Some Remarks on Counting Propositional Logic

TL;DR

This paper clarifies the expressive power of counting propositional logic's univariate fragment, providing methods to measure probabilities and linking it to stochastic experiments and dyadic distributions.

Contribution

It introduces an effective procedure for probability measurement and establishes a precise connection between counting logic and dyadic distribution events.

Findings

Effective procedure for measuring probabilities of counting formulas

Counting logic can simulate events with dyadic distributions

Clarifies the expressive power of the univariate fragment

Abstract

Counting propositional logic was recently introduced in relation to randomized computation and shown able to logically characterize the full counting hierarchy. In this paper we aim to clarify the intuitive meaning and expressive power of its univariate fragment. On the one hand, we provide an effective procedure to measure the probability of counting formulas. On the other, we make the connection between this logic and stochastic experiments explicit, proving that the counting language can simulate any (and only) event associated with dyadic distributions.