[Paper Review] Learning Pairwise Disjoint Simple Languages from Positive Examples

Alexis Linard, Rick Smetsers+4 more|arXiv (Cornell University)|Jun 6, 2017

Algorithms and Data Compression6 references3 citations

TL;DR

This paper proposes two compression-based clustering methods to learn pairwise disjoint simple regular languages from positive-only examples, leveraging pumping lemma decompositions and tandem repeats to group strings by shared structural patterns. The approach successfully identifies distinct print-job patterns in industrial printer data, demonstrating practical applicability for language discovery in real-world settings.

ABSTRACT

A classical problem in grammatical inference is to identify a deterministic finite automaton (DFA) from a set of positive and negative examples. In this paper, we address the related - yet seemingly novel - problem of identifying a set of DFAs from examples that belong to different unknown simple regular languages. We propose two methods based on compression for clustering the observed positive examples. We apply our methods to a set of print jobs submitted to large industrial printers.

Motivation & Objective

• To address the challenge of learning multiple disjoint regular languages when only positive examples are available, without negative examples.
• To discover underlying structural patterns in strings that belong to different simple regular languages, such as repetition patterns or tandem repeats.
• To develop clustering methods that group strings based on shared pumping decompositions, enabling language inference without prior labeling.
• To apply the method to real industrial data—print jobs from large-scale printers—where each job belongs to a distinct, simple language category.
• To validate the approach on a dataset of 23,000+ print jobs, showing effective recovery of language patterns like (ab)+, (abc)+, and a(bc)+a.

Proposed method

• Use the pumping lemma to identify possible decompositions of strings into uv^i w, where v is a repeatable infix that can be pumped.
• Extract tandem repeats from strings as candidate repeating units, treating them as structural signatures for clustering.
• Apply compression-based similarity measures (e.g., using code word length functions) to compare strings and group those with similar decompositions.
• Generalize decompositions hierarchically from uv^{≥2}w to uv^{+}w to uv^{*}w, enabling progressive clustering at different levels of abstraction.
• Use equivalence checking to validate that different representations of the same language (e.g., a(baa)+b vs. ab(aab)+) describe the same language.
• Cluster strings based on shared pumping decompositions and tandem repeat structures, assigning each to a language family.

Experimental results

Research questions

• RQ1 Can we learn multiple pairwise disjoint regular languages from only positive examples, without negative examples?
• RQ2 Can pumping lemma-based decompositions and tandem repeats serve as reliable structural features for clustering strings into distinct languages?
• RQ3 How effective is compression-based similarity in grouping strings that belong to the same simple regular language?
• RQ4 Can this method recover meaningful, human-interpretable language patterns (e.g., (ab)+, a(bc)+a) from real industrial print job data?
• RQ5 What is the impact of language simplicity on the success of the proposed clustering approach?

Key findings

• The method successfully identified 12 distinct print-job patterns from a dataset of over 23,000 strings, including (ab)+, (abc)+, and a(bc)+a.
• Strings sharing the same tandem repeat or pumping decomposition were consistently grouped into the same language cluster, indicating structural coherence.
• The compression-based similarity measure effectively captured shared structural patterns, enabling accurate clustering without labeled data.
• The hierarchical generalization of decompositions from uv^{≥2}w to uv^{*}w allowed robust clustering across varying repetition levels.
• The approach demonstrated practical viability in an industrial setting, recovering interpretable language patterns from real-world print job sequences.
• Equivalence checking confirmed that different representations of the same language (e.g., a(baa)+b and ab(aab)+) were correctly identified as equivalent.