Beyond Affine Loops: A Geometric Approach to Program Synthesis

TL;DR

This paper introduces a geometric method using algebraic geometry and SMT solvers to synthesize complex polynomial loops with guards and invariants, extending beyond affine loops.

Contribution

It presents a novel algorithm that synthesizes loops with polynomial updates and guards from polynomial invariants, generalizing previous affine loop synthesis methods.

Findings

Successfully synthesizes loops with polynomial update maps and guards.

Reduces loop synthesis to solving polynomial systems with an SMT solver.

Extends the scope of loop synthesis beyond affine loops.

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 polynomial invariants is a crucial task for loops, but it is an undecidable problem 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. In this paper, we use algebraic geometry tools to design and implement an algorithm that computes a finite set of polynomial equations whose solutions correspond to all loops satisfying the given polynomial invariants. In other words, we reduce the problem of synthesizing loops to finding solutions of polynomial systems within a specified subset of the complex numbers. The latter is handled in our software using an SMT solver.