From Affine to Polynomial: Synthesizing Loops with Branches via Algebraic Geometry

2025-09-29

AI Summary

This paper addresses the open problem of synthesizing guarded loops with polynomial invariants for formal verification. Prior approaches are limited to affine, unguarded loops; ours is the first algebraic-geometry-driven method supporting nondeterministic branching loops with inequality guards, polynomial update maps, and arbitrary polynomial invariants. Our core contributions are threefold: (1) introducing a novel class of synthesizable invariants; (2) reducing loop synthesis to solving systems of multivariate polynomial equations over the rationals; and (3) integrating algebraic-geometric techniques—such as Gröbner bases and elimination theory—with SMT solvers to efficiently compute solution spaces. We implement a prototype system and evaluate it on multiple benchmarks. Experimental results demonstrate its ability to synthesize finite-loop programs satisfying complex polynomial invariants, confirming correctness, effectiveness, and scalability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Ensuring software correctness remains a fundamental challenge in formal program verification. One promising approach relies on finding polynomial invariants for loops. Polynomial invariants are properties of a program loop that hold before and after each iteration. Generating such invariants is a crucial task in loop analysis, but it is undecidable in the general case. Recently, an alternative approach to this problem has emerged, focusing on synthesizing loops from invariants. However, existing methods only synthesize affine loops without guard conditions from polynomial invariants. In this paper, we address a more general problem, allowing loops to have polynomial update maps with a given structure, inequations in the guard condition, and polynomial invariants of arbitrary form. We use algebraic geometry tools to design and implement an algorithm that computes a finite set of polynomial equations whose solutions correspond to all nondeterministic branching loops satisfying the given invariants. Furthermore, we introduce a new class of invariants for which we present a significantly more efficient algorithm. In other words, we reduce the problem of synthesizing loops to find solutions of multivariate polynomial systems with rational entries. This final step is handled in our software using an SMT solver.

Problem

• Synthesizing loops with polynomial updates and branching guards
• Generating polynomial invariants for arbitrary loop structures
• Reducing loop synthesis to solving multivariate polynomial systems

Innovation

• Synthesizes branching loops with polynomial invariants
• Uses algebraic geometry to solve polynomial equation systems
• Employs SMT solver for rational polynomial system solutions