The Descriptive Complexity Approach to LOGCFL

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 in first-order logic with linear order. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of arithmetical predicates for plus and times (equivalently, 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 reductions without arithmetic. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with predicates for plus and times than without. Considering a particular groupoidal quantifier, we prove that first-order logic with the “majority of pairs” quantifier 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.