Eduardo Skapinakis | Searcharxiv

arXiv2026

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations

This paper presents a high-level report on an ongoing project aiming to leverage implicit approaches based on discrete ordinary differential equations (ODEs) to study multiple complexity classes, even beyond small circuit and polynomial-time classes. Stimulated by recent ODE-based characterizations of polynomial-time functions (FP) and classes over the reals, the research project outlined here pushes this investigation further into counting and alternation. Specifically, we present a uniform framework, built upon a single base algebra and a unified family of schemas, where complexity levels, such as those of the polynomial and counting hierarchies, are captured simply by the nesting depth of ODE operators. Crucially, our approach starts from a base class much weaker than FP, thus strengthening existing recursion-theoretic treatments and establishing a natural connection to descriptive complexity. Moreover, by isolating three elementary schemas, our framework makes the computational content of linearity restrictions completely transparent while extending ODE-based implicit complexity to previously unaddressed counting classes, such as oplusP. More generally, this work establishes a clear bridge between differentiation and counting, offering a fresh perspective on the relationships between different complexity classes, which remains the object of ongoing and future research.

cs.CC↗

arXiv2026

Topological Criteria for Hypothesis Testing with Finite-Precision Measurements

We establish topological necessary and sufficient conditions under which a pair of statistical hypotheses can be consistently distinguished when i.i.d. observations are recorded only to finite precision. To accommodate finite-precision data, we introduce finite-precision tests: tests whose decision regions are open in the sample-space topology. We first show that, both for classical and finite-precision tests, the existence of such tests with finite-sample error control, asymptotic error control, or uniform convergence of the errors are all equivalent. A pair of null- and alternative hypotheses $H_0$ and $H_1$ admits a consistent finite-precision test if and only if both are $F_σ$ in the weak topology on the space of probability measures $W := H_0\cup H_1$. The hypotheses admit uniform error control under $H_i$ if and only if $H_i$ is closed in $W$, and admit uniformly consistent testing with bounded precision under metric separation of $H_0$ and $H_1$. These criteria imply that, without regularity assumptions, conditional independence is not consistently testable from finite-precision data when the conditioning space has no isolated points - strengthening existing impossibility results to Polish sample spaces and showing that even pointwise consistency cannot be obtained. We introduce an equicontinuity assumption on the family of conditional distributions under which we recover consistent finite-precision testability of conditional independence with uniform error control under the null, provided sample spaces are Polish and the conditioning space is locally compact. The equicontinuity assumption is itself a finite-precision-testable hypothesis, so the resulting test for conditional independence is, in a precise sense, assumption-free.

math.ST↗