Quantitative Axiomatisation of Shortest-Distinguishing-Word Distance for Regular Expressions

Alapfogalmak

This paper presents a sound and complete quantitative axiomatisation of the shortest-distinguishing-word distance between regular languages.

Kivonat

The paper focuses on providing a quantitative axiomatisation for reasoning about the behavioural distance of regular expressions. The key contributions are:

Instantiation of the abstract coalgebraic framework for behavioural distances to the case of deterministic automata, yielding the shortest-distinguishing-word distance.
Presentation of a quantitative equational theory REG that axiomatises the shortest-distinguishing-word distance between regular expressions.
Proof of soundness of the axiomatisation, showing that the quantitative algebra of regular expressions with the shortest-distinguishing-word distance is a model of REG.
Proof of completeness, demonstrating that the axiomatisation REG is complete with respect to the shortest-distinguishing-word distance on regular expressions. The completeness proof relies on the order-theoretic and Banach space structures carried by the sets of pseudometrics.
Observation that in the presence of the infinitary continuity rule, the fixpoint introduction rule from Salomaa's axiomatisation of language equivalence becomes derivable in REG.

Statisztikák

None.

Idézetek

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Főbb Kivonatok

A Complete Quantitative Axiomatisation of Behavioural Distance of Regular Expressions

by Wojc... : arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13352.pdf

Mélyebb kérdések

What other notions of behavioural distances for different types of transition systems could be axiomatised using a similar approach

In addition to the shortest-distinguishing-word distance for regular expressions, similar approaches could be used to axiomatize other notions of behavioral distances for various types of transition systems. For example, behavioral distances for probabilistic transition systems, timed automata, or even non-deterministic finite automata could be axiomatized using a similar quantitative equational logic framework. By defining appropriate lifting functions and pseudometrics tailored to the specific characteristics of these transition systems, one could establish sound and complete axiomatizations for their behavioral distances. The key lies in adapting the quantitative equational theories to capture the nuances of each type of transition system and defining the appropriate rules for reasoning about their behavioral distances.

How could the completeness proof be adapted to handle other types of behavioural distances beyond the shortest-distinguishing-word distance

To adapt the completeness proof for other types of behavioral distances beyond the shortest-distinguishing-word distance, one would need to tailor the proof to the specific properties and characteristics of the distance metric in question. The key would be to establish the necessary properties of the distance metric, such as continuity, non-expansiveness, and convergence, and then demonstrate how these properties can be leveraged in the completeness proof. By carefully analyzing the structure of the distance metric and its relationship to the transition system it represents, one could extend the completeness proof to cover a broader range of behavioral distances. This adaptation would involve modifying the rules of inference, the properties of the distance metric, and the structure of the transition system to align with the specific behavioral distance being considered.

What are the potential applications of quantitative axiomatisations of behavioural distances in areas such as program verification, model checking, or software engineering

The quantitative axiomatizations of behavioral distances have significant implications in various areas of computer science and software engineering.

Program Verification: By quantitatively reasoning about the behavioral distances between different program states or executions, one can enhance program verification techniques. This can help in identifying subtle differences in program behaviors, detecting errors, and ensuring the correctness of software systems.

Model Checking: Quantitative axiomatizations of behavioral distances can improve the efficiency and accuracy of model checking algorithms. By quantifying the dissimilarity between different system behaviors, model checking tools can better analyze system properties, verify correctness, and detect anomalies.

Software Engineering: In software engineering, quantitative axiomatizations of behavioral distances can aid in software testing, debugging, and maintenance. By quantifying the differences in system behaviors, developers can identify and resolve issues more effectively, leading to higher-quality software products.

Overall, the application of quantitative axiomatizations of behavioral distances can enhance the reliability, efficiency, and robustness of computational systems across various domains.