Showing papers on "Indexed language published in 2017"
TL;DR: This work considers d-dimensional contextual array grammars and investigates their computational power when using various control mechanisms – matrices, regular control languages, and tissue P systems, which work like regular control Languages, but may end up with a final check for the non-applicability of some rules.
7 citations
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TL;DR: This paper shows that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L, and shows that the language of primitives, and primitive sets, in free groups of rank higher than two is context- sensitive.
Abstract: L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives in the free group on two generators, the Bridson-Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth.
6 citations
Posted Content•
TL;DR: In this paper, a pumping lemma for the word/tree languages generated by higher-order grammars has been shown for order-2 tree languages and order-3 word languages.
Abstract: We study a pumping lemma for the word/tree languages generated by higher-order grammars. Pumping lemmas are known up to order-2 word languages (i.e., for regular/context-free/indexed languages), and have been used to show that a given language does not belong to the classes of regular/context-free/indexed languages. We prove a pumping lemma for word/tree languages of arbitrary orders, modulo a conjecture that a higher-order version of Kruskal's tree theorem holds. We also show that the conjecture indeed holds for the order-2 case, which yields a pumping lemma for order-2 tree languages and order-3 word languages.
6 citations
06 Mar 2017
TL;DR: The decidability properties of restricted linear indexed grammars have been studied in the literature as mentioned in this paper, and it is known that the Parikh image of every language in the family is semi-linear.
Abstract: The family, \({\mathsf{{\mathcal {L}}(IND_{LIN})}}\), of languages generated by linear indexed grammars has been studied in the literature. It is known that the Parikh image of every language in \({\mathsf{{\mathcal {L}}(IND_{LIN})}}\) is semi-linear. However, there are bounded semi-linear languages that are not in \({\mathsf{{\mathcal {L}}(IND_{LIN})}}\). Here, we look at larger families of (restricted) indexed languages and study their properties, their relationships, and their decidability properties.
6 citations
TL;DR: It is proved in this paper that every language described by a grammar with contexts can be recognized in deterministic linear space.
3 citations
TL;DR: It is shown that if L is an indexed language, then α is a morphic word, i.e., α can be generated by iterating a morphism under a coding, and this implies that the infinite words determined by indexed languages are exactly the morphic words.
Abstract: We prove a pumping lemma in order to characterize the infinite words determined by indexed languages. An infinite language L determines an infinite word α if every string in L is a prefix of α. If L is regular or context-free, it is known that α must be ultimately periodic. We show that if L is an indexed language, then α is a morphic word, i.e., α can be generated by iterating a morphism under a coding. Since the other direction, that every morphic word is determined by some indexed language, also holds, this implies that the infinite words determined by indexed languages are exactly the morphic words. The pumping lemma which we use to obtain this result generalizes one recently proved for the class ET0L.
2 citations
01 Jul 2017
TL;DR: This work proves a pumping lemma for word/tree languages of arbitrary orders, modulo a conjecture that a higher-order version of Kruskal's tree theorem holds, and shows that the conjecture indeed holds for the order-2 case.
Abstract: We study a pumping lemma for the word/tree languages generated by higher-order grammars. Pumping lemmas are known up to order-2 word languages (i.e., for regular/context-free/indexed languages), and have been used to show that a given language does not belong to the classes of regular/context-free/indexed languages. We prove a pumping lemma for word/tree languages of arbitrary orders, modulo a conjecture that a higher-order version of Kruskal's tree theorem holds. We also show that the conjecture indeed holds for the order-2 case, which yields a pumping lemma for order-2 tree languages and order-3 word languages.
2 citations