The Computational Complexity Of Learning Minimal Regular Expressions
Regular expressions are a powerful tool for matching patterns in textual data. However, designing an optimal regular expression to match a specific pattern can be challenging. This article explores the problem of automatically inducing minimal regular expressions from examples, known as regular expression induction.
The Challenge of Learning Regular Expressions
Manual coding of regular expressions requires significant domain expertise and trial-and-error. Automatically learning concise regular expressions from data would allow their powerful pattern matching capabilities to be leveraged more easily. However, regular expression induction poses considerable algorithmic challenges.
The space of possible regular expressions matching a dataset is often infinitely large. Searching this vast space to find the smallest, most general regular expression capable of matching provided examples is extraordinarily difficult.
Formal Definitions and Problem Statement
Formally, the regular expression induction problem takes as input a finite set of positive string examples S+ and negative examples S-, over an alphabet Σ. The objective is to infer a minimal regular expression R that matches all strings in S+ but none in S-. R must also generalize to match unseen positive string examples.
For complexity analysis, regular expression size is measured by the number of symbol occurrences. The induction process should favor smaller regular expressions based on Occam’s razor principle – the simplest explanation is most likely to generalize correctly.
Algorithms for Inferring Regular Expressions
Occam’s Razor Approach
One approach to finding small regular expressions is to systematically search the space of expressions ordered by size. This begins with the simplest possible regular expressions, incrementally adding complexity until the examples are matched. However, this search space grows extremely rapidly with longer input examples.
Using Grammatical Inference
More sophisticated algorithms frame regex induction as grammatical inference. By iteratively merging common sub-patterns, these methods infer formal grammars describing the structure of positive examples. The final inferred grammar can be converted into a regular expression.
Efficient implementations of grammatical inference use tree-based data structures and clever optimizations to avoid exploring gigantic search spaces. However, worst-case performance remains exponential for typical formulations.
Computational Complexity Analysis
Hardness Results
Regular expression induction has been shown to be NP-hard even for simple alphabet and dataset sizes. Intuitively, because an infinitely large language can be represented by small regexes, search spaces grow extremely rapidly. There is no known polynomial-time algorithm capable of solving all instances.
Efficiency of Algorithms
That said, efficient heuristics exist that perform well in practical cases. By carefully restricting permitted regular expression operators and maximizing reuse of subexpressions, some grammatical inference algorithms achieve reasonable worst-case guarantees. Empirically their typical case efficiency on real-world problems is good.
More expressive regex features like backreferences and lookaround assertions result in much higher inference complexity. As capabilities grow, so too do the inherent difficulties for induction algorithms.
Regular Expression Induction in Practice
Applications and Use Cases
Despite the fundamental computational challenges, regular expression induction tools have proved useful for tasks like reverse-engineering textual formats, extracting structured data from documents, and clustering/classification based on textual patterns.
Induced regular expressions can help construct robust parsers for legacy file formats without formal specifications. They also facilitate building data extractors for web sites or other unstructured data sources.
Current Implementations and Limitations
A number of regex induction systems have been built, including ReLIE, REGEXPERT, and infer.NET. Typical limitations include restriction to particular regex subsets, fragility when input examples contain noise, and inability to handle complex ambiguities.
For extremely large datasets, search spaces become intrinsically prohibitive. Approximation algorithms must be developed that sacrifice provable minimality or completeness for reasonable runtime on practical problems.
Open Problems and Future Work
Many opportunities remain for advancing the state of the art in regular expression induction. Developing faster and more robust algorithms, handling richer regex features, and tackling issues around scalability are active research areas.
Principled methods of effectively guiding the search through intractably large spaces could expand the practical viability. Integration of human domain knowledge also offers a promising way forward for the most challenging induction tasks currently beyond automation.
