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Automated Formal Analysis of Equivalence and Similarity of Probabilistic Program Outputs


Core Concepts
The authors present a new method for static equivalence and similarity refutation analyses of probabilistic program pairs. The method is fully automated, applicable to infinite-state probabilistic programs, and provides formal guarantees on the correctness of its results.
Abstract
The paper presents a new method for static analysis of equivalence and similarity of output distributions defined by pairs of probabilistic programs. The key aspects of the method are: Equivalence Refutation: The method searches for a function f over program outputs whose expected value differs between the two programs. It computes an upper expectation supermartingale (UESM) for f in the first program and a lower expectation submartingale (LESM) for f in the second program. The UESM, LESM, and function f together provide a formal certificate that the output distributions of the two programs are not equivalent. Similarity Refutation: The method extends the above approach to also refute similarity of output distributions, by requiring the function f to be 1-Lipschitz continuous. This allows the method to compute a lower bound on the Kantorovich distance between the output distributions. The authors present fully automated algorithms for both equivalence and similarity refutation, based on the above proof rules. The algorithms simultaneously compute the function f, the UESM, and the LESM via a constraint solving-based approach. The method is applicable to numerical probabilistic programs with polynomial arithmetic expressions and both discrete and continuous sampling. The experimental evaluation demonstrates the ability of the method to refute equivalence and compute lower bounds on Kantorovich distance for a variety of program pairs.
Stats
The output distributions of the two programs in Figure 1 differ by at least 999.5 in Kantorovich distance.
Quotes
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Deeper Inquiries

How can the method be extended to handle probabilistic programs with conditioning?

The method can be extended to handle probabilistic programs with conditioning by incorporating the conditioning aspect into the analysis. This would involve considering the impact of conditioning on the output distributions of the programs. Specifically, the analysis would need to account for how conditioning affects the probability distributions and how it influences the equivalence or similarity between the output distributions of the programs. By incorporating conditioning into the analysis, the method can provide a more comprehensive assessment of the relational properties of probabilistic programs that involve conditioning.

Can the lower bounds on Kantorovich distance computed by the method be tightened further?

While the lower bounds on Kantorovich distance computed by the method provide valuable insights into the difference between output distributions, there may be opportunities to tighten these bounds further. One approach to potentially tightening the lower bounds is to refine the analysis by considering additional factors or metrics that could provide more precise information about the similarity or dissimilarity of the output distributions. This could involve incorporating more complex metrics or refining the analysis process to capture more nuanced aspects of the output distributions. By enhancing the analysis methodology and considering more detailed information, it may be possible to achieve tighter lower bounds on the Kantorovich distance.

What are the implications of the presented approach for the design and compilation of probabilistic programming languages?

The presented approach for equivalence and similarity refutation of probabilistic programs has significant implications for the design and compilation of probabilistic programming languages. By offering a method for automated formal analysis of relational properties of probabilistic program pairs, the approach enhances the correctness and reliability of probabilistic programming languages. From a design perspective, the method provides a systematic way to verify equivalence and similarity of output distributions, which can inform the development of probabilistic programming languages with built-in verification mechanisms. This can lead to more robust and trustworthy probabilistic programming languages that are better equipped to handle complex probabilistic models and ensure the correctness of their output distributions. In terms of compilation, the approach enables bug detection in probabilistic program compilers by comparing the source code to its intermediate representation without program execution. This can help identify and rectify compilation errors that may lead to incorrect output distributions. By providing formal guarantees on the correctness of results, the approach enhances the reliability of compilation processes for probabilistic programming languages. Overall, the approach contributes to the advancement of probabilistic programming languages by offering a method for automated verification and analysis of relational properties.
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