Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals

TL;DR

This paper explores the expressivity, complexity, and axiomatization of probabilistic inclusion logic and its relation to existential second-order logic with additive real arithmetic, revealing PTIME data complexity and NP limitations.

Contribution

It identifies a fragment of existential second-order logic matching probabilistic inclusion logic and provides a complete axiomatization for the latter.

Findings

Probabilistic inclusion logic captures exactly a fragment of existential second-order logic with additive real arithmetic.

The logic has PTIME data complexity.

Full existential second-order logic with additive real arithmetic expresses only NP properties.

Abstract

Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural fragment of existential second-order logic with additive real arithmetic that captures exactly the expressivity of probabilistic inclusion logic. We furthermore relate these formalisms to linear programming, and doing so obtain PTIME data complexity for the logics. Moreover, on finite structures, we show that the full existential second-order logic with additive real arithmetic can only express NP properties. Lastly, we present a sound and complete axiomatization for probabilistic inclusion logic at the atomic level.