Remo Nitschke | Search arXiv

arXiv2026

Contributions to the hierarchy of probabilistic languages

We reconsider the theory of probabilistic formal languages generated by n-gram models and by probabilistic context-free grammars (PCFGs). The expected hierarchy of probabilistic grammars is established by proving that every probabilistic language generated by an n-gram model is also generated by some PCFG, while some probabilistic languages generated by PCFGs cannot be generated by any $n$-gram model. We introduce the notion of fully connected PCFGs, namely PCFGs in Chomsky normal form where every production rule only involving non-terminals has non-zero probability. Our main result shows that any probabilistic language generated by an $n$-gram model differs from any probabilistic language generated by a fully connected PCFG. Therefore, the class of probabilistic languages generated by $n$-gram models is not a subset of the class generated by fully connected PCFGs.

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arXiv2025

Time-series Random Process Complexity Ranking Using a Bound on Conditional Differential Entropy

Conditional differential entropy provides an intuitive measure for relatively ranking time-series complexity by quantifying uncertainty in future observations given past context. However, its direct computation for high-dimensional processes from unknown distributions is often intractable. This paper builds on the information theoretic prediction error bounds established by Fang et al. \cite{fang2019generic}, which demonstrate that the conditional differential entropy \textbf{$h(X_k \mid X_{k-1},...,X_{k-m})$} is upper bounded by a function of the determinant of the covariance matrix of next-step prediction errors for any next step prediction model. We add to this theoretical framework by further increasing this bound by leveraging Hadamard's inequality and the positive semi-definite property of covariance matrices. To see if these bounds can be used to rank the complexity of time series, we conducted two synthetic experiments: (1) controlled linear autoregressive processes with additive Gaussian noise, where we compare ordinary least squares prediction error entropy proxies to the true entropies of various additive noises, and (2) a complexity ranking task of bio-inspired synthetic audio data with unknown entropy, where neural network prediction errors are used to recover the known complexity ordering. This framework provides a computationally tractable method for time-series complexity ranking using prediction errors from next-step prediction models, that maintains a theoretical foundation in information theory.

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