A Compositional Atlas for Algebraic Circuits
Benjie Wang, Denis Deratani Mauá, Guy Van den Broeck, YooJung Choi
TL;DR
This work presents a unifying, algebraic framework for tractable inference on circuits over arbitrary semirings by decomposing queries into Aggregation ($\oplus$), Product ($\otimes$), and Elementwise Mapping. It introduces key circuit properties—$\bm{X}$-determinism, $\bm{X}$-compatibility, and $\bm{X}$-support-compatibility—that ensure tractability is preserved under composition, and provides general tractability results applicable to diverse problems such as marginal MAP, probabilistic logic programming, and causal inference. Through case studies on Algebraic Model Counting (AMC) and causal backdoor/frontdoor adjustments, the framework yields new tractability conditions and tighter complexity bounds, unifying prior results and enabling analysis of novel queries. The approach offers a blueprint for designing knowledge-compiled representations with the structural properties needed for efficient, exact inference across a broad family of semirings, with implications for learning and program synthesis in probabilistic and causal settings.
Abstract
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.