Abstract
Operator Precedence Grammars (OPGs) define a deterministic class of context-free languages, which extend input-driven languages and still enjoy many properties: they are closed w.r.t. Boolean operations, concatenation and Kleene star; the emptiness problem is decidable; they are recognized by a suitable model of pushdown automaton; they can be characterized in terms of a monadic second-order logic. Also, they admit efficient parallel parsing.
In this paper we introduce a subclass of OPGs, namely Free Grammars (FrGs); we prove some of its basic properties, and that, for each such grammar G, a first-order logic formula \(\psi \) can effectively be built so that L(G) is the set of all and only strings satisfying \(\psi \).
FrGs were originally introduced for grammatical inference of programming languages. Our result can naturally boost their applicability; to this end, a tool is made freely available for the semiautomatic construction of FrGs.
Work partially supported by MIUR project PRIN 2010LYA9RH-006.
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Notes
- 1.
This less usual but equivalent definition of axioms as a set has been adopted for parenthesis languages [19] and other input-driven languages; we chose it for this paper to simplify some notations and constructions.
- 2.
An example language that cannot be generated with an \(\doteq \)-acyclic OPM is the following: \(\mathcal {L} = \{a^n {(b c)}^n \mid n\ge 0\} \cup \{b^n {(c a)}^n \mid n\ge 0\} \cup \{c^n (a b)^n \mid n\ge 0\} \).
- 3.
The grammars presented here have been produced by the Flup tool (the whole package, which includes various utilities for the general class of OPLs, is available at [1]). In the future we plan to couple Flup with an additional tool that minimizes the original grammar by applying the classical procedure introduced in [19].
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Lonati, V., Mandrioli, D., Panella, F., Pradella, M. (2015). First-Order Logic Definability of Free Languages. In: Beklemishev, L., Musatov, D. (eds) Computer Science -- Theory and Applications. CSR 2015. Lecture Notes in Computer Science(), vol 9139. Springer, Cham. https://doi.org/10.1007/978-3-319-20297-6_20
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