98 research outputs found
MAT learners for recognizable tree languages and tree series
We review a family of closely related query learning algorithms for unweighted and weighted tree automata, all of which are based on adaptations of the minimal adequate teacher (MAT) model by Angluin. Rather than presenting
new results, the goal is to discuss these algorithms in sufficient detail to make their similarities and differences transparent to the reader interested in grammatical inference of tree automata
Minimizing Tree Automata for Unranked Trees
International audienceAutomata for unranked trees form a foundation for XML schemas, querying and pattern languages. We study the problem of efficiently minimizing such automata. We start with the unranked tree automata (UTAs) that are standard in database theory, assuming bottom-up determinism and that horizontal recursion is represented by deterministic finite automata. We show that minimal UTAs in that class are not unique and that minimization is NP-hard. We then study more recent automata classes that do allow for polynomial time minimization. Among those, we show that bottom-up deterministic stepwise tree automata yield the most succinct representations
Hyper-Minimization for Deterministic Weighted Tree Automata
Hyper-minimization is a state reduction technique that allows a finite change
in the semantics. The theory for hyper-minimization of deterministic weighted
tree automata is provided. The presence of weights slightly complicates the
situation in comparison to the unweighted case. In addition, the first
hyper-minimization algorithm for deterministic weighted tree automata, weighted
over commutative semifields, is provided together with some implementation
remarks that enable an efficient implementation. In fact, the same run-time O(m
log n) as in the unweighted case is obtained, where m is the size of the
deterministic weighted tree automaton and n is its number of states.Comment: In Proceedings AFL 2014, arXiv:1405.527
MSOL-Definability Equals Recognizability for Halin Graphs and Bounded Degree k-Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. These algorithms are constructed as finite state tree automata, and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e. every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. We prove this conjecture for a number of special cases in a stronger form. That is, we show that each recognizable property is definable in MSOL, i.e. the counting operation is not needed in our expressions. We give proofs for Halin graphs, bounded degree k-outerplanar graphs and some related graph classes. We furthermore show that the conjecture holds for any graph class that admits tree decompositions that can be defined in MSOL, thus providing a useful tool for future proofs
Об одном достаточном условии нерегулярности языков
The article deals with a proof of one sufficient condition for the irregularity of languages. This condition is related to the properties of certain relations on the set of natural numbers, namely relations possessing the property, referred to as strong separability. In turn, this property is related to the possibility of decomposition of an arithmetic vector space into a direct sum of subspaces. We specify languages in some finite alphabet through the properties of a vector that shows the number of occurrences of each letter of the alphabet in the language words and is called the word distribution vector in the word. The main result of the paper is the proof of the theorem according to which a language given in such a way that the vector of distribution of letters in each word of the language belongs to a strongly separable relation on the set of natural numbers is not regular. Such an approach to the proof of irregularity is based on the Myhill-Nerode theorem known in the theory of formal languages, according to which the necessary and sufficient condition for the regularity of a language consists in the finiteness of the index of some equivalence relation defined by the language.The article gives a definition of a strongly separable relation on the set of natural numbers and examines examples of such relations. Also describes a construction covering a considerably wide class of strongly separable relations and connected with decomposition of the even-dimensional vector space into a direct sum of subspaces of the same dimension. Gives the proof of the lemma to assert an availability of an infinite sequence of vectors, any two terms of which are pairwise disjoint, i.e. one belongs to some strongly separable relation, and the other does not. Based on this lemma, there is a proof of the main theorem on the irregularity of a language defined by a strongly separable relation.This result sheds additional light on the effectiveness of regularity / irregularity analysis tools based on the Myhill-Neroud theorem. In addition, the proved theorem and analysis of some examples of strongly separable relations allows us to establish non-trivial connections between the theory of formal languages and the theory of linear spaces, which, as analysis of sources shows, is relevant.In terms of development of the obtained results, the problem of the general characteristic of strongly separable relations is of interest, as well as the analysis of other properties of numerical sets that are important from the point of view of regularity / irregularity analysis of languages.Данная статья посвящена доказательству одного достаточного условия нерегулярности языков. Это условие связано со свойствами некоторых отношений на множестве натуральных чисел, а именно отношений, обладающих свойством, названное сильной отделимостью. В свою очередь, это свойство связано с возможностью разложения арифметического векторного пространства в прямую сумму подпространств. Мы задаем языки в некотором конечном алфавите через свойства вектора, показывающего числа вхождений каждой буквы алфавита в слова языка и называемого вектором распределения букв в слове. Основной результат статьи состоит в доказательстве теоремы, согласно которой язык, задаваемый таким образом, что вектор распределения букв в каждом слове языка принадлежит сильно отделимому отношению на множестве натуральных чисел, нерегулярен. Такой подход к доказательству нерегулярности основан на известной в теории формальных языков теореме Майхилла-Нероуда, согласно которой необходимое и достаточное условие регулярности языка состоит в конечности индекса некоторого отношения эквивалентности, определяемого языком.В статье дается определение сильно отделимого отношения на множестве натуральных чисел и рассматриваются примеры таких отношений. Дается также описание конструкции, покрывающей весьма широкий класс сильно отделимых отношений и связанной с разложением векторного пространства четной размерности в прямую сумму подпространств одинаковой размерности. Доказывается лемма, утверждающая существование бесконечной последовательности векторов, любые два члена которой попарно дизъюнктны, т.е. один принадлежит некоторому сильно отделимому отношению, а другой нет. На основании этой леммы доказывается основная теорема о нерегулярности языка, определяемым сильно отделимым отношением.Этот результат проливает дополнительный свет на эффективность инструментов анализа регулярности/нерегулярности языков, базирующихся на теореме Майхилла-Нероуда. Кроме того, доказанная теорема и анализ некоторых примеров сильно отделимых отношений позволяет установить нетривиальные связи между теорией формальных языков и теорией линейных пространств, что, как показывает анализ источников, является актуальной проблематикой.В плане развития полученных результатов интерес представляет задача общей характеристики сильно отделимых отношений, а также анализ других свойств числовых множеств, важных с точки зрения анализа регулярности/нерегулярности языков
Languages, machines, and classical computation
3rd ed, 2021. A circumscription of the classical theory of computation building up from the Chomsky hierarchy. With the usual topics in formal language and automata theory
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