1,052 research outputs found

    Multigraphs without large bonds are wqo by contraction

    Full text link
    We show that the class of multigraphs with at most pp connected components and bonds of size at most kk is well-quasi-ordered by edge contraction for all positive integers p,kp,k. (A bond is a minimal non-empty edge cut.) We also characterize canonical antichains for this relation and show that they are fundamental

    Symblicit algorithms for optimal strategy synthesis in monotonic Markov decision processes

    Full text link
    When treating Markov decision processes (MDPs) with large state spaces, using explicit representations quickly becomes unfeasible. Lately, Wimmer et al. have proposed a so-called symblicit algorithm for the synthesis of optimal strategies in MDPs, in the quantitative setting of expected mean-payoff. This algorithm, based on the strategy iteration algorithm of Howard and Veinott, efficiently combines symbolic and explicit data structures, and uses binary decision diagrams as symbolic representation. The aim of this paper is to show that the new data structure of pseudo-antichains (an extension of antichains) provides another interesting alternative, especially for the class of monotonic MDPs. We design efficient pseudo-antichain based symblicit algorithms (with open source implementations) for two quantitative settings: the expected mean-payoff and the stochastic shortest path. For two practical applications coming from automated planning and LTL synthesis, we report promising experimental results w.r.t. both the run time and the memory consumption.Comment: In Proceedings SYNT 2014, arXiv:1407.493

    A classification of separable Rosenthal compacta and its applications

    Full text link
    The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last one is devoted to applications.Comment: 55 pages, no figure

    On Saturated kk-Sperner Systems

    Get PDF
    Given a set XX, a collection FP(X)\mathcal{F}\subseteq\mathcal{P}(X) is said to be kk-Sperner if it does not contain a chain of length k+1k+1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if X|X| is sufficiently large with respect to kk, then the minimum size of a saturated kk-Sperner system FP(X)\mathcal{F}\subseteq\mathcal{P}(X) is 2k12^{k-1}. We disprove this conjecture by showing that there exists ε>0\varepsilon>0 such that for every kk and Xn0(k)|X| \geq n_0(k) there exists a saturated kk-Sperner system FP(X)\mathcal{F}\subseteq\mathcal{P}(X) with cardinality at most 2(1ε)k2^{(1-\varepsilon)k}. A collection FP(X)\mathcal{F}\subseteq \mathcal{P}(X) is said to be an oversaturated kk-Sperner system if, for every SP(X)FS\in\mathcal{P}(X)\setminus\mathcal{F}, F{S}\mathcal{F}\cup\{S\} contains more chains of length k+1k+1 than F\mathcal{F}. Gerbner et al. proved that, if Xk|X|\geq k, then the smallest such collection contains between 2k/212^{k/2-1} and O(logkk2k)O\left(\frac{\log{k}}{k}2^k\right) elements. We show that if Xk2+k|X|\geq k^2+k, then the lower bound is best possible, up to a polynomial factor.Comment: 17 page

    Generating all finite modular lattices of a given size

    Get PDF
    Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 24, semimodular lattices up to size 22, and lattices of size 19. We also show that 2n32^{n-3} is a lower bound for the number of nonisomorphic modular lattices of size nn.Comment: Preprint, 12 pages, 2 figures, 1 tabl
    corecore