Regular Expressions with Backreferences and Lookaheads Capture the Class of Languages Accepted by Nondeterministic Log-Space Turing Machines

Core Concepts

Regular expressions with backreferences and lookaheads (REWBLk) capture the class of languages accepted by nondeterministic log-space Turing machines (NLOG).

Abstract

The paper investigates the expressiveness of regular expressions with backreferences and lookaheads (REWBLk). The key findings are:

REWBLk can represent the NLOG-complete language for the reachability problem of directed acyclic graphs (DAGs), showing that REWB (regular expressions with backreferences) already contains an NLOG-complete language.

REWBLk is as expressive as NLOG, the class of languages accepted by nondeterministic log-space Turing machines. This is shown by translating REWBLk expressions to log-space nondeterministic Turing machines and vice versa.

The membership problem for REWBLk is PSPACE-complete. This is demonstrated by encoding the PSPACE-complete problem of checking the truth of quantified Boolean formulas (QBF) into the membership problem of REWBLk.

The paper also shows that REWB(+) and REWB(-), which are subclasses of REWBLk with only positive or negative lookaheads, can represent unary non-indexed languages. This implies that these subclasses are incomparable to the class of indexed languages.

The paper utilizes log-space nested-oracles nondeterministic Turing machines to naturally handle the nested lookaheads in REWBLk. It also leverages the Immerman-Szelepcsényi theorem to show that the class of languages accepted by these machines coincides with NLOG.

Overall, the paper provides a comprehensive understanding of the expressiveness and complexity of regular expressions with backreferences and lookaheads, establishing their tight connection to the NLOG complexity class.

Stats

The language Lreach = {s # x1 →y1# · · · #xn →yn# t : s, t, xi, yi ∈V∗, and there is a path from s to t} is NLOG-complete.
The language L1exp = {a2k : k ∈N} and L2exp = {a22k
: k ∈N} are represented by REWB(+) and REWB(-) expressions, respectively.

Quotes

"Backreferences and lookaheads are vital features to make classical regular expressions (REGEX) practical."
"REWBLk coincides with NLOG, the class of languages accepted by log-space nondeterministic Turing machines (NTMs)."
"The membership problem of REWBLk is PSPACE-complete."

Key Insights Distilled From

Regular Expressions with Backreferences and Lookaheads Capture NLOG

by Yuya Uezato at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17492.pdf

Deeper Inquiries

How can the results in this paper be leveraged to develop more efficient regular expression engines that fully support backreferences and lookaheads

The results presented in this paper can be instrumental in the development of more efficient regular expression engines that fully support backreferences and lookaheads. By establishing the connection between REWBLk and NLOG, it opens up avenues for optimizing the processing of complex regular expressions. Here are some ways these results can be leveraged:

Algorithm Optimization: Understanding that REWBLk coincides with NLOG allows for the development of algorithms that can efficiently process regular expressions with backreferences and lookaheads. By leveraging the properties of NLOG, algorithms can be designed to handle these features in a more streamlined and optimized manner.

Memory Management: With the insight that REWBLk is equivalent to NLOG, memory management techniques can be tailored to handle the complexities of backreferences and lookaheads more effectively. This can lead to more efficient memory utilization and faster processing of regular expressions.

Parallel Processing: The connection to NLOG can also guide the implementation of parallel processing techniques for regular expression evaluation. By distributing the workload across multiple processors or cores, the processing of complex regular expressions can be accelerated.

Error Handling: The theoretical foundations provided in the paper can inform the development of robust error-handling mechanisms in regular expression engines. By understanding the computational complexity of REWBLk, developers can anticipate potential bottlenecks and design error recovery strategies accordingly.

In essence, the insights gained from the research can be translated into practical strategies for enhancing the performance and capabilities of regular expression engines that support backreferences and lookaheads.

What other language classes or complexity-theoretic properties can be related to REWBLk beyond the NLOG and PSPACE connections shown in this work

Beyond the connections established with NLOG and PSPACE, there are several other language classes and complexity-theoretic properties that can be related to REWBLk. Some of these connections include:

Context-Sensitive Languages: REWBLk's increased expressiveness places it beyond the realm of context-free languages. Exploring its relationship with context-sensitive languages can provide further insights into the computational power of REWBLk.

Linear Bounded Automata (LBA): Investigating whether REWBLk can be simulated by LBAs, which have limited memory capabilities, can shed light on the space complexity of REWBLk and its computational boundaries.

Decidability and Undecidability: Analyzing the decidability and undecidability properties of problems related to REWBLk, such as the emptiness problem or equivalence problem, can provide a deeper understanding of the computational limits of REWBLk.

Hierarchy of Complexity Classes: Studying where REWBLk falls within the hierarchy of complexity classes, such as the polynomial hierarchy or the exponential hierarchy, can offer valuable insights into its computational complexity and expressive power.

By exploring these connections, researchers can gain a more comprehensive understanding of the capabilities and limitations of REWBLk in the broader landscape of formal languages and complexity theory.

Are there any practical applications or domains where the increased expressiveness of REWBLk compared to classical regular expressions would be particularly beneficial

The increased expressiveness of REWBLk compared to classical regular expressions can have significant practical applications in various domains where complex pattern matching and text processing are required. Some of the practical applications where the enhanced capabilities of REWBLk would be particularly beneficial include:

Natural Language Processing (NLP): In NLP tasks such as information extraction, sentiment analysis, and text summarization, the ability to define intricate patterns using backreferences and lookaheads can improve the accuracy and efficiency of text processing algorithms.

Data Mining and Information Retrieval: In data mining applications, the advanced features of REWBLk can be utilized to extract specific patterns or structures from large datasets. This can aid in tasks such as entity recognition, document classification, and data clustering.

Bioinformatics: In bioinformatics, where analyzing biological sequences and structures is crucial, the enhanced expressiveness of REWBLk can facilitate the identification of complex genetic patterns, regulatory elements, and protein sequences.

Cybersecurity: In cybersecurity applications, the ability to define sophisticated rules for detecting patterns in network traffic, identifying malicious code, and analyzing log files can be invaluable. REWBLk's capabilities can enhance the efficiency and accuracy of intrusion detection systems and security analytics.

By leveraging the advanced features of REWBLk in these domains, practitioners and researchers can tackle complex pattern matching challenges more effectively and extract valuable insights from textual data.