Improved descriptional complexity results on generalized forbidding grammars
TL;DR: This work studies for what sizes of generalized forbidding grammars one can obtain the computational power of Turing machines and shows that for sizes (2, 6, 8, 6), this result is specifically shown.
About: This article is published in Discrete Applied Mathematics.The article was published on 2021-01-22. It has received 4 citations till now. The article focuses on the topics: Formal language & Type (model theory).
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17 Jul 2019
TL;DR: The computational completeness of MSSC grammars with degrees (2, 1), ( 2, 0) and (3, 0), one of the first approaches ever proposed in regulated rewriting, is considered.
Abstract: Matrix grammars are one of the first approaches ever proposed in regulated rewriting, prescribing that rules have to be applied in a certain order. Typical descriptional complexity measures incorporate the number of nonterminals or the length, i.e., the number of rules per matrix. In simple semi-conditional (SSC) grammars, the derivations are controlled by a permitting string or by a forbidden string associated to each rule. The maximum length i of permitting strings and the maximum length j of forbidden strings are called the degree of such grammars. Matrix SSC grammars (MSSC) put matrix grammar control on SSC rules. We consider the computational completeness of MSSC grammars with degrees (2, 1), (2, 0) and (3, 0). The results are important in the following aspects. (i) With permitting strings alone, it is unknown if SSC grammars are computational complete, while MSSC grammars describe \(\textsf {RE}\) even with severe further restrictions on their descriptional complexity. (ii) Matrix grammars with appearance checking with three nonterminals are computationally complete; however, the length is unbounded. With our constructions for MSSC grammars, we can even bound the length.
3 citations
24 Aug 2020
TL;DR: In this paper, the authors consider GFID systems where each insertion-deletion rule is associated with a set of words and the rule can be applied to a string only if every word of the set is not a subword of the string.
Abstract: We consider generalized forbidding insertion-deletion systems (GFID) where each insertion-deletion rule is associated with a set \(\mathcal {F}\) of words and the rule can be applied to a string only if every word of \(\mathcal {F}\) is not a subword of the string. The parameters in the size \((k;n,i',i'';m,j',j'')\) of a GFID system denote (from left to right) the maximum length of a word in \(\mathcal {F}\), the maximal length of an insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; the last three parameters follow a similar pattern with respect to deletion. We show that GFID systems of sizes \((k;n,i',i'';m,j',j'')\), where \(k=2\) and \(n\,+\,i'\,+\,i''=m\,+\,j'\,+\,j''=2\), with \(n,m> 0\) and \(i',i'',j',j''\in \{0,1\}\), describe all recursively enumerable languages, by explaining algorithms that transform a given type-0 grammar in some normal form to a GFID system of the required size.
3 citations
14 Sep 2020
TL;DR: In this paper, the authors studied the computational completeness of generalized forbidding matrix grammars, i.e., a matrix is applicable to a sentential form w only if none of the words in its forbidding set occurs as a subword in w.
Abstract: Matrix grammars are one of the first approaches ever proposed in regulated rewriting, prescribing that rules have to be applied in a certain order. In traditional regulated rewriting, the most interesting case shows up when all rules are context-free. Typical descriptional complexity measures incorporate the number of nonterminals or the length, i.e., the number of rules per matrix. When viewing matrices as program fragments, it becomes natural to consider additional applicability conditions for such matrices. Here, we focus on forbidding sets, i.e., a matrix is applicable to a sentential form w only if none of the words in its forbidding set occurs as a subword in w. This gives rise to further natural descriptional complexity measures: How long could words in forbidding sets be? How many words could be in any forbidding set? How many matrices contain non-empty forbidding contexts? As context-free grammars with forbidding sets are known as generalized forbidding grammars, we call this variant of matrix grammars also generalized forbidding. In this paper, we attempt to answer the above four questions while studying the computational completeness of generalized forbidding matrix grammars. We also link our research to processing strings with membrane computing and discuss appropriate variations of \(\textsf {P}\) systems.
1 citations
16 Aug 2021
TL;DR: The authors review computational completeness using only a small amount of possibly scarce resources in formal languages and present a review of these results in the form of an essay, with a focus on formal languages.
Abstract: Throughout the history of Formal Languages, one of the research directions has always been to describe computational completeness using only a small amount of possibly scarce resources. We review some of these results in the form of an essay.
1 citations
References
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Book•
01 Jan 1989
TL;DR: This book presents 25 different regulating mechanisms by definitions, examples and basic facts, especially concerning hierarchies, as well as selective substitution grammars as one common generalization.
Abstract: It is well-known that context-free grammars cannot cover all aspects of natural languages, progamming languages and other related fields. Therefore a lot of mechanisms have been introduced which control the application of context-free rules. This book presents 25 different regulating mechanisms by definitions, examples and basic facts, especially concerning hierarchies. Matrix, programmed, and random context grammars as typical representants are studied in more detail. Besides their algebraic and decidability properties a comparison is made with respect to syntactic complexity measures and pure versions. Further topics are combinations of some control mechanisms, regulated L systems, automata characterizations, Szilard languages, and grammar forms of regulated grammars as well as selective substitution grammars as one common generalization.
847 citations
182 citations
Book•
25 Aug 2010TL;DR: This book provides an extensive overview of the formal language landscape between CFG and PTIME, moving from Tree Adjoining Grammars to Multiple Context-Free grammars and then to Range Concatenation Grammar while explaining available parsing techniques for these formalisms.
Abstract: Given that context-free grammars (CFG) cannot adequately describe natural languages, grammar formalisms beyond CFG that are still computationally tractable are of central interest for computational linguists. This book provides an extensive overview of the formal language landscape between CFG and PTIME, moving from Tree Adjoining Grammars to Multiple Context-Free Grammars and then to Range Concatenation Grammars while explaining available parsing techniques for these formalisms. Although familiarity with the basic notions of parsing and formal languages is helpful when reading this book, it is not a strict requirement. The presentation is supported with many illustrations and examples relating to the different formalisms and algorithms, and chapter summaries, problems and solutions. The book will be useful for students and researchers in computational linguistics and in formal language theory.
134 citations
TL;DR: Quelques nouvelles formes normales pour grammaires de type 0 sont presentees, dans tous les casx aucun symbole non terminal additionnel n'est necessaire.
Abstract: Quelques nouvelles formes normales pour grammaires de type 0 sont presentees. Chaque langage recursivement enumerable peut etre engendre par une grammaire ou les regles «context-free» sont de forme S→v, ou S est le symbole initial non terminal. En ce qui concerne les regles «non context-free», on a l'une des cinq situations suivantes: ou bien deux regles du type AB→e, CD→e, ou deux regles du type AB→e, CC→e, ou deux regles du type AA→e, BBB→e, ou une regle du type ABBBA→e, ou regle du type ABC→e. Dans tous les casx aucun symbole non terminal additionnel n'est necessaire
81 citations
23 May 2001
TL;DR: It is shown that the number of non-terminal symbols used in the appearance checking mode can be restricted to two, and in the case of graph controlled (and programmed grammars) with appearance checking this number can be reduced to three.
Abstract: We improve the results elaborated in [6] on the number of non-terminal symbols needed in matrix grammars, programmed grammars, and graph-controlled grammars with appearance checking for generating arbitrary recursively enumerable languages. Of special interest is the result that the number of non-terminal symbols used in the appearance checking mode can be restricted to two. In the case of graph controlled (and programmed grammars) with appearance checking also the number of non-terminal symbols can be reduced to three (and four, respectively); in the case of matrix grammars with appearance checking we either need four non-terminal symbols with three of them being used in the appearance checking mode or else again we only need two nonterminal symbols being used in the appearance checking mode, but in that case we cannot bound the total number of non-terminal symbols.
75 citations