[Paper Review] The separation problem for regular languages by piecewise testable languages

Lorijn van Rooijen, Marc Zeitoun|arXiv (Cornell University)|Mar 8, 2013

semigroups and automata theory13 references7 citations

TL;DR

This paper presents a polynomial-time algorithm to determine whether two regular languages can be separated by a piecewise testable language (PT-language), a class of languages definable by forbidden subwords. The method uses a refined automaton construction over the power set of the original alphabet to represent topological closures, enabling efficient intersection checks and yielding a direct, efficient solution to the separation problem for PT-languages.

ABSTRACT

Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. We present and discuss the separation problem for regular languages. We then give a direct polynomial time algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a $BΣ1(

Motivation & Objective

• To solve the separation problem for regular languages using piecewise testable languages as separators.
• To provide a direct, efficient algorithm that avoids reliance on complex profinite semigroup theory.
• To establish a polynomial-time decision procedure for PT-separability of two regular languages.
• To refine and make constructive an algebraic argument from Almeida and Zeitoun using automaton-based closure representations.
• To improve upon prior complexity bounds by achieving polynomial time in both the number of states and the alphabet size, under certain conditions.

Proposed method

• Construct an extended automaton over the alphabet $2^A$ (power set of the original alphabet) to represent the topological closure of a regular language.
• Use the closure automaton to test whether the closures of two regular languages intersect, which determines separability.
• Leverage the fact that emptiness and intersection of closure automata can be checked in NLOGSPACE, enabling polynomial-time complexity.
• Apply a refined version of Almeida's algebraic argument to show that non-separability implies a common $(\vec{u}, \vec{B})$-path in both automata.
• Define and use the concept of $(\vec{u}, \vec{B})$-adequate sequences to extract structural properties of the languages.
• Prove that if both languages contain $(\vec{u}, \vec{B})$-adequate sequences, then both automata admit a $(\vec{u}, \vec{B})$-path, implying non-separability.

Experimental results

Research questions

• RQ1 Can we decide in polynomial time whether two regular languages are separable by a piecewise testable language?
• RQ2 What is the structural condition under which two regular languages cannot be separated by a piecewise testable language?
• RQ3 How can the topological closure of a regular language be effectively represented for algorithmic purposes?
• RQ4 Is there a direct, constructive algorithm for PT-separability that avoids reliance on profinite semigroup theory?
• RQ5 What is the precise relationship between the existence of a common $(\vec{u}, \vec{B})$-path and the non-separability of two languages by a PT-language?

Key findings

• The paper presents a polynomial-time algorithm to decide whether two regular languages are separable by a piecewise testable language.
• The algorithm is based on constructing closure automata over the power set of the input alphabet, which allows efficient intersection and emptiness checks.
• The construction works for both deterministic and nondeterministic finite automata, with time complexity polynomial in the number of states and exponential in the alphabet size.
• Non-separability is characterized by the existence of a common $(\vec{u}, \vec{B})$-path in the automata of both languages.
• The proof establishes that if both languages contain $(\vec{u}, \vec{B})$-adequate sequences, then both automata must contain a $(\vec{u}, \vec{B})$-path.
• The method provides a constructive and direct refinement of an algebraic argument from profinite semigroup theory, making it more accessible and efficient for algorithmic use.