Aperiodicity, Star-freeness, and First-order Logic Definability of Operator Precedence Languages

TL;DR

This paper extends the classic equivalence between star-freeness, aperiodicity, and first-order logic definability from regular languages to operator-precedence languages, a broad class of deterministic context-free languages used in programming language syntax.

Contribution

It introduces OP-expressions to characterize OPLs and proves aperiodic OPLs are first-order definable, generalizing fundamental formal language theory results.

Findings

Aperiodic OPLs are shown to be first-order definable.

Established the equivalence among star-freeness, aperiodicity, and first-order logic for OPLs.

Proposed a grammar transformation method to prove these properties.

Abstract

A classic result in formal language theory is the equivalence among non-counting, or aperiodic, regular languages, and languages defined through star-free regular expressions, or first-order logic. Past attempts to extend this result beyond the realm of regular languages have met with difficulties: for instance it is known that star-free tree languages may violate the non-counting property and there are aperiodic tree languages that cannot be defined through first-order logic. We extend such classic equivalence results to a significant family of deterministic context-free languages, the operator-precedence languages (OPL), which strictly includes the widely investigated visibly pushdown, alias input-driven, family and other structured context-free languages. The OP model originated in the '60s for defining programming languages and is still used by high performance compilers; its rich algebraic properties have been investigated initially in connection with grammar learning and recently completed with further closure properties and with monadic second order logic definition. We introduce an extension of regular expressions, the OP-expressions (OPE) which define the OPLs and, under the star-free hypothesis, define first-order definable and non-counting OPLs. Then, we prove, through a fairly articulated grammar transformation, that aperiodic OPLs are first-order definable. Thus, the classic equivalence of star-freeness, aperiodicity, and first-order definability is established for the large and powerful class of OPLs. We argue that the same approach can be exploited to obtain analogous results for visibly pushdown languages too.