Core Concepts

The paper proves the universality of regular realizability problems for several classes of finite relations on the set of non-negative integers, where the relations are described in a specific format. The universality is shown up to reductions using NP-oracles.

Abstract

The paper studies the regular realizability problems, which are a class of algorithmic problems parametrized by languages (called filters). An instance of the problem is a regular language described by a finite automaton, and the task is to verify the non-emptiness of the intersection between the regular language and the filter.
The main results are:
Universality of the regular realizability problems for several classes of finite relations on the set of non-negative integers. The relations are described in a format proposed by P. Wolf and H. Fernau, which allows for arbitrary order and repetitions in the list of relation elements.
The universality is shown up to reductions using NP-oracles. This corresponds to the previous results of P. Wolf and H. Fernau about the decidability of regular realizability problems for many graph-theoretic properties.
To achieve the universality results, the paper introduces a new model of Boolean function computation, which generalizes the size of decision trees and branching programs. This model is used to analyze the regular realizability problems for relation descriptions.
The paper also separately analyzes the case of relations that are invariant under bijections of the non-negative integers, as this is relevant for graph-theoretic properties that are invariant under isomorphisms.
The key technical tools used in the proofs are efficient asymptotically good codes and a framework for representing families of finite sets by directed graphs.

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by Alexander Ru... at **arxiv.org** 04-02-2024

Deeper Inquiries

The universality results for regular realizability problems have potential applications beyond the graph-theoretic domain. One significant application is in the field of computational biology. These results can be utilized in bioinformatics for analyzing biological sequences and structures. By encoding biological data into the format of finite relations on non-negative integers, researchers can apply the universality results to solve complex algorithmic problems related to DNA sequences, protein structures, and genetic interactions. This can lead to advancements in genomics, proteomics, and systems biology by providing efficient solutions to problems such as sequence alignment, structural prediction, and network analysis.

The new model of Boolean function computation introduced in the paper opens up avenues for further development and application in various areas of theoretical computer science. One potential direction is in the study of computational complexity theory. The model can be used to analyze the complexity of Boolean functions and subsets of finite sets, providing insights into the computational power of different types of algorithms. Additionally, the model can be extended to investigate the complexity of decision trees, branching programs, and other computational models, leading to a deeper understanding of the inherent complexity of computational problems.
Furthermore, the model can be applied in the design and analysis of cryptographic algorithms. By studying the complexity of Boolean functions and their representations as subsets of finite sets, researchers can develop secure encryption and decryption schemes based on the hardness of computing certain functions. This can contribute to the advancement of cryptography and data security in various applications.

There are connections between the techniques used in this paper and the study of descriptional complexity of formal languages. The encoding of finite relations on non-negative integers can be seen as a form of descriptional complexity, where the challenge lies in representing complex structures using simple encodings. The use of efficient asymptotically good codes to represent families of subsets by directed graphs is a technique that resonates with the concept of succinct representations in descriptional complexity.
Moreover, the analysis of regular realizability problems and the construction of automata over infinite alphabets can be related to the study of succinctness and expressiveness in formal language theory. By exploring the trade-offs between the complexity of automata and the representational power of languages, researchers can gain insights into the inherent limitations and capabilities of different computational models. This can lead to advancements in understanding the complexity of formal languages and the efficiency of algorithms for language recognition and manipulation.

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