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Topology and Ambiguity in Omega Context Free Languages
TL;DR: In this paper, the authors studied the relationship between the topological complexity of an omega context free language and its degree of ambiguity, and showed that omega context-free languages with respect to B\"uchi pushdown automata have a maximum degree of ambiguities.
Abstract: We study the links between the topological complexity of an omega context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel omega context free languages which are recognized by B\"uchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation of taking the omega power of a finitary context free language. We prove also that taking the adherence or the delta-limit of a finitary language preserves neither unambiguity nor inherent ambiguity. On the other side we show that methods used in the study of omega context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by B\"uchi 2-tape automata and we get first results in that direction.
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TL;DR: The Borel hierarchy of the class of context free languages accepted by Buchi 1-counter automata has been shown to be the same as the Wadge hierarchy of contextsafe languages with a Buchi or a Muller acceptance condition.
Abstract: We show that the Borel hierarchy of the class of context free $\omega$-languages, or even of the class of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Borel hierarchy of the class of $\omega$-languages accepted by Turing machines with a Buchi acceptance condition. In particular, for each recursive non-null ordinal $\alpha$, there exist some ${\bf \Sigma}^0_\alpha$-complete and some ${\bf \Pi}^0_\alpha$-complete $\omega$-languages accepted by Buchi 1-counter automata. And the supremum of the set of Borel ranks of context free $\omega$-languages is an ordinal $\gamma_2^1$ that is strictly greater than the first non-recursive ordinal $\omega_1^{\mathrm{CK}}$. We then extend this result, proving that the Wadge hierarchy of context free $\omega$-languages, or even of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Wadge hierarchy of $\omega$-languages accepted by Turing machines with a Buchi or a Muller acceptance condition.
43 citations
TL;DR: This is a survey of results about versions of fine hierarchies and many-one reducibilities that appear in different parts of theoretical computer science, to identify the unifying notions, techniques and ideas.
28 citations
01 Jan 2014
TL;DR: The Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages are considered, which form the second level of the Chomsky hierarchy of languages of infinite words.
Abstract: We survey recent results on the topological complexity of context-free omega-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of ambiguity, and the special case of omega-powers.
12 citations
Posted Content•
TL;DR: In this paper, the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets was investigated, and it was shown that any tree language accepted by an unambiguous Buchi tree automaton must be Borel.
Abstract: We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer $n \geq 1$, there is a $D_{\omega^n}({\bf \Sigma}^1_1)$-complete tree language L_n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Buchi tree automaton must be Borel. Then we consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i, k)$. We prove that the $D_{\omega^n}({\bf \Sigma}^1_1)$-complete tree languages L_n are Wadge reducible to the game tree language $W_{(i, k)}$ for $k-i \geq 2$. In particular these languages $W_{(i, k)}$ are not in any class $D_{\alpha}({\bf \Sigma}^1_1)$ for $\alpha < \omega^\omega$.
9 citations
Posted Content•
TL;DR: The Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages are considered, which form the second level of the Chomsky hierarchy of languages of infinite words.
Abstract: We survey recent results on the topological complexity of context-free omega-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of ambiguity, and the special case of omega-powers.
4 citations
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01 Jan 1987
TL;DR: In this article, the authors present a largely balanced approach, which combines many elements of the different traditions of the subject, and includes a wide variety of examples, exercises, and applications, in order to illustrate the general concepts and results of the theory.
Abstract: Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.
3,340 citations
02 Jan 1991
TL;DR: This chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree.
Abstract: Publisher Summary This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.
1,475 citations
Book•
01 Jan 1980TL;DR: Descriptive set theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets.
Abstract: Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern 'effective' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory.
1,086 citations
Book•
[...]
01 Apr 1997
TL;DR: The Department of Languages provides a vital component in the undergraduate liberal arts education offered at the University of WisconsinEau Claire and serves at home and abroad as ambassadors for a diverse, peacefully interacting global society.
Abstract: The Department of Languages provides a vital component in the undergraduate liberal arts education offered at the University of WisconsinEau Claire. The department promotes multilingualism as the key to intercultural understanding. It does so through courses in foreign language, culture, and literature, including English language for native speakers of other languages; major and minor programs in a number of world languages; study abroad opportunities and internships in many countries; and student/faculty collaborative research. Graduates from the Department of Languages serve at home and abroad as ambassadors for a diverse, peacefully interacting global society.
456 citations