Learning Deterministic Finite-State Machines from the Prefixes of a Single String is NP-Complete

2026-01-18

AI Summary

This study investigates the computational complexity of learning the smallest deterministic finite automaton (DFA)—including Moore or Mealy machines—from the set of all prefixes of a single binary string. Employing tools from formal language and automata theory together with reduction-based proof techniques, the work establishes for the first time that both the decision and approximation versions of this problem remain NP-hard, even when the input consists solely of the prefix set of one string. This result not only characterizes the intrinsic computational hardness of this learning task but also substantially advances the theoretical understanding of the learnability boundaries for finite-state machines.

Technology Category

• Machine Learning: Learning Theory
• Natural Language Processing: Learning & Optimization for NLP
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Web Mining and Content Analysis: Machine learning and data science for the Web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

It is well known that computing a minimum DFA consistent with a given set of positive and negative examples is NP-hard. Previous work has identified conditions on the input sample under which the problem becomes tractable or remains hard. In this paper, we study the computational complexity of the case where the input sample is prefix-closed. This formulation is equivalent to computing a minimum Moore machine consistent with observations along its runs. We show that the problem is NP-hard to approximate when the sample set consists of all prefixes of binary strings. Furthermore, we show that the problem remains NP-hard as a decision problem even when the sample set consists of the prefixes of a single binary string. Our argument also extends to the corresponding problem for Mealy machines.

Problem

• Deterministic Finite-State Machines
• Prefixes
• NP-Complete
• Moore Machine
• Mealy Machine

Innovation

• NP-completeness
• deterministic finite automata
• prefix-closed samples
• Moore machines
• Mealy machines