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Beyond Affine Loops: A Geometric Approach to Program Synthesis

Erdenebayar Bayarmagnai, Fatemeh Mohammadi, Rémi Prébet

TL;DR

The paper tackles synthesizing polynomial loops from given polynomial invariants by building a geometric pipeline that translates loop synthesis into solving a finite polynomial system. It introduces a structured representation $\mathscr{C}(\mathbf{a},h,\mathbf{f};\bm{g})$ whose vanishing describes all admissible loop updates that satisfy the invariants, and provides Algorithm GenerateLoops to derive the defining polynomials. The authors explore solving strategies—structure-exploitation via irreducible decompositions, numerical homotopy methods, and SMT-based search for rational/integer solutions—demonstrating practicality with a Macaulay2 prototype and Z3 integration. This approach significantly broadens loop synthesis beyond affine updates, enabling polynomial updates with guards and suggesting pathways for handling more complex control flow and invariants in the future.

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.

Beyond Affine Loops: A Geometric Approach to Program Synthesis

TL;DR

The paper tackles synthesizing polynomial loops from given polynomial invariants by building a geometric pipeline that translates loop synthesis into solving a finite polynomial system. It introduces a structured representation whose vanishing describes all admissible loop updates that satisfy the invariants, and provides Algorithm GenerateLoops to derive the defining polynomials. The authors explore solving strategies—structure-exploitation via irreducible decompositions, numerical homotopy methods, and SMT-based search for rational/integer solutions—demonstrating practicality with a Macaulay2 prototype and Z3 integration. This approach significantly broadens loop synthesis beyond affine updates, enabling polynomial updates with guards and suggesting pathways for handling more complex control flow and invariants in the future.

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.
Paper Structure (11 sections, 4 theorems, 29 equations, 2 tables, 2 algorithms)

This paper contains 11 sections, 4 theorems, 29 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From loops to polynomial systems
  4. Polynomial system solving
  5. Exploit structure
  6. Numerical methods
  7. Satisfiability Modulo Theories (SMT) Solvers
  8. Implementation and Experiments
  9. Implementation details
  10. Experimental results
  11. Conclusion

Key Result

Proposition 2.2

Let $X\subseteq\mathbb{C}^n$ be an algebraic variety and consider a polynomial map $F : \mathbb{C}^n \longrightarrow \mathbb{C}^n$. We define for all $m \in \mathbb{N}$. Then, there exists $N\in \mathbb{N}$ such that $X_N=X_{N+1}$, and for any such index $X_N = S_{(F,X)}$.

Theorems & Definitions (17)

  • Example 1
  • Definition 2.1
  • Proposition 2.2
  • Example 2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • proof
  • ...and 7 more