A Proof Theory of (ω-)Context-Free Languages via Non-Wellfounded Proofs

Core Concepts

The authors develop a non-wellfounded proof system, µHKA, that is sound and complete for the equational theory of context-free languages. They then use this result to obtain an alternative proof of completeness for an infinitary axiomatization of the same theory. Finally, they extend the system to handle ω-context-free languages.

Abstract

The paper investigates the proof theory of regular expressions with fixed points, which can be used to represent (ω-)context-free grammars. The authors start with a hypersequential system HKA for regular expressions and define its extension µHKA by adding least fixed points. They prove the soundness and completeness of the non-wellfounded proofs in µHKA for the standard language model of context-free languages. The authors then apply proof-theoretic techniques to recover an infinitary axiomatization of the resulting equational theory, which is complete for inclusions of context-free languages. This provides an alternative proof of the completeness result of Grathwohl, Henglein, and Kozen. Finally, the authors extend the syntax by adding greatest fixed points, obtaining µνHKA, which can compute ω-context-free languages. They show the soundness and completeness of the corresponding system using a mixture of proof-theoretic and game-theoretic techniques.

Stats

The paper does not contain any key metrics or important figures to support the author's arguments.

Quotes

"We investigate the proof theory of regular expressions with fixed points, construed as a notation for (ω-)context-free grammars." "Establishing axiomatisations and proof systems for classes of formal languages has been a difficult challenge." "Inclusion of CFLs is Π0 1-complete, so any recursive (hence also cyclic) axiomatisation must necessarily be incomplete."

Key Insights Distilled From

A proof theory of (omega-)context-free languages, via non-wellfounded proofs

by Anupam Das,A... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16231.pdf

Deeper Inquiries

How can the proof-theoretic techniques developed in this paper be applied to other classes of formal languages beyond context-free and ω-context-free languages

The proof-theoretic techniques developed in the paper can be applied to other classes of formal languages beyond context-free and ω-context-free languages by adapting the system to accommodate the specific syntax and semantics of the target language class. The key lies in defining a suitable set of rules and axioms that capture the essence of the language class in question. For instance, for regular languages, the system can be modified to handle regular expressions and their corresponding language semantics. By extending the proof system with appropriate rules and constraints, it can be tailored to reason about regular languages and their properties. Similarly, for more complex language classes such as context-sensitive languages or recursively enumerable languages, the proof system can be further extended to handle the intricacies of these languages.

What are the potential applications of the non-wellfounded proof systems for (ω-)context-free languages in areas such as program verification or model checking

The non-wellfounded proof systems for (ω-)context-free languages have several potential applications in areas such as program verification and model checking. In program verification, these proof systems can be used to reason about the correctness and behavior of programs that manipulate strings or sequences of symbols. By establishing soundness and completeness results for these proof systems, one can verify properties of programs that involve context-free or ω-context-free languages, ensuring their correctness and reliability. Additionally, in model checking, these proof systems can be utilized to analyze and verify models of systems that exhibit complex language behaviors. By leveraging the non-wellfounded proof techniques, one can efficiently verify properties of models that involve context-free or ω-context-free languages, enhancing the reliability and accuracy of the model checking process.

Can the game-theoretic approach used to handle the greatest fixed points be extended to develop a unified framework for reasoning about various types of formal languages and their associated proof systems

The game-theoretic approach used to handle the greatest fixed points can be extended to develop a unified framework for reasoning about various types of formal languages and their associated proof systems. By generalizing the game-theoretic techniques to accommodate different types of fixed points and language classes, a unified framework can be established to handle a wide range of formal languages and proof systems. This unified framework can provide a systematic and coherent approach to reasoning about language properties, proof systems, and formal language classes, offering a versatile tool for researchers and practitioners in the field of formal language theory. By extending the game-theoretic approach to encompass diverse language classes and proof systems, a unified methodology can be developed to address the complexities and challenges of reasoning about formal languages in a comprehensive and structured manner.