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Showing papers on "Indexed language published in 2015"


Posted Content
TL;DR: It is shown that the computation of downward closures can be reduced to checking a certain unboundedness property and proved that downward closures are computable for every language class with effectively semilinear Parikh images that are closed under rational transductions.
Abstract: The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes. This work presents a simple general method for computing downward closures. For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property. This result is used to prove that downward closures are computable for (i) every language class with effectively semilinear Parikh images that are closed under rational transductions, (ii) matrix languages, and (iii) indexed languages (equivalently, languages accepted by higher-order pushdown automata of order 2).

35 citations


Book ChapterDOI
06 Jul 2015
TL;DR: For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property in this paper.
Abstract: The downward closure of a word language is the set of all not necessarily contiguous subwords of its members It is well-known that the downward closure of any language is regular While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes This work presents a simple general method for computing downward closures For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property This result is used to prove that downward closures are computable for i every language class with effectively semilinear Parikh images that are closed under rational transductions, ii matrix languages, and iii indexed languages equivalently, languages accepted by higher-order pushdown automata of orderi¾?2

29 citations


Posted Content
TL;DR: In this article, it was shown that the set of all solutions in reduced words is an EDT0L language, which is a proper subclass of indexed languages, and that it is an indexed language in the sense of Aho.
Abstract: We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been been open for some years, apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products $\star_{1 \leq i \leq s}F_i$, where each $F_i$ is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by Ferte, Marin and Senizergues (ToCS 2014), which is a very restricted case. Our general result became possible due to the recent recompression technique of Jez. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper (arXiv 1405.5133). For example, we improve the complexity from quadratic nondeterministic space in (arXiv 1405.5133) to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: NSPACE($n\log n$). The conjectured complexity is NP, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured NP.

17 citations


Posted Content
TL;DR: It is shown that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language, an indexed language in the sense of Aho.
Abstract: We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $\mathsf{NSPACE}(n\log n)$ here.

13 citations


Journal ArticleDOI
TL;DR: This paper proposes a more general model, in which context specifications may be two-sided, that is, both the left and the right contexts can be specified by the corresponding operators.

11 citations


Proceedings ArticleDOI
01 Dec 2015
TL;DR: It is shown that the family of arbitrary sticker languages, generated from arbitrary sticker systems, is included in theFamily of Watson-Crick linear languages,generated from Watson-crick linear grammars.
Abstract: Deoxyribonucleic acid, or popularly known as DNA, continues to inspire many theoretical computing models, such as sticker systems and Watson-Crick grammars Sticker systems are the abstraction of ligation processes performed on DNA, while Watson-Crick grammars are models motivated from Watson-Crick finite automata and Chomsky grammars Both of these theoretical models benefit from the Watson-Crick complementarity rule In this paper, we establish the results on the relationship between Watson-Crick linear grammars, which is included in Watson-Crick context-free grammars, and sticker systems We show that the family of arbitrary sticker languages, generated from arbitrary sticker systems, is included in the family of Watson-Crick linear languages, generated from Watson-Crick linear grammars

5 citations


Book ChapterDOI
06 Jul 2015
TL;DR: In this article, it was shown that the set of all solutions in reduced words is an EDT0L language, which is a proper subclass of indexed languages, and the conjectured complexity was shown to be optimal with respect to space complexity.
Abstract: We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been open for some years [9, 10], apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products $$\star _{1 \le i \le s}F_i$$, where each $$F_i$$ is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by [8], which is a very restricted case. Our general result became possible due to the recent recompression technique of Jei¾?. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper [5]. For example, we improve the complexity from quadratic nondeterministic space in [5] to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: $$\mathsf {NSPACE}n\log n$$. The conjectured complexity is $$\mathsf {NP}$$, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured $$\mathsf {NP}$$.

2 citations