Author:
Mandrioli Dino,Pradella Matteo,Reghizzi Stefano Crespi
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.
Publisher
Centre pour la Communication Scientifique Directe (CCSD)
Subject
General Computer Science,Theoretical Computer Science
Cited by
2 articles.
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