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A Fixed Point Iteration Technique for Proving Correctness of Slicing for Probabilistic Programs

Torben Amtoft, Anindya Banerjee

TL;DR

The paper addresses proving correctness of a slicing transformation for probabilistic programs by using fixed-point iterations on continuous semantic functionals. It shows that the naive fixed-point approach can fail when loops cause misalignment between source and target iterations, and introduces an exclusion-set technique to restructure the source iteration so that it progresses in lockstep with the target. By defining refined iterates that start above the bottom on a carefully chosen exclusion set and verifying three key properties, the authors provide a general recipe to derive $R\;\phi\;\gamma$ from $R\;\phi'_k\;\gamma'_k$, with convergence to the desired fixpoints. The method is illustrated on a running probabilistic program, connects to prior work on slicing in probabilistic settings, and offers a broader perspective on when fixed-point proofs require nonstandard starting points. This approach potentially extends to other program transformations where loop-induced divergence in fixed-point iterations obstructs direct proofs.

Abstract

When proving the correctness of a method for slicing probabilistic programs, it was previously discovered by the authors that for a fixed point iteration to work one needs a non-standard starting point for the iteration. This paper presents and explores this technique in a general setting; it states the lemmas that must be established to use the technique to prove the correctness of a program transformation, and sketches how to apply the technique to slicing of probabilistic programs.

A Fixed Point Iteration Technique for Proving Correctness of Slicing for Probabilistic Programs

TL;DR

The paper addresses proving correctness of a slicing transformation for probabilistic programs by using fixed-point iterations on continuous semantic functionals. It shows that the naive fixed-point approach can fail when loops cause misalignment between source and target iterations, and introduces an exclusion-set technique to restructure the source iteration so that it progresses in lockstep with the target. By defining refined iterates that start above the bottom on a carefully chosen exclusion set and verifying three key properties, the authors provide a general recipe to derive from , with convergence to the desired fixpoints. The method is illustrated on a running probabilistic program, connects to prior work on slicing in probabilistic settings, and offers a broader perspective on when fixed-point proofs require nonstandard starting points. This approach potentially extends to other program transformations where loop-induced divergence in fixed-point iterations obstructs direct proofs.

Abstract

When proving the correctness of a method for slicing probabilistic programs, it was previously discovered by the authors that for a fixed point iteration to work one needs a non-standard starting point for the iteration. This paper presents and explores this technique in a general setting; it states the lemmas that must be established to use the technique to prove the correctness of a program transformation, and sketches how to apply the technique to slicing of probabilistic programs.

Paper Structure

This paper contains 13 sections, 1 theorem, 29 equations, 5 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Setting
  2. Goal
  3. Example Setting
  4. Example
  5. Slicing the Example
  6. Basic Approach
  7. Problem
  8. Refined Approach
  9. General Result
  10. An Application
  11. The Intuition Behind the Exclusion Set
  12. Perspectives
  13. Rectifying Lemma 6.13 in Amtoft+Banerjee:TOPLAS-2020

Key Result

theorem 1

With $X \subseteq \mathcal{U}$ given, for $k \geq 0$ define $\phi'_k = F^k(\phi'_0)$ with $\phi'_0$ given by (thus the starting point for the iteration may be above the standard starting point $\bot$), and similarly for $k \geq 0$ define $\gamma'_k = G^k(\gamma'_0)$ where $\gamma'_0$ is given by: Now assume that the below 3 properties all hold: Then we do have ${R}\;{\phi}\;{\gamma}$.

Figures (5)

  • Figure 1: The source code of our running example.
  • Figure 2: The source control-flow graph of our running example.
  • Figure 3: The target control-flow graph of our example program.
  • Figure 4: The target code of our example program.
  • Figure 5: The modified source control-flow graph of our example program.

Theorems & Definitions (2)

  • theorem 1
  • proof