Polar: An Algebraic Analyzer for (Probabilistic) Loops

TL;DR

Polar is an algebraic framework that automates the analysis of classical and probabilistic loops by solving algebraic recurrences, enabling precise invariant inference and sensitivity analysis.

Contribution

It introduces a comprehensive algebraic approach for analyzing probabilistic loops, including exact recurrence solving and invariant inference, with soundness and completeness within certain restrictions.

Findings

Analyzes probabilistic loops with if-statements and polynomial arithmetic.

Computes exact closed-forms of higher-order moments.

Provides sound incomplete techniques for moments computation.

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.