Operator precedence and the visibly pushdown property

Abstract

Floyd's operator precedence grammars and languages (FG, FL) are a classical subclass of deterministic context-free (DCF) grammars and languages. We prove that several recently introduced language families motivated by the needs of model checking and of specifying XML-like languages are proper subsets of FL. The main cases considered include visibly pushdown languages (VPL) and balanced languages (BALAN), which are characterized by restricted precedence relations. FL have all the closure properties available for regular languages and generally viewed as necessary for application to model checking: reversal, prefixing and suffixing, concatenation, Kleene star, and boolean operations. All but the last results are new, and some require complex proofs, due to the necessary changes of syntax structure. Thus FL are the largest known subfamily of DCF having the same closure properties as VPL. FG, unlike VPL grammars, which are intended for abstract syntax modelling, are structurally adequate to specify real programming languages.

First-Order Logic Definability of Free Languages

“…Previous literature [8,17] assumed that all precedence matrices of OPLs are -cycle free, i.e., they do not contain sequences of relations a 1= a 2= . .…”

“…The FrG G depicted in Figure 1 with its OPM generates unparenthesized arithmetic expressions with the usual precedences of × w.r.t. +, which instead cannot be expressed as a VPL [8]. This grammar is obtained from the maxgrammar associated with the OPM by taking only those nonterminals that have letter n in both left and right sets.…”

“…Although they are still in use in this peculiar application field, thanks to their simplicity and the efficiency of their parsers [13], their theoretical investigation has been interrupted for a long time and only recently we resumed it in a long term research plan [8]. This led to discover many important properties of this class of languages which can be exploited in different modern applications.…”

Locally Chain-Parsable Languages

“…It suffices a little thought to see that IDL have the local parsability property, a fact also stemming from the fact that IDL are included in OPL [6]. Yet, the rigid alphabetic 3-partition severely reduces their generative capacity.…”

“…Yet, the rigid alphabetic 3-partition severely reduces their generative capacity. If we allow the parser decision whether to push, pop, or stay, to be based on a pair of adjacent terminal characters (more precisely on the precedence relation , ,= between them), instead of just one as in the IDL, we obtain the OPL family, which has essentially the same closure and decidability properties [6,9]. Loosely speaking, we may say that the input that drives the automaton for OPL is a terminal string of length two.…”