The descriptive complexity approach to LOGCFL

TL;DR

This paper explores the descriptive complexity of LOGCFL by analyzing subclasses defined by groupoidal quantifiers, revealing new insights into their expressive power and relationships with other complexity classes.

Contribution

It extends the theory relating monoidal quantifiers to NC1, characterizes LOGCFL subclasses, and introduces aperiodic nondeterministic finite automata as a technical tool.

Findings

No single unary groupoidal quantifier with FO captures all context-free languages.

A variant of Greibach's language is LOGCFL-complete under certain projections.

FO with unary groupoidal quantifiers is more expressive with the BIT predicate.

Abstract

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.