A Compositional Atlas for Algebraic Circuits

2024-12-07

Neural Information Processing Systems

AI Summary

This paper addresses the tractability of combinatorial circuit reasoning over algebraic structures. It introduces a unified algebraic framework that models heterogeneous queries—including marginal MAP inference, probabilistic answer set programming, and causal backdoor adjustment—as compositions of semiring-based aggregation, product, and pointwise mapping operators. Methodologically, it establishes general tractability criteria grounded in circuit structural properties—such as marginal determinism and operator compatibility—as well as constraints on pointwise mappings. This work provides the first systematic unification of tractability conditions across diverse combinatorial query classes. Beyond subsuming existing circuit reasoning problems, it yields several novel tractability results, substantially expanding the theoretical frontier and practical scope of tractable queries in algebraic circuit reasoning.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Compositional inference in algebraic circuits
• Tractability conditions for operators
• Unified analysis for diverse queries

Innovation

• Algebraic perspective for inference
• Semiring-based operator combination
• Circuit properties for tractability