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

Abstract

Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the 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 secondorder logic with additive real arithmetic can only express NP properties.

The Implication Problem for Functional Dependencies and Variants of Marginal Distribution Equivalences

“…As for the second question, the implication problems for probabilistic independence and marginal identity have already been studied separately: the problem for probabilistic independence has been axiomatized in [12], and the problem for marginal identity (over finite probability distributions) has recently been axiomatized in [17]. Obtaining an axiomatization for both of these dependencies together is appealing, given how commonly IID assumption is used in probability theory and statistics.…”

“…The implication problem for probabilistic independence has been axiomatized over 30 years ago in [12]. More recently, the implication problem for marginal identity (over finite probability distributions) was axiomatized in [17]. The study of joint implication problems for different probabilistic and relational atoms have potential for practical applications because various probabilistic dependency notions appear in many situations.…”

“…For example, consider marginal identity together with probabilistic independence. The chase algorithm that was used for proving the completeness of the axiomatization of marginal identity in [17] uses inclusion dependencies that contain index variables for counting multiplicities of certain tuples. There does not seem to be a simple way of extending this procedure to also include probabilistic independencies.…”