Chomsky Hierarchy Research Articles

ON PERIODIC POINTS OF FREE INVERSE MONOID HOMOMORPHISMS

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

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.

Cohesiveness in promise problems

Promise problems have been introduced in 1985 by S. Even e.a. as a generalization of decision problems. Using a very general approach we study solvability and unsolvability conditions for promise problems of set and language families. We show, that cores of unsolvability are completely determined by partitions of cohesive sets. We prove the existence of cores in unsolvable promise problems assuming certain closure properties for the given set family. Connections to immune sets and complexity cores are presented. Furthermore, results about cohesiveness with respect to the language families from the Chomsky hierarchy are given.

The Complexity of Szilard Languages of Matrix Grammars Revisited

The regulated rewriting mechanism is one of the most efficient methods to augment the Chomsky hierarchy with a large variety of language classes. In this paper we investigate the derivation mechanism in regulated rewriting grammars such as matrix grammars, by studying their Szilard languages. We focus on the complexity of Szilard languages associated with unrestricted and leftmost-like derivations in matrix grammars, with or without appearance checking. The reason is twofold. First, to relate these classes of languages to parallel complexity classes such as NC1 and AC1, and, second, to improve some previous results. We prove that unrestricted Szilard languages and certain leftmost Szilard languages of context-free matrix grammars, without appearance checking, can be accepted by indexing alternating Turing machines in logarithmic time and space. Consequently, these classes are included in UE*-uniform NC1. Unrestricted Szilard languages of matrix grammars with appearance checking can be accepted by deterministic Turing machines in On log n time and Olog n space. Leftmost-like Szilard languages of context-free matrix grammars, with appearance checking, can be recognized by nondeterministic Turing machines by using the same time and space resources. Hence, all these classes are included in AC1.

What Complexity Differences Reveal About Domains in Language*

An important distinction between phonology and syntax has been overlooked. All phonological patterns belong to the regular region of the Chomsky Hierarchy, but not all syntactic patterns do. We argue that the hypothesis that humans employ distinct learning mechanisms for phonology and syntax currently offers the best explanation for this difference.

Graph transformations and software engineering: Success stories and lost chances

Textual as well as visual and diagrammatic notations are essential in software engineering, and are used in many different contexts. Chomsky grammars are the key tool to handle textual notations, and find many applications for textual languages. Visual and diagrammatic languages add spatial dimensions that reduce the applicability of textual grammars and call for new tools.Graph transformation systems have been studied for over 40 years and are a powerful tool to deal with syntax, semantics and transformation of diagrammatic notations. The enormous importance of visual and diagrammatic languages and the strong support that graph transformation provide to the manipulation of diagrammatic notations would suggest a big success of graph transformation in software engineering.Graph transformation systems find their application both as language generating devices and specification means for system evolution, and thus can have many applications in software engineering. In this paper we discuss the main features of graph transformation and how they can help software engineers. We look back to the many attempts to use graph transformations in software engineering in the last 15 years, identify some success stories, and discuss to what extent graph transformation succeeded, when they have not succeeded yet, what are the main causes of failures, and how they can help software engineering in the next 15 years.

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 neurobiology of syntax: beyond string sets.

The human capacity to acquire language is an outstanding scientific challenge to understand. Somehow our language capacities arise from the way the human brain processes, develops and learns in interaction with its environment. To set the stage, we begin with a summary of what is known about the neural organization of language and what our artificial grammar learning (AGL) studies have revealed. We then review the Chomsky hierarchy in the context of the theory of computation and formal learning theory. Finally, we outline a neurobiological model of language acquisition and processing based on an adaptive, recurrent, spiking network architecture. This architecture implements an asynchronous, event-driven, parallel system for recursive processing. We conclude that the brain represents grammars (or more precisely, the parser/generator) in its connectivity, and its ability for syntax is based on neurobiological infrastructure for structured sequence processing. The acquisition of this ability is accounted for in an adaptive dynamical systems framework. Artificial language learning (ALL) paradigms might be used to study the acquisition process within such a framework, as well as the processing properties of the underlying neurobiological infrastructure. However, it is necessary to combine and constrain the interpretation of ALL results by theoretical models and empirical studies on natural language processing. Given that the faculty of language is captured by classical computational models to a significant extent, and that these can be embedded in dynamic network architectures, there is hope that significant progress can be made in understanding the neurobiology of the language faculty.

Formal language theory: refining the Chomsky hierarchy.

The first part of this article gives a brief overview of the four levels of the Chomsky hierarchy, with a special emphasis on context-free and regular languages. It then recapitulates the arguments why neither regular nor context-free grammar is sufficiently expressive to capture all phenomena in the natural language syntax. In the second part, two refinements of the Chomsky hierarchy are reviewed, which are both relevant to the extant research in cognitive science: the mildly context-sensitive languages (which are located between context-free and context-sensitive languages), and the sub-regular hierarchy (which distinguishes several levels of complexity within the class of regular languages).

On the properties of language classes defined by bounded reaction automata

Reaction automata are a formal model that has been introduced to investigate the computing powers of interactive behaviors of biochemical reactions (Okubo et al. (2012) [19]). Reaction automata are language acceptors with multiset rewriting mechanism whose basic frameworks are based on reaction systems introduced in Ehrenfeucht and Rozenberg (2007) [8].In this paper we continue the investigation of reaction automata with a focus on the formal language theoretic properties of subclasses of reaction automata, called linear-bounded reaction automata (LRAs) and exponentially-bounded reaction automata (ERAs). Besides LRAs, we newly introduce an extended model (denoted by λ-LRAs) by allowing λ-moves in the accepting process of reaction, and investigate the closure properties of language classes accepted by both LRAs and λ-LRAs. Further, we establish new relationships of language classes accepted by LRAs and by ERAs with the Chomsky hierarchy. The main results include the following: (i)the class of languages accepted by λ-LRAs forms an AFL with additional closure properties,(ii)any recursively enumerable language can be expressed as a homomorphic image of a language accepted by an LRA,(iii)the class of languages accepted by ERAs coincides with the class of context-sensitive languages.

Splicing systems and the Chomsky hierarchy

In this paper, we prove decidability properties and new results on the position of the family of languages generated by (circular) splicing systems within the Chomsky hierarchy. The two main results of the paper are the following. First, we show that it is decidable, given a circular splicing language and a regular language, whether they are equal. Second, we prove the language generated by an alphabetic splicing system is context-free. Alphabetic splicing systems are a generalization of simple and semi-simple splicing systems already considered in the literature.

Reaction automata

Reaction systems are a formal model that has been introduced to investigate the interactive behaviors of biochemical reactions. Based on the formal framework of reaction systems, we propose new computing models called reaction automata that feature (string) language acceptors with multiset manipulation as a computing mechanism, and show that reaction automata are computationally Turing universal. Further, some subclasses of reaction automata with space complexity are investigated and their language classes are compared to the ones in the Chomsky hierarchy.

Computing by Folding

The present paper introduces a new computing paradigm based on the idea of string folding. Comparisons between the computational power of the proposed model with the classical families of languages from the Chomsky hierarchy are studied. Some preliminary results are reported and some conjectures are discussed. In this respect, the proposed model is promising not only because of the expected theoretical results, but also because of the possible indirect applications in various fields (as for instance, mathematical linguistics, DNA computing, computing using light, and so on).

Recent Progresses in the Linguistic Modeling of Biological Sequences Based on Formal Language Theory

Treating genomes just as languages raises the possibility of producing concise generalizations about information in biological sequences. Grammars used in this way would constitute a model of underlying biological processes or structures, and that grammars may, in fact, serve as an appropriate tool for theory formation. The increasing number of biological sequences that have been yielded further highlights a growing need for developing grammatical systems in bioinformatics. The intent of this review is therefore to list some bibliographic references regarding the recent progresses in the field of grammatical modeling of biological sequences. This review will also contain some sections to briefly introduce basic knowledge about formal language theory, such as the Chomsky hierarchy, for non-experts in computational linguistics, and to provide some helpful pointers to start a deeper investigation into this field.

Growing Grammars and Length-reducing Automata

Growing context-sensitive grammars were introduced in 1986 as a restricted variant of context-sensitive grammars, where all productions are length increasing. Several interesting properties of these grammars have been shown since then, including polynomial time complexity of the membership problem and machine model characterizations. Various characterizations of the model, efficient recognition algorithm and the properties of its deterministic variant (possessing a characterization by string-rewriting systems) justify the practical value. Moreover, as pointed out by McNaughton in 1999, growing context-sensitive grammars complement the Chomsky hierarchy in a very natural way. This article reviews results on this topic and proposes some open problems.

PARIKH MATRIX MAPPING AND LANGUAGES

Restricting the Parikh Matrix mapping to a language rather than to an alphabet rises a set of problems that seem interesting to us. Moreover, amiability (or M -equivalence as it is named by other authors) is exploited in order to characterise certain types of languages. The paper also proposes a series of results establishing relations between classes of languages defined by Parikh matrix mappings and the Chomsky hierarchy. Finally, a result related to word composition concludes the paper, showing that for every two arbitrary words there exist two other words such that their composition (two by two) is amiable.

Morphic characterizations of languages in Chomsky hierarchy with insertion and locality

This paper concerns new characterizations of regular, context-free, and recursively enumerable languages, using insertion systems with lower complexity. This is achieved by using both strictly locally testable languages and morphisms. The representation is in a similar way to the Chomsky-Schu¨tzenberger representation of context-free languages. Specifically, each recursively enumerable language L can be represented in the form L=h(L(γ)∩R), where γ is an insertion system of weight (3,3), R is a strictly 2-testable language, and h is a projection. A similar representation can be obtained for context-free languages, using insertion systems of weight (2,0) and strictly 2-testable languages, as well as for regular languages, using insertion systems of weight (1,0) and strictly 2-testable languages.

What artificial grammar learning reveals about the neurobiology of syntax

In this paper we examine the neurobiological correlates of syntax, the processing of structured sequences, by comparing FMRI results on artificial and natural language syntax. We discuss these and similar findings in the context of formal language and computability theory. We used a simple right-linear unification grammar in an implicit artificial grammar learning paradigm in 32 healthy Dutch university students (natural language FMRI data were already acquired for these participants). We predicted that artificial syntax processing would engage the left inferior frontal region (BA 44/45) and that this activation would overlap with syntax-related variability observed in the natural language experiment. The main findings of this study show that the left inferior frontal region centered on BA 44/45 is active during artificial syntax processing of well-formed (grammatical) sequence independent of local subsequence familiarity. The same region is engaged to a greater extent when a syntactic violation is present and structural unification becomes difficult or impossible. The effects related to artificial syntax in the left inferior frontal region (BA 44/45) were essentially identical when we masked these with activity related to natural syntax in the same subjects. Finally, the medial temporal lobe was deactivated during this operation, consistent with the view that implicit processing does not rely on declarative memory mechanisms that engage the medial temporal lobe. In the context of recent FMRI findings, we raise the question whether Broca’s region (or subregions) is specifically related to syntactic movement operations or the processing of hierarchically nested non-adjacent dependencies in the discussion section. We conclude that this is not the case. Instead, we argue that the left inferior frontal region is a generic on-line sequence processor that unifies information from various sources in an incremental and recursive manner, independent of whether there are any processing requirements related to syntactic movement or hierarchically nested structures. In addition, we argue that the Chomsky hierarchy is not directly relevant for neurobiological systems.

Theoretical and computational properties of transpositions

Transposable genetic elements are prevalent across many living organisms from bacteria to large mammals. Given the linear primary structure of genetic material, this process is natural to study from a theoretical perspective using formal language theory. We abstract the process of genetic transposition to operations on languages and study it combinatorially and computationally. It is shown that the power of such systems is large relative to the classic Chomsky Hierarchy. However, we are still able to algorithmically determine whether or not a string is a possible product of the iterated application of the operations.

Left-forbidding cooperating distributed grammar systems

A left-forbidding grammar, introduced in this paper, is a context-free grammar, where a set of nonterminal symbols is attached to each context-free production. Such a production can rewrite a nonterminal provided that no symbol from the attached set occurs to the left of the rewritten nonterminal in the current sentential form. The present paper discusses cooperating distributed grammar systems with left-forbidding grammars as components and gives some new characterizations of language families of the Chomsky hierarchy. In addition, it also proves that twelve nonterminals are enough for cooperating distributed grammar systems working in the terminal derivation mode with two left-forbidding components (including erasing productions) to characterize the family of recursively enumerable languages.