115 research outputs found

    The Wadge Hierarchy of Deterministic Tree Languages

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    We provide a complete description of the Wadge hierarchy for deterministically recognisable sets of infinite trees. In particular we give an elementary procedure to decide if one deterministic tree language is continuously reducible to another. This extends Wagner's results on the hierarchy of omega-regular languages of words to the case of trees.Comment: 44 pages, 8 figures; extended abstract presented at ICALP 2006, Venice, Italy; full version appears in LMCS special issu

    Decision Problems for Deterministic Pushdown Automata on Infinite Words

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    The article surveys some decidability results for DPDAs on infinite words (omega-DPDA). We summarize some recent results on the decidability of the regularity and the equivalence problem for the class of weak omega-DPDAs. Furthermore, we present some new results on the parity index problem for omega-DPDAs. For the specification of a parity condition, the states of the omega-DPDA are assigned priorities (natural numbers), and a run is accepting if the highest priority that appears infinitely often during a run is even. The basic simplification question asks whether one can determine the minimal number of priorities that are needed to accept the language of a given omega-DPDA. We provide some decidability results on variations of this question for some classes of omega-DPDAs.Comment: In Proceedings AFL 2014, arXiv:1405.527

    Index problems for game automata

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    For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognize this language with a non-deterministic, alternating, or weak alternating parity automaton. These questions are known as, respectively, the non-deterministic, alternating, and weak Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognizable by deterministic automata (the alternating variant trivializes). We investigate a wider class of regular languages, recognizable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognize languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is, the alternating index problem does not trivialize any more. Our main contribution is that all three index problems are decidable for languages recognizable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognized by a game automaton

    Baire reductions and good Borel reducibilities

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    In reference [8] we have considered a wide class of "well-behaved" reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [8] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ\xi functions).Comment: 21 page

    A Characterisation of Pi^0_2 Regular Tree Languages

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    We show an algorithm that for a given regular tree language L decides if L is in Pi^0_2, that is if L belongs to the second level of Borel Hierarchy. Moreover, if L is in Pi^0_2, then we construct a weak alternating automaton of index (0, 2) which recognises L. We also prove that for a given language L, L is recognisable by a weak alternating (1, 3)-automaton if and only if it is recognisable by a weak non-deterministic (1, 3)-automaton

    Concurrent games with tail objectives

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    AbstractWe study infinite stochastic games played by two players over a finite state space, with objectives specified by sets of infinite traces. The games are concurrent (players make moves simultaneously and independently), stochastic (the next state is determined by a probability distribution that depends on the current state and chosen moves of the players) and infinite (proceed for an infinite number of rounds). The analysis of concurrent stochastic games can be classified into: quantitative analysis, analyzing the optimum value of the game and ε-optimal strategies that ensure values within ε of the optimum value; and qualitative analysis, analyzing the set of states with optimum value 1 and ε-optimal strategies for the states with optimum value 1. We consider concurrent games with tail objectives, i.e., objectives that are independent of the finite-prefix of traces, and show that the class of tail objectives is strictly richer than that of the ω-regular objectives. We develop new proof techniques to extend several properties of concurrent games with ω-regular objectives to concurrent games with tail objectives. We prove the positive limit-one property for tail objectives. The positive limit-one property states that for all concurrent games if the optimum value for a player is positive for a tail objective Φ at some state, then there is a state where the optimum value is 1 for the player for the objective Φ. We also show that the optimum values of zero-sum (strictly conflicting objectives) games with tail objectives can be related to equilibrium values of nonzero-sum (not strictly conflicting objectives) games with simpler reachability objectives. A consequence of our analysis presents a polynomial time reduction of the quantitative analysis of tail objectives to the qualitative analysis for the subclass of one-player stochastic games (Markov decision processes)

    Topological Complexity of Sets Defined by Automata and Formulas

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    In this thesis we consider languages of infinite words or trees defined by automata of various types or formulas of various logics. We ask about the highest possible position in the Borel or the projective hierarchy inhabited by sets defined in a given formalism. The answer to this question is called the topological complexity of the formalism.It is shown that the topological complexity of Monadic Second Order Logic extended with the unbounding quantifier (introduced by Bojańczyk to express some asymptotic properties) over ω-words is the whole projective hierarchy. We also give the exact topological complexities of related classes of languages recognized by nondeterministic ωB-, ωS- and ωBS-automata studied by Bojańczyk and Colcombet, and a lower complexity bound for an alternating variant of ωBS-automata.We present the series of results concerning bi-unambiguous languages of infinite trees, i.e. languages recognized by unambiguous parity tree automata whose complements are also recognized by unambiguous parity automata. We give an example of a bi-unambiguous tree language G that is analytic-complete. We present an operation σ on tree languages with the property that σ(L) is topologically harder than any language in the sigma-algebra generated by the languages continuously reducible to L. If the operation is applied to a bi-unambiguous language than the result is also bi-unambiguous. We then show that the application of the operation can be iterated to obtain harder and harder languages. We also define another operation that enables a limit step iteration. Using the operations we are able to construct a sequence of bi-unambiguous languages of increasing topological complexity, of length at least ω square.W niniejszej rozprawie rozważane są języki nieskończonych słów lub drzew definiowane poprzez automaty różnych typów lub formuły różnych logik. Pytamy o najwyższą możliwą pozycję w hierarchii borelowskiej lub rzutowej zajmowaną przez zbiory definiowane w danym formalizmie. Odpowiedź na to pytanie jest nazywana złożonością topologiczną formalizmu.Przedstawiony został dowód, że złożonością topologiczną Logiki Monadycznej Drugiego Rzędu rozszerzonej o kwantyfikator Unbounding (wprowadzony przez Bojańczyka w celu umożliwienia wyrażania własności asymptotycznych) na słowach nieskończonych jest cała hierarchia rzutowa. Obliczone zostały również złożoności topologiczne klas języków rozpoznawanych przez niedeterministyczne ωB-, ωS- i ωBS-automaty rozważane przez Bojańczyka i Colcombet'a, oraz zostało podane dolne ograniczenie złożoności wariantu alternującego ωBS-automatów.Zaprezentowane zostały wyniki dotyczące języków podwójnie jednoznacznych, tzn. języków rozpoznawanych przez jednoznaczne automaty parzystości na drzewach, których dopełnienia również są rozpoznawane przez jednoznaczne automaty parzystości. Podany został przykład podwójnie jednoznacznego języka drzew G, który jest analityczny-zupełny. Została wprowadzona operacja σ na językach drzew taka, że język σ(L) jest topologicznie bardziej złożony niż jakikolwiek język należący do sigma-algebry generowanej przez języki redukujące się w sposób ciągły do języka L. W wyniku zastosowania powyższej operacji do języka podwójnie jednoznacznego otrzymujemy język podwójnie jednoznaczny. Zostało pokazane, że kolejne iteracje aplikacji powyższej operacji dają coraz bardziej złożone języki. Została również wprowadzona druga operacja, która umożliwia krok graniczny iteracji. Używając obydwu powyższych operacji można skonstruować ciąg długości ω kwadrat złożony z języków podwójnie jednoznacznych o coraz większej złożoności
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