Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference queries become tractable, such as model counting and most probable configuration. Recent works have explored analyzing probabilistic and causal inference queries as compositions of basic operators to derive tractability conditions. In this paper, we take an algebraic perspective for compositional inference, and show that a large class of queries - including marginal MAP, probabilistic answer set programming inference, and causal backdoor adjustment - correspond to a combination of basic operators over semirings: aggregation, product, and elementwise mapping. Using this framework, we uncover simple and general sufficient conditions for tractable composition of these operators, in terms of circuit properties (e.g., marginal determinism, compatibility) and conditions on the elementwise mappings. Applying our analysis, we derive novel tractability conditions for many such compositional queries. Our results unify tractability conditions for existing problems on circuits, while providing a blueprint for analysing novel compositional inference queries.

翻译：基于和积结构的电路已成为一种普遍表示方法，用于紧凑编码从布尔函数到概率分布的知识。通过对此类电路结构施加约束，某些推理查询（如模型计数和最可能配置）变得可高效计算。近期研究探索将概率与因果推理查询分析为基本算子的组合，以推导可计算性条件。本文从代数视角研究组合推理，证明一大类查询——包括边缘最大后验概率、概率答案集编程推理及因果后门调整——对应于半环上基本算子（聚合、乘积及逐元映射）的组合。利用此框架，我们基于电路特性（如边缘确定性、兼容性）和逐元映射条件，揭示了这些算子可组合计算的简洁普适充分条件。应用我们的分析，我们为许多此类组合查询推导出新的可计算性条件。本成果统一了现有电路问题的可计算性条件，同时为分析新型组合推理查询提供了蓝图。