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.

翻译：概率小组的语义学是逻辑分析概率依赖性的逻辑框架。 我们的焦点是概率包容逻辑及其延伸的共性、复杂性和直观性。 我们发现存在二阶逻辑的自然碎片,其中含有精确反映概率包容逻辑的直观性的累加实际算术。 我们还将这些形式主义与线性编程联系起来,从而获得逻辑方面的PTIME数据复杂性。 此外,关于有限结构,我们显示,带有累加实际算术的完全存在二阶逻辑只能表达NP特性。 最后,我们提出了原子一级概率包容逻辑的健全和完整的共和性。