Polar: An Algebraic Analyzer for (Probabilistic) Loops

2026-02-16

Principles of Verification

AI Summary

This work proposes an algebraic method for the automated analysis of loop programs featuring conditional branching, polynomial arithmetic, and common probability distributions. By transforming loop semantics into linear recurrence systems, the approach automatically computes closed-form expressions for higher-order moments of program variables, derives strongest polynomial invariants, and quantifies sensitivity to unknown parameters. Within a restricted yet practically relevant programming model, the analysis is both sound and complete; for more complex cases, it offers efficient, sound—but incomplete—approximation strategies. To the best of our knowledge, this is the first framework enabling fully automated algebraic reasoning for such probabilistic and classical loop programs.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Security and Privacy: Large-scale security measurements

Abstract

We present the Polar framework for fully automating the analysis of classical and probabilistic loops using algebraic reasoning. The central theme in Polar comes with handling algebraic recurrences that precisely capture the loop semantics. To this end, our work implements a variety of techniques to compute exact closed-forms of recurrences over higher-order moments of variables, infer invariants, and derive loop sensitivities with respect to unknown parameters. Polar can analyze probabilistic loops containing if-statements, polynomial arithmetic, and common probability distributions. By translating loop analysis into linear recurrence solving, Polar uses the derived closed-forms of recurrences to compute the strongest polynomial invariant or to infer parameter sensitivity. Polar is both sound and complete within well-defined programming model restrictions. Lifting any of these restrictions results in significant hardness limits of computation. To overcome computational burdens for the sake of efficiency, Polar also provides incomplete but sound techniques to compute moments of combinations of variables.

Problem

• probabilistic loops
• algebraic recurrences
• loop analysis
• polynomial invariants
• parameter sensitivity

Innovation

• algebraic recurrence
• probabilistic loops
• closed-form solution
• polynomial invariant
• parameter sensitivity