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

TL;DR

This paper proves that learning the smallest deterministic finite-state machine from the prefixes of a single string is NP-hard, highlighting the computational difficulty even in seemingly simple cases.

Contribution

It establishes NP-hardness for the problem of learning minimal automata from prefix-closed samples, including the special case of a single string, extending to Moore and Mealy machines.

Findings

NP-hard to approximate from all prefixes of binary strings

NP-hard decision problem even with prefixes of a single string

Results extend to Moore and Mealy machine 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.