68 research outputs found

    Quotient Complexity Of Closed Languages

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    The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-013-9515-7A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by factor we mean contiguous subsequence, and by subword we mean scattered subsequence. We study the state complexity (which we prefer to call quotient complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star, and reversal in each of the four classes of closed languages. We show that repeated applications of positive closure and complement to a closed language result in at most four distinct languages, while Kleene closure and complement give at most eight.Natural Sciences and Engineering Research Council of Canada [OGP0000871]VEGA grant [2/0183/11][APVV-0035-10

    Boundary-Border Extensions of the Kuratowski Monoid

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    The Kuratowski monoid K\mathbf{K} is generated under operator composition by closure and complement in a nonempty topological space. It satisfies 2≤∣K∣≤142\leq|\mathbf{K}|\leq14. The Gaida-Eremenko (or GE) monoid KF\mathbf{KF} extends K\mathbf{K} by adding the boundary operator. It satisfies 4≤∣KF∣≤344\leq|\mathbf{KF}|\leq34. We show that when ∣K∣<14|\mathbf{K}|<14 the GE monoid is determined by K\mathbf{K}. When ∣K∣=14|\mathbf{K}|=14 if the interior of the boundary of every subset is clopen, then ∣KF∣=28|\mathbf{KF}|=28. This defines a new type of topological space we call Kuratowski disconnectedKuratowski\ disconnected. Otherwise ∣KF∣=34|\mathbf{KF}|=34. When applied to an arbitrary subset the GE monoid collapses in one of 7070 possible ways. We investigate how these collapses and KF\mathbf{KF} interdepend, settling two questions raised by Gardner and Jackson. Computer experimentation played a key role in our research.Comment: 48 pages, 9 figure

    Regular tree languages in low levels of the Wadge Hierarchy

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    In this article we provide effective characterisations of regular languages of infinite trees that belong to the low levels of the Wadge hierarchy. More precisely we prove decidability for each of the finite levels of the hierarchy; for the class of the Boolean combinations of open sets BC(Σ10)BC(\Sigma_1^0) (i.e. the union of the first ω\omega levels); and for the Borel class Δ20\Delta_2^0 (i.e. for the union of the first ω1\omega_1 levels)

    Acta Cybernetica : Volume 17. Number 1.

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    Abstract GSOS Rules and a Modular Treatment of Recursive Definitions

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    Terminal coalgebras for a functor serve as semantic domains for state-based systems of various types. For example, behaviors of CCS processes, streams, infinite trees, formal languages and non-well-founded sets form terminal coalgebras. We present a uniform account of the semantics of recursive definitions in terminal coalgebras by combining two ideas: (1) abstract GSOS rules l specify additional algebraic operations on a terminal coalgebra; (2) terminal coalgebras are also initial completely iterative algebras (cias). We also show that an abstract GSOS rule leads to new extended cia structures on the terminal coalgebra. Then we formalize recursive function definitions involving given operations specified by l as recursive program schemes for l, and we prove that unique solutions exist in the extended cias. From our results it follows that the solutions of recursive (function) definitions in terminal coalgebras may be used in subsequent recursive definitions which still have unique solutions. We call this principle modularity. We illustrate our results by the five concrete terminal coalgebras mentioned above, e.\,g., a finite stream circuit defines a unique stream function
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