Core Concepts

A Myhill-Nerode type theorem for higher-dimensional automata (HDAs) states that a language is regular if and only if it has finite prefix quotient.

Abstract

The paper establishes a Myhill-Nerode type theorem for higher-dimensional automata (HDAs), stating that a language is regular if and only if it has finite prefix quotient. HDAs extend standard automata with additional structure to distinguish between interleavings and concurrency.
The key highlights and insights are:
The authors introduce a stronger equivalence relation "strong equivalence" (≈L) on ipomsets, which takes concurrency of events into account, in contrast to the standard "weak equivalence" (∼L) used in the classical Myhill-Nerode theorem.
They construct an HDA MN(L) whose cells are ≈L-equivalence classes of ipomsets, and show that MN(L) recognizes the language L. If the prefix quotient of L is finite, then the essential part of MN(L) is also finite.
The authors show that there exist regular languages that cannot be recognized by deterministic HDAs, in contrast to the classical Myhill-Nerode theorem for finite automata. They develop a language-internal characterization of deterministic languages.
The paper also develops analogues of the Myhill-Nerode construction and of determinism for HDAs with interfaces (iHDAs), which allow for more principled handling of non-accessible parts.
Overall, the paper extends the classical Myhill-Nerode theorem to the higher-dimensional setting of HDAs, highlighting the additional challenges and insights that arise from the richer structure of HDAs compared to standard automata.

Stats

None.

Quotes

None.

Deeper Inquiries

In order to extend the Myhill-Nerode construction for HDAs to handle infinite-state systems, we need to consider the concept of infinite equivalence classes. In the context of regular languages and finite automata, the Myhill-Nerode theorem relies on the idea of partitioning the set of all strings into equivalence classes based on the behavior of the automaton. Each equivalence class represents a unique state in the minimal DFA that recognizes the language.
For infinite-state systems, the challenge lies in dealing with an infinite number of possible equivalence classes. One approach is to consider a coarser equivalence relation that merges certain states together to form larger equivalence classes. This can help in reducing the number of distinct states while still capturing the essential behavior of the system.
Another approach is to use techniques from formal language theory and automata theory that are specifically designed to handle infinite structures, such as ω-automata. These automata are capable of recognizing languages over infinite words and can be used to model systems with infinite behaviors.
Overall, extending the Myhill-Nerode construction to handle infinite-state systems requires a careful consideration of the unique characteristics of such systems and the development of appropriate theoretical frameworks to analyze their language recognition capabilities.

The non-determinizability result for HDAs has significant implications for the practical applications of these models in various fields such as concurrency theory, formal verification, and system modeling.
Complexity of Analysis: The non-determinizability of certain HDAs implies that there are regular languages that cannot be recognized by deterministic HDAs. This complexity in recognizing certain languages can make the analysis and verification of systems modeled using HDAs more challenging.
Modeling Concurrency: HDAs are particularly useful for modeling systems with concurrency and interleaving behaviors. The non-determinizability result highlights the inherent complexity of concurrency in system modeling and the need for sophisticated techniques to handle such scenarios.
Verification and Validation: The non-determinizability of HDAs underscores the importance of rigorous verification and validation techniques for systems modeled using these automata. It emphasizes the need for advanced tools and methodologies to ensure the correctness and reliability of concurrent systems.
Algorithm Design: The non-determinizability result can also impact the design of algorithms for analyzing HDAs. It may necessitate the development of specialized algorithms that can handle the non-deterministic nature of certain languages recognized by HDAs.
Overall, the non-determinizability result adds a layer of complexity to the practical applications of HDAs, requiring researchers and practitioners to develop advanced techniques and tools to effectively model, analyze, and verify systems with concurrent behaviors.

There are several other language-theoretic properties of HDAs that could be characterized in a similar way to the Myhill-Nerode theorem. Some of these properties include:
Equivalence Relations: Just like the Myhill-Nerode theorem characterizes regular languages based on equivalence classes of strings, similar theorems could be developed to characterize other language classes based on different equivalence relations in the context of HDAs.
Closure Properties: Properties related to closure under certain operations, such as union, intersection, complementation, and concatenation, could be explored for HDAs. Understanding how these operations interact with the language recognized by HDAs can provide insights into their expressive power.
Decidability and Complexity: Investigating the decidability and complexity of language-theoretic problems for HDAs, such as language inclusion, equivalence, and emptiness, can shed light on the computational properties of these automata and their languages.
Hierarchy of Language Classes: Exploring the hierarchy of language classes recognized by HDAs, similar to the Chomsky hierarchy for formal languages, can help classify the expressive power of different types of HDAs and understand their computational capabilities.
By delving into these language-theoretic properties, researchers can gain a deeper understanding of the capabilities and limitations of HDAs in modeling and analyzing complex systems with concurrency and interleaving behaviors.

0