innsikt - Algorithms and Data Structures - # Transformations of Muller Automata and Games

Optimal Transformations of Deterministic and History-Deterministic Muller Automata into Parity and Rabin Automata

Q: How can the insights from the alternating cycle decomposition be leveraged to develop efficient algorithms for minimizing history-deterministic automata beyond the coBüchi case

The insights from the alternating cycle decomposition (ACD) can be instrumental in developing efficient algorithms for minimizing history-deterministic automata beyond the coBüchi case. By leveraging the structural information provided by the ACD, algorithms can be designed to identify and exploit patterns in the alternating cycle decomposition of automata. This can lead to more targeted and optimized approaches for minimizing history-deterministic automata, especially for languages beyond the coBüchi case. The ACD offers a compact representation of the accepting and rejecting cycles of automata, capturing essential structural properties that can be utilized in the minimization process. By analyzing the alternating cycle decomposition, algorithms can identify redundancies, symmetries, and other structural characteristics that can be exploited to minimize history-deterministic automata efficiently. This can involve techniques such as identifying common substructures, optimizing state transitions, and streamlining the overall automaton design based on the insights provided by the ACD. Overall, the ACD serves as a valuable tool for understanding the structure of automata and can guide the development of algorithms that optimize the minimization of history-deterministic automata, extending beyond the coBüchi case to handle a broader range of languages and acceptance conditions.

Q: What are the implications of the optimality results presented in this work for the succinctness of history-deterministic automata compared to deterministic ones

The optimality results presented in the paper have significant implications for the succinctness of history-deterministic automata compared to deterministic ones. The findings suggest that history-deterministic automata can achieve minimality in terms of state complexity, even when compared to deterministic automata. This indicates that history-deterministic automata can be as succinct as deterministic ones, if not more so, in recognizing certain languages. The ACD-based transformations introduced in the paper offer a systematic and optimal way to convert automata while preserving determinism or history-determinism. These transformations ensure that the resulting automata are minimal in terms of state complexity, providing a clear advantage in terms of succinctness. By leveraging the ACD and the optimality results, it is possible to identify new classes of languages where history-deterministic automata are provably more succinct than their deterministic counterparts. In practical terms, the optimality results imply that history-deterministic automata can offer a more efficient and concise representation of certain language classes, making them a valuable tool in automata theory and language recognition. The ACD-based transformations not only ensure optimality but also provide insights into the structural properties of automata that can be leveraged for further optimizations and advancements in the field.

Q: Can the ACD-based transformations be used to identify new classes of languages where history-deterministic automata are provably more succinct

While the paper primarily focuses on transformations preserving determinism or history-determinism, there is potential to extend similar optimal transformations to the non-deterministic case. By leveraging the structural insights from the alternating cycle decomposition (ACD), it is possible to develop efficient algorithms for transforming non-deterministic automata into equivalent forms while preserving key properties and minimizing state complexity. The ACD provides fundamental information on the structure of automata, offering insights into the interplay between the underlying graph and the acceptance condition. By utilizing this information, algorithms can be designed to optimize the transformation of non-deterministic automata, ensuring that the resulting automata are minimal and efficient in recognizing the specified languages. By extending the ACD-based transformations to the non-deterministic case, it is possible to achieve similar optimality results and ensure that the transformed automata maintain essential properties while minimizing state complexity. This approach can lead to advancements in automata theory and language recognition, providing efficient solutions for transforming and optimizing non-deterministic automata in various applications and scenarios.

Grunnleggende konsepter

The paper presents optimal transformations of deterministic and history-deterministic Muller automata into equivalent deterministic parity automata and history-deterministic Rabin automata, while preserving the structural properties of the original automata.

Sammendrag

The paper studies transformations of automata and games using Muller conditions into equivalent ones using parity or Rabin conditions. The authors present two main transformations:

A transformation that turns a deterministic Muller automaton into an equivalent deterministic parity automaton. This transformation is shown to be optimal, in the sense that the resulting parity automaton is minimal amongst those that can be derived from the original automaton by duplication of states.
A transformation that provides an equivalent history-deterministic Rabin automaton from a Muller automaton. This Rabin automaton is also shown to be minimal amongst history-deterministic Rabin automata that can be derived from the original Muller automaton.

The authors introduce the notions of locally bijective morphisms and history-deterministic mappings to formally capture the correctness and optimality of these transformations.

The proposed transformations are based on a novel data structure called the alternating cycle decomposition (ACD), which extends and generalizes the Zielonka tree. The ACD provides a compact representation of the accepting and rejecting cycles of a Muller automaton, capturing the interplay between the structure of the underlying graph and the acceptance condition.

Beyond the transformations, the paper also provides several structural results about Muller transition systems using the ACD, including characterizations for relabelling automata with different acceptance conditions and a comprehensive study of a normal form for parity automata.

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Sitater

"The optimal number of colours needed by a parity automaton to recognise a language L reveals a fundamental piece of information about it, called its parity index."
"Parity languages are exactly Muller languages corresponding to families F ⊆2Γ+ of subsets of colours such that both F and its complement are closed under union."
"Solving parity games is both in NP and co-NP (more precisely, the problem is in UP ∩co-UP)."

Viktige innsikter hentet fra

From Muller to Parity and Rabin Automata: Optimal Transformations Preserving (History) Determinism

by Anto... klokken arxiv.org 04-23-2024

https://arxiv.org/pdf/2305.04323.pdf

From Muller to Parity and Rabin Automata: Optimal Transformations Preserving (History) Determinism

Dypere Spørsmål

How can the insights from the alternating cycle decomposition be leveraged to develop efficient algorithms for minimizing history-deterministic automata beyond the coBüchi case

The insights from the alternating cycle decomposition (ACD) can be instrumental in developing efficient algorithms for minimizing history-deterministic automata beyond the coBüchi case. By leveraging the structural information provided by the ACD, algorithms can be designed to identify and exploit patterns in the alternating cycle decomposition of automata. This can lead to more targeted and optimized approaches for minimizing history-deterministic automata, especially for languages beyond the coBüchi case.
The ACD offers a compact representation of the accepting and rejecting cycles of automata, capturing essential structural properties that can be utilized in the minimization process. By analyzing the alternating cycle decomposition, algorithms can identify redundancies, symmetries, and other structural characteristics that can be exploited to minimize history-deterministic automata efficiently. This can involve techniques such as identifying common substructures, optimizing state transitions, and streamlining the overall automaton design based on the insights provided by the ACD.
Overall, the ACD serves as a valuable tool for understanding the structure of automata and can guide the development of algorithms that optimize the minimization of history-deterministic automata, extending beyond the coBüchi case to handle a broader range of languages and acceptance conditions.

What are the implications of the optimality results presented in this work for the succinctness of history-deterministic automata compared to deterministic ones

The optimality results presented in the paper have significant implications for the succinctness of history-deterministic automata compared to deterministic ones. The findings suggest that history-deterministic automata can achieve minimality in terms of state complexity, even when compared to deterministic automata. This indicates that history-deterministic automata can be as succinct as deterministic ones, if not more so, in recognizing certain languages.
The ACD-based transformations introduced in the paper offer a systematic and optimal way to convert automata while preserving determinism or history-determinism. These transformations ensure that the resulting automata are minimal in terms of state complexity, providing a clear advantage in terms of succinctness. By leveraging the ACD and the optimality results, it is possible to identify new classes of languages where history-deterministic automata are provably more succinct than their deterministic counterparts.
In practical terms, the optimality results imply that history-deterministic automata can offer a more efficient and concise representation of certain language classes, making them a valuable tool in automata theory and language recognition. The ACD-based transformations not only ensure optimality but also provide insights into the structural properties of automata that can be leveraged for further optimizations and advancements in the field.

Can the ACD-based transformations be used to identify new classes of languages where history-deterministic automata are provably more succinct

While the paper primarily focuses on transformations preserving determinism or history-determinism, there is potential to extend similar optimal transformations to the non-deterministic case. By leveraging the structural insights from the alternating cycle decomposition (ACD), it is possible to develop efficient algorithms for transforming non-deterministic automata into equivalent forms while preserving key properties and minimizing state complexity.
The ACD provides fundamental information on the structure of automata, offering insights into the interplay between the underlying graph and the acceptance condition. By utilizing this information, algorithms can be designed to optimize the transformation of non-deterministic automata, ensuring that the resulting automata are minimal and efficient in recognizing the specified languages.
By extending the ACD-based transformations to the non-deterministic case, it is possible to achieve similar optimality results and ensure that the transformed automata maintain essential properties while minimizing state complexity. This approach can lead to advancements in automata theory and language recognition, providing efficient solutions for transforming and optimizing non-deterministic automata in various applications and scenarios.