35,681 research outputs found

    On asymptotics of the beta-coalescents

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    We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1,b)(1,b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b=1b=1, which corresponds to the Bolthausen--Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a,b)(a,b)-coalescents with 0<a<10<a<1 leads to a simplified derivation of the known (2a)(2-a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1,b)(1,b)-coalescent by exploiting the method of sequential approximations.Comment: 25 pages, submitted for publicatio

    Visibly Linear Dynamic Logic

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    We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown languages over finite words. In VLDL one can, e.g., express that a function resets a variable to its original value after its execution, even in the presence of an unbounded number of intermediate recursive calls. We prove that VLDL describes exactly the ω\omega-visibly pushdown languages. Thus it is strictly more expressive than LTL and able to express recursive properties of programs with unbounded call stacks. The main technical contribution of this work is a translation of VLDL into ω\omega-visibly pushdown automata of exponential size via one-way alternating jumping automata. This translation yields exponential-time algorithms for satisfiability, validity, and model checking. We also show that visibly pushdown games with VLDL winning conditions are solvable in triply-exponential time. We prove all these problems to be complete for their respective complexity classes.Comment: 25 Page

    Sampling from Stochastic Finite Automata with Applications to CTC Decoding

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    Stochastic finite automata arise naturally in many language and speech processing tasks. They include stochastic acceptors, which represent certain probability distributions over random strings. We consider the problem of efficient sampling: drawing random string variates from the probability distribution represented by stochastic automata and transformations of those. We show that path-sampling is effective and can be efficient if the epsilon-graph of a finite automaton is acyclic. We provide an algorithm that ensures this by conflating epsilon-cycles within strongly connected components. Sampling is also effective in the presence of non-injective transformations of strings. We illustrate this in the context of decoding for Connectionist Temporal Classification (CTC), where the predictive probabilities yield auxiliary sequences which are transformed into shorter labeling strings. We can sample efficiently from the transformed labeling distribution and use this in two different strategies for finding the most probable CTC labeling