shown that their generative power lies strictly between the context-free languages of finite index and the matrix languages of finite index. Moreover, we studied extended context-free Petri nets with place capacities. A possible extension of the concept is to use capacity functions that allow an unbounded number of some nonterminals.

The investigation shows that for every grammar controlled by a cf Petri net with $k$-place capacity, $k\geq 1$, there exists an equivalent grammar controlled by a cf Petri net with 1-place capacity, i.e., the families of languages generated by cf Petri nets with place capacities do not form a hierarchy with respect to the place capacities.


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> [*Several new methods have been recently proposed for performing valid inference after model selection. An older method is sampling splitting: use part of the data for model selection and the rest for inference. In this paper we revisit sample splitting combined with the bootstrap (or the Normal approximation). We show that this leads to a simple, assumption-free approach to inference and we establish results on the accuracy of the method. In fact, we find new bounds on the accuracy of the bootstrap and the Normal approximation for general nonlinear parameters with increasing dimension which we then use to assess the accuracy of regression inference. We define new parameters that measure variable importance and that can be inferred with greater accuracy than the usual regression coefficients. Finally, we elucidate an inference-prediction trade-off: splitting increases the accuracy and robustness of inference but can decrease the accuracy of the predictions.*]{}

“Investigators who use \[regression\] are not paying adequate attention to the connection - if any - between the models and the phenomena they are studying. ... By the time the models are deployed, the scientific position is nearly hopeless. Reliance on models in such cases is Panglossian ...”

—David Freedman

Introduction
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