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

2008 ◽  
Vol 19 (04) ◽  
pp. 859-871 ◽  
Author(s):  
GHEORGHE PĂUN ◽  
MARIO J. PÉREZ-JIMÉNEZ ◽  
TAKASHI YOKOMORI
Keyword(s):  
Dna Computing ◽  
Research Direction ◽  
Regular Languages ◽  
Turing Computability ◽  
Chomsky Hierarchy ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
Language L ◽  
Context Free

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.

LANGUAGE FAMILIES DEFINED BY A CILIATE BIO-OPERATION: HIERARCHIES AND DECISION PROBLEMS

2005 ◽  
Vol 16 (04) ◽  
pp. 645-662 ◽  
Author(s):  
JÜRGEN DASSOW ◽  
MARKUS HOLZER
Keyword(s):  
Inverted Repeat ◽  
Decision Problems ◽  
Membership Problem ◽  
Chomsky Hierarchy ◽  
Context Sensitive ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
The Status ◽  
Emptiness Problem ◽  
Context Free

We formalize the hairpin inverted repeat excision, which is known in ciliate genetics as an operation on words and languages by defining [Formula: see text] as the set of all words xαyRαRz where w = xαyαRz and the pointer α is in P. We extend this concept to language families which results in families [Formula: see text]. For [Formula: see text] and [Formula: see text] be the families of finite, regular, context-free, context-sensitive or recursively enumerable language, respectively, we determine the hierarchy of the families [Formula: see text] and compare these families with those of the Chomsky hierarchy. Furthermore, we present the status of decidability of the membership problem, emptiness problem and finiteness problem for the families [Formula: see text].

scholarly journals On Derivation Languages of a Class of Splicing Systems

2018 ◽  
Vol 23 (4) ◽  
pp. 981-993 ◽  
Author(s):  
Kalpana Mahalingam ◽  
Prithwineel Paul ◽  
Erkki Mäkinen
Keyword(s):  
Chomsky Hierarchy ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
The Family ◽  
Finite Set ◽  
Morphic Image ◽  
And Control ◽  
Splicing Systems ◽  
Context Free ◽  
Free Language

Derivation languages are language theoretical tools that describe halting derivation processes of a generating device. We consider two types of derivation languages, namely Szilard and control languages for splicing systems where iterated splicing is done in non-uniform way defined by Mitrana, Petre and Rogojin in 2010. The families of Szilard (rules and labels are mapped in a one to one manner) and control (more than one rule can share the same label) languages generated by splicing systems of this type are then compared with the family of languages in the Chomsky hierarchy. We show that context-free languages can be generated as Szilard and control languages and any non-empty context-free language is a morphic image of the Szilard language of this type of system with finite set of rules and axioms. Moreover, we show that these systems with finite set of axioms and regular set of rules are capable of generating any recursively enumerable language as a control language.

Scattered Context Grammars with One Non-Context-Free Production are Computationally Complete

2021 ◽  
Vol 179 (4) ◽  
pp. 361-384
Author(s):  
Zbyněk Křivka ◽  
Alexander Meduna
Keyword(s):  
Open Problem ◽  
Recursively Enumerable Language ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
Context Free

This paper investigates the reduction of scattered context grammars with respect to the number of non-context-free productions. It proves that every recursively enumerable language is generated by a scattered context grammar that has no more than one non-context-free production. An open problem is formulated.

PC GRAMMAR SYSTEMS WITH CLUSTERS OF COMPONENTS

2011 ◽  
Vol 22 (01) ◽  
pp. 203-212 ◽  
Author(s):  
ERZSÉBET CSUHAJ-VARJÚ ◽  
MARION OSWALD ◽  
GYÖRGY VASZIL
Keyword(s):  
Future Research ◽  
Open Problems ◽  
Recursively Enumerable Language ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
Grammar Systems ◽  
Context Free

We introduce PC grammar systems where the components form clusters and the query symbols refer to clusters not individual grammars, i.e., the addressee of the query is not precisely identified. We prove that if the same component replies to all queries issued to a cluster in a rewriting step, then non-returning PC grammar systems with 3 clusters and 7 context-free components are able to generate any recursively enumerable language. We also provide open problems and directions for future research.

scholarly journals BOND-FREE LANGUAGES: FORMALIZATIONS, MAXIMALITY AND CONSTRUCTION METHODS

2005 ◽  
Vol 16 (05) ◽  
pp. 1039-1070 ◽  
Author(s):  
LILA KARI ◽  
STAVROS KONSTANTINIDIS ◽  
PETR SOSÍK
Keyword(s):  
Polynomial Time ◽  
Structural Characterization ◽  
Regular Languages ◽  
Important Case ◽  
Similarity Relation ◽  
Construction Methods ◽  
Large Sets ◽  
Code Property ◽  
Context Free

The problem of negative design of DNA languages is addressed, that is, properties and construction methods of large sets of words that prevent undesired bonds when used in DNA computations. We recall a few existing formalizations of the problem and then define the property of sim-bond-freedom, where sim is a similarity relation between words. We show that this property is decidable for context-free languages and polynomial-time decidable for regular languages. The maximality of this property also turns out to be decidable for regular languages and polynomial-time decidable for an important case of the Hamming similarity. Then we consider various construction methods for Hamming bond-free languages, including the recently introduced method of templates, and obtain a complete structural characterization of all maximal Hamming bond-free languages. This result is applicable to the θ-k-code property introduced by Jonoska and Mahalingam.

MORPHIC CHARACTERIZATIONS OF LANGUAGE FAMILIES IN TERMS OF INSERTION SYSTEMS AND STAR LANGUAGES

2011 ◽  
Vol 22 (01) ◽  
pp. 247-260 ◽  
Author(s):  
FUMIYA OKUBO ◽  
TAKASHI YOKOMORI
Keyword(s):  
Unique Feature ◽  
Recursively Enumerable ◽  
Context Free Language ◽  
Similar Characterization ◽  
Language L ◽  
Finite Set ◽  
Context Free ◽  
String Rewriting ◽  
Free Language ◽  
Recursively Enumerable Languages

Insertion systems have a unique feature in that only string insertions are allowed, which is in marked contrast to a variety of the conventional computing devices based on string rewriting. This paper will mainly focus on those systems whose insertion operations are performed in a context-free fashion, called context-free insertion systems, and obtain several characterizations of language families with the help of other primitive languages (like star languages) as well as simple operations (like projections, weak-codings). For each k ≥ 1, a language L is a k-star language if L = F+ for some finite set F with the length of each string in F is no more than k. The results of this kind have already been presented in [10] by Păun et al., while the purpose of this paper is to prove enhanced versions of them. Specifically, we show that each context-free language L can be represented in the form L = h(L(γ)∩F+), where γ is an insertion system of weight (3, 0) (at most three symbols are inserted in a context-free manner), h is a projection, and F+ is a 2-star language. A similar characterization can be obtained for recursively enumerable languages, where insertion systems of weight (3, 3) and 2-star languages are involved.

STRICT LOCAL TESTABILITY WITH CONSENSUS EQUALS REGULARITY, AND OTHER PROPERTIES

2013 ◽  
Vol 24 (06) ◽  
pp. 747-763 ◽  
Author(s):  
STEFANO CRESPI REGHIZZI ◽  
PIERLUIGI SAN PIETRO
Keyword(s):  
Regular Languages ◽  
Context Sensitive ◽  
Language Definition ◽  
Context Free

A recent language definition device named consensual is based on agreement between similar words. Considering a language over a bipartite alphabet made by pairs of unmarked/marked letters, the match relation specifies when such words agree. Thus a set (the “base”) over the bipartite alphabet consensually specifies another language that includes any terminal word such that a set of corresponding matching words is in the base. We show that all and only the regular languages are consensually generated by a strictly locally testable base; the result is based on a generalization of Medvedev's homomorphic characterization of regular languages. Consensually context-free languages strictly include the base family. The consensual and the base families collapse together if the base is context-sensitive.

scholarly journals Formal language theory: refining the Chomsky hierarchy

2012 ◽  
Vol 367 (1598) ◽  
pp. 1956-1970 ◽  
Author(s):  
Gerhard Jäger ◽  
James Rogers
Keyword(s):  
Natural Language ◽  
Formal Language ◽  
Regular Languages ◽  
Language Theory ◽  
Context Free Grammar ◽  
Chomsky Hierarchy ◽  
Context Sensitive ◽  
Four Levels ◽  
Context Free ◽  
Natural Language Syntax

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 LEFTMOST DERVIATION IN MATRIX GRAMMARS

1999 ◽  
Vol 10 (01) ◽  
pp. 61-79 ◽  
Author(s):  
JÜRGEN DASSOW ◽  
HENNING FERNAU ◽  
GHEORGHE PĂUN
Keyword(s):  
Systematic Study ◽  
Regular Language ◽  
Formal Languages ◽  
Open Problems ◽  
Recursively Enumerable ◽  
Leftmost Derivation ◽  
Context Free ◽  
Recursively Enumerable Languages ◽  
Context Free Grammars

Matrix grammars are one of the classical topics of formal languages, more specifically, regulated rewriting. Although this type of control on the work of context-free grammars is one of the earliest, matrix grammars still raise interesting questions (not to speak about old open problems in this area). One such class of problems concerns the leftmost derivation (in grammars without appearance checking). The main point of this paper is the systematic study of all possibilities of defining leftmost derivation in matrix grammars. Twelve types of such a restriction are defined, only four of which being discussed in literature. For seven of them, we find a proof of a characterization of recursively enumerable languages (by matrix grammars with arbitrary context-free rules but without appearance checking). Other three cases characterize the recursively enumerable languages modulo a morphism and an intersection with a regular language. In this way, we solve nearly all problems listed as open on page 67 of the monograph [7], which can be seen as the main contribution of this paper. Moreover, we find a characterization of the recursively enumerable languages for matrix grammars with the leftmost restriction defined on classes of a given partition of the nonterminal alphabet.

Insertion-deletion systems with substitutions I

Computability ◽  
2021 ◽  
pp. 1-27
Author(s):  
Martin Vu ◽  
Henning Fernau
Keyword(s):  
Single Letter ◽  
Context Dependency ◽  
Insertions And Deletions ◽  
Context Sensitive ◽  
Recursively Enumerable ◽  
Substitution Site ◽  
New Type ◽  
Left And Right ◽  
Context Free

In this paper, we discuss the addition of substitutions as a further type of operations to (in particular, context-free) insertion-deletion systems, i.e., in addition to insertions and deletions we allow single letter replacements to occur. We investigate the effect of the addition of substitution rules on the context dependency of such systems, thereby also obtaining new characterizations of and even normal forms for context-sensitive (CS) and recursively enumerable (RE) languages and their phrase-structure grammars. More specifically, we prove that for each RE language, there is a system generating this language that only inserts and deletes strings of length two without considering the context of the insertion or deletion site, but which may change symbols (by a substitution operation) by checking a single symbol to the left of the substitution site. When we allow checking left and right single-letter context in substitutions, even context-free insertions and deletions of single letters suffice to reach computational completeness. When allowing context-free insertions only, checking left and right single-letter context in substitutions gives a new characterization of CS. This clearly shows the power of this new type of rules.

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