Free Grammars and Languages

Floyd’s Operator Precedence languages (OPLs) were originally introduced to support deterministic parsing of programming languages [11]; then, interest in them decayed for several decades, probably due to the advent of more expressive grammars, such as LR ones which also allow for efficient deterministic parsing. Recently, however, we renewed our interest in this family of languages on the basis of two major properties thereof: 1) their “locality property”, i.e., the fact that partial strings can be parsed independently of the context in which they occur within a whole string; this enables more effective parallel and incremental parsing techniques than for other deterministic languages [3,4]; and 2) the fact that, to the best of our knowledge, OPLs are the largest family closed w.r.t. Boolean operations, concatenation, Kleene * and other classical operations [7]; furthermore they are recognized by a peculiar automata family [12] and are characterized in terms of classical monadic second order (MSO) logic [13]. This latter property entitles OPLs as a best candidate for extending the application of powerful verification techniques such as model-checking far beyond the original class of regular languages and even other recent families such as Visibly Pushdown Languages (VPLs) [2] which are strictly contained within OPLs too. In this paper we introduce (more precisely, resume) a subclass of OPLs, namely free languages (FrLs) which were defined in [9,8] with the main motivation of grammar inference. FrLs constitute a kind of algebra within the structure defined by any given operator precedence matrix (OPM) [8]. Besides briefly outlining a “specialized” version of automata explicitly tailored for FrLs, we offer a new logic characterization in terms of first-order logic, as opposed to the traditional but more complex one in terms of MSO logic. FrLs however, lose some closure properties and have some “distinguishing” limits in terms of generative power, which nevertheless covers various relevant and heterogeneous cases: thus, they better lend themselves to act as “basic skeleton” for describing the structural part of a language to be complemented either with a suitable intersection with “regular control” or with additional restrictions formalized in terms of first-order properties conjuncted with the original ones associated with a grammar (FrG), somewhat in the spirit of the classical Chomsky-Schutzenberger characterization of CF languages.