All about unambiguous polynomial closure

TL;DR

This paper investigates unambiguous polynomial closure in language classes, showing reductions in membership problems, equivalences with deterministic closures, and decidability of separation and covering for finite classes, along with logical characterizations.

Contribution

It establishes key properties of unambiguous polynomial closure, including problem reductions, equivalences, and decidability results, expanding understanding of language class closures.

Findings

Membership problem reduces to base class C

Unambiguous polynomial closure equals alternating deterministic closure

Separation and covering are decidable for finite classes

Abstract

We study a standard operator on classes of languages: unambiguous polynomial closure. We prove that for every class C of regular languages satisfying mild properties, the membership problem for its unambiguous polynomial closure UPol(C) reduces to the same problem for C. We also show that unambiguous polynomial closure coincides with alternating left and right deterministic closure. Moreover, we prove that if additionally C is finite, the separation and covering problems are decidable for UPol(C). Finally, we present an overview of the generic logical characterizations of the classes built using unambiguous polynomial closure.