complete string rewrite system. We prove that, if the monoid is cancellative, then our inclusion problem is polynomially reducible to the problem of testing equivalence of straight-line programs in the same monoid. As an application, we obtain a polynomial time algorithm for testing if a context-free language is included in a Dyck language (the best previous algorithm for this problem was doubly exponential).

Representing real numbers in a generalized numeration system

We show how to represent an interval of real numbers in an abstract numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show that our framework can be applied to the rational base numeration systems.

Refinement of Representation Theorems for Context-Free Languages

In this paper, we obtain some refinement of representation theorems for context-free languages by using Dyck languages, insertion systems, strictly locally testable languages, and morphisms. For instance, we improved the Chomsky-Schutzenberger representation theorem and show that each context-free language L can be represented in the form L = h (D ∩ R), where D is a Dyck language, R is a strictly 3-testable language, and h is a morphism. A similar representation for context-free languages can be obtained, using insertion systems of weight (3, 0) and strictly 4-testable languages.

REPRESENTATIONS AND CHARACTERIZATIONS OF LANGUAGES IN CHOMSKY HIERARCHY BY MEANS OF INSERTION-DELETION SYSTEMS

Insertion-deletion operations are much investigated in linguistics and in DNA computing and several characterizations of Turing computability and characterizations or representations of languages in Chomsky hierarchy were obtained in this framework. In this note we contribute to this research direction with a new characterization of this type, as well as with representations of regular and context-free languages, mainly starting from context-free insertion systems of as small as possible complexity. For instance, each recursively enumerable language L can be represented in a way similar to the celebrated Chomsky-Schützenberger representation of context-free languages, i.e., in the form L = h(L(γ) ∩ D), where γ is an insertion system of weight (3, 0) (at most three symbols are inserted in a context of length zero), h is a projection, and D is a Dyck language. A similar representation can be obtained for regular languages, involving insertion systems of weight (2,0) and star languages, as well as for context-free languages – this time using insertion systems of weight (3, 0) and star languages.

On complexity functions of infinite words associated with generalized Dyck languages

In this article, we construct a family of infinite words, generated by countable automata and also generated by substitutions over infinite alphabets, closely related to parenthesis languages and we study their complexity functions. We obtain a family of binary infinite words m ( b ) , indexed on the number b ≥ 1 of parenthesis types, such that the growth order of the complexity function of m ( b ) is n ( log n ) 2 if b = 1 and n 1 + log 2 b b if b ≥ 2 .

Chomsky-Schützenberger Type Characterizations of Poly-Slender and Parikh Slender Context-Free Languages

In this paper we propose a Chomsky-Schützenberger type characterization of k-poly-slender context-free languages, as the homomorphical image of an intersection of a Dyck language and a (2k + 1)-poly-slender regular language. A stronger result is provided, namely the homomorphism and the Dyck language are determined irrespective of the given poly-slender context-free language, when considering the family of all poly-slender context-free languages. A similar characterization is obtained for Parikh slender context-free languages.

Lexicographical Generation of a Generalized Dyck Language

We consider a generalization of the Dyck language. We have a set of opening and a set of closing brackets and a relation that gives the pairs of brackets that can be used in the generalized Dyck language. This Dyck language is a handy coding for several classes of labeled trees.We present an algorithm that generates all words of length 2n of the generalized Dyck language lexicographically. Thereby, each word is computed from its predecessor according to the lexicographical order without any knowledge about the words generated before. Additionally, we introduce a condition on the relation for the generalized Dyck language to be simply generated, which means that an algorithm needs to read only the suffix to be changed in order to compute the successor of a word according to the lexicographical order. We present an algorithm that checks whether a Dyck language is simply generated or not.For an arbitrary relation, we compute the sth moments of the random variable describing the length of the suffix to be cha...

Testing membership in parenthesis languages

AbstractWe continue the investigation of properties defined by formal languages. This study was initiated by Alon et al. [1], who described an algorithm for testing properties defined by regular languages. Alon et al. also considered several context free languages, and in particular Dyck languages, which contain strings of properly balanced parentheses. They showed that the first Dyck language, which contains strings over a single type of pairs of parentheses, is testable in time independent of n, where n is the length of the input string. However, the second Dyck language, defined over two types of parentheses, requires Ω (log n) queries. Here we describe a sublinear‐time algorithm for testing all Dyck languages. Specifically, the running time of our algorithm is Õ(n2/3/ϵ3), where ϵ is the given distance parameter. Furthermore, we improve the lower bound for testing Dyck languages to \documentclass{article}\usepackage{amsfonts,amsmath}\pagestyle{empty}\begin{document}$\tilde{\Omega}$\end{document}(n1/11) for constant ϵ. We also describe a testing algorithm for the context free language LREV = {uurvvr : u, v ∈ Σ*}, where Σ is a fixed alphabet. The running time of our algorithm is \documentclass{article} \usepackage{amsfonts,amsmath}\pagestyle{empty} \begin{document}$\tilde{O}(\sqrt{n/\epsilon}$ \end{document}, which almost matches the lower bound given by Alon et al. [1]. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 98–138, 2003

On the enumeration and generation of generalized Dyck words

We provide a new algebraic grammar for generalized Dyck languages as introduced by Labelle and Yeh (Discrete Math. 82 (1990) 1–6). The study of these languages leads to the particular sublanguages of words without proper factors belonging to the studied language. A random generation scheme is shown for generalized Dyck languages, which leads to some asymptotic results. In the two-letter case, for which the words correspond to ‘rational slope Dyck paths’, more exact and asymptotic enumerative results are obtained, including the asymptotic average area to integer or 3 2 slope Dyck paths.

Context-Free Recognition with Weighted Automata

We introduce the definition of language recognition with weighted automata, a generalization of the classical definition of recognition with unweighted acceptors. We show that, with our definition of recognition, weighted automata can be used to recognize a class of languages that strictly includes regular languages. The class of languages accepted depends on the weight set which has the algebraic structure of a semiring. We give a generic linear time algorithm for recognition with weighted automata and describe examples with various weight sets illustrating the recognition of several classes of context-free languages. We prove, in particular, that the class of languages equivalent to the language of palindromes can be recognized by weighted automata over the (+,ċ)-semiring, and that the class of languages equivalent to the Dyck language of first order D 1 '*can be recognized by weighted automata over the real tropical semiring. We also prove that weighted automata over the real tropical semiring can be used to recognize regular expressions.

HNN EXTENSIONS OF SEMILATTICES

The main purpose of this paper is to investigate properties of an HNN extension of a semilattice, to give its equivalent characterizations and to discuss similarities with free groups. An HNN extension of a semilattice is shown to be a universal object in a certain category and an F-inverse cover over a free group for every inverse semigroup in the category. We also show that a graph with respect to a certain subset of an HNN extension of a semilattice is a tree and that this property characterizes an HNN extension of a semilattice. Moreover, we look into three subclasses: the class of full HNN extensions of semilattices with an identity, the class of universally E-unitary inverse semigroups and the class of HNN extensions of finite semilattices. The first class consists of factorizable E-unitary inverse semigroups whose maximal group homomorphic images are free. We obtain a generalization of the Nielsen–Schreier subgroup theorem to this class. The second consists of inverse semigroups presented by relations on Dyck words. An inverse semigroup in the third class has a relatively easy finite presentation using a Dyck language and has solvable word problem.