非周期性、无星性与算子优先语言的一阶逻辑可定义性
Aperiodicity, Star-freeness, and First-order Logic Definability of Operator Precedence Languages
AI总结：
本文将非周期性、无星性和一阶可定义性的经典等价关系扩展至算子优先语言，通过引入OP表达式和文法变换完成证明，并指出该方法同样适用于可见下推语言。
摘要：
形式语言理论中的一个经典结果是：非计数（或非周期）正则语言、通过无星正则表达式定义的语言以及一阶逻辑定义的语言三者之间是等价的。过去尝试将这一结果扩展到正则语言范围之外时遇到了困难：例如，已知无星树语言可能违反非计数性质，并且存在无法通过一阶逻辑定义的非周期树语言。我们将此类经典等价结果扩展到确定性上下文无关语言的一个重要家族，即算子优先语言（OPL），该家族严格包含被广泛研究的可见下推（又名输入驱动）家族及其他结构化上下文无关语言。OP模型起源于20世纪60年代，用于定义编程语言，至今仍被高性能编译器使用；其丰富的代数性质最初在与文法学习的关联中被研究，最近又通过进一步的闭包性质和一元二阶逻辑定义得以完善。我们引入了正则表达式的扩展，即定义OPL的OP表达式（OPE），并在无星假设下定义了一阶可定义和非计数的OPL。然后，我们通过一种相当复杂的文法变换证明，非周期OPL是一阶可定义的。因此，对于庞大而强大的OPL类，确立了无星性、非周期性和一阶可定义性的经典等价性。我们认为，同样方法也可用于获取可见下推语言的类似结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
