Compositionality of the Kantorovich Lifting and an Application to Up-To Techniques

Core Concepts

The Kantorovich lifting of functors can be made compositional under certain conditions, which is an essential ingredient for adopting up-to techniques to the setting of quantale-valued behavioural distances.

Abstract

The paper focuses on the Kantorovich lifting of functors, which generalizes the traditional notions of behavioural equivalence to a quantitative setting. The main contributions are:

Compositionality results for the Kantorovich lifting, showing that the lifting of a composed functor coincides with the composition of the liftings, under certain conditions.
Description of how to lift distributive laws in the case where one of the two functors is polynomial. These results are essential for adopting up-to techniques to the case of quantale-valued behavioural distances.
Up-to techniques are a coinductive technique for efficiently showing lower bounds for behavioural distances. The authors show how the compositionality results can be used to enable up-to techniques for Kantorovich liftings, complementing previous work on Wasserstein liftings.
The authors apply their technique to several examples, such as trace metrics for probabilistic automata and trace semantics for systems with exceptions, demonstrating how up-to techniques can help reduce the size or even make finite the witnesses yielding upper bounds.

Stats

There are no key metrics or important figures used to support the author's key logics.

Quotes

"Behavioural distances of transition systems modelled as coalgebras for endofunctors generalize the traditional notions of behavioural equivalence to a quantitative setting, in which states are equipped with a measure of how (dis)similar they are."
"Up-to techniques are a well-known coinductive technique for efficiently showing lower bounds for behavioural distances."

Key Insights Distilled From

Behavioural Metrics: Compositionality of the Kantorovich Lifting and an Application to Up-To Techniques

by Keri... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19632.pdf

Deeper Inquiries

How can the compositionality results be extended to a broader class of functors beyond polynomial functors

The compositionality results presented in the paper for polynomial functors can be extended to a broader class of functors by considering more general distributive laws and evaluation maps. By carefully defining suitable evaluation functions for a wider range of functors and ensuring compatibility with the monad structure, it is possible to establish compositionality for a larger class of functors. This extension would involve constructing appropriate EM-laws and ensuring that they lift correctly to the category of V-Graphs. By following a systematic approach similar to the one outlined for polynomial functors, the compositionality results can be generalized to encompass a wider variety of functor types.

What are the limitations of the up-to techniques presented in the paper, and how could they be further improved or generalized

The up-to techniques presented in the paper have certain limitations that could be addressed for further improvement and generalization. One limitation is the reliance on specific evaluation maps and distributive laws, which may not always be straightforward to define for complex systems or non-standard functors. To overcome this limitation, a more systematic method for constructing evaluation maps and ensuring their compatibility with the monad structure could be developed. Additionally, the scalability of the up-to techniques for larger state spaces or more intricate systems could be a challenge, and optimizing the computational efficiency of these techniques would be beneficial. Generalizing the up-to techniques to handle a broader range of behavioural distances and coalgebraic structures would also enhance their applicability in various scenarios.

What are the potential applications of the Kantorovich lifting and up-to techniques in other areas of computer science beyond the examples provided

The Kantorovich lifting and up-to techniques have potential applications in various areas of computer science beyond the examples provided in the paper. One application could be in the analysis of complex systems such as distributed algorithms, network protocols, or machine learning models. By utilizing behavioural distances and up-to techniques, researchers can efficiently compare system behaviors, identify discrepancies, and verify correctness properties. These techniques could also be valuable in software testing, anomaly detection, and optimization of system performance. Furthermore, in the field of artificial intelligence, the Kantorovich lifting and up-to techniques could be utilized for measuring similarity between data distributions, evaluating model outputs, and enhancing the interpretability of machine learning algorithms. Overall, the versatility and effectiveness of these techniques make them applicable in a wide range of computer science domains for behavioral analysis and verification purposes.