Range concatenation grammar (RCG) is a grammar formalism developed by Pierre Boullier [1] in 1998 as an attempt to characterize a number of phenomena of natural language, such as Chinese numbers and German word order scrambling, which are outside the bounds of the mildly context-sensitive languages.
[2] From a theoretical point of view, any language that can be parsed in polynomial time belongs to the subset of RCG called positive range concatenation grammars, and reciprocally.
[4] Though intended as a variant on Groenink's literal movement grammars (LMGs), RCGs treat the grammatical process more as a proof than as a production.
Whereas LMGs produce a terminal string from a start predicate, RCGs aim to reduce a start predicate (which predicates of a terminal string) to the empty string, which constitutes a proof of the terminal strings membership in the language.
A Positive Range Concatenation Grammar (PRCG) is a tuple
, where: A Negative Range Concatenation Grammar (NRCG) is defined like a PRCG, but with the addition that some predicates occurring in the right-hand side of a clause can have the form
A Range Concatenation Grammar is a positive or a negative one.
Although PRCGs are technically NRCGs, the terms are used to highlight the absence (PRCG) or presence (NRCG) of negative predicates.
Variables bind to ranges, not to arbitrary strings of nonterminals.
When instantiating a clause, where an argument consists of multiple elements from
The strings of predicates being rewritten represent constraints that the string being tested has to satisfy (if positive), or in the case of negative predicates not satisfy.
Like LMGs, RCG clauses have the general schema
consist of strings of terminal symbols and/or variable symbols, which pattern match against actual argument values like in LMG.
Adjacent variables constitute a family of matches against partitions, so that the argument
, with two variables, matches the literal string
These would give rise to three different instantiations of the clause containing that argument
Predicate terms come in two forms, positive (which produce the empty string on success), and negative (which produce the empty string on failure/if the positive term does not produce the empty string).
are terminal strings, if there is a rule
in the grammar that the predicate string matches, the predicate string is replaced by
, substituting for the matched variables in each
are terminal symbols, the predicate string
For the individual rewrite steps, when multiple alternative variable matches are possible, any rewrite which could lead the whole proof to succeed is considered.
, the proof is considered a success, regardless of how many other ways to fail exist.
RCGs are capable of recognizing the non-linear index language
as follows: Letting x, y, and z be variable symbols:
{\displaystyle {\begin{aligned}S(xyz)&\to A(x,y,z)\\A(ax,ay,az)&\to A(x,y,z)\\A(bx,by,bz)&\to A(x,y,z)\\A(\epsilon ,\epsilon ,\epsilon )&\to \epsilon \end{aligned}}}
Or, using the more correct dotted notation for ranges:
Every context-free grammar (CFG) can be converted into a range concatenation grammar: The intersection and union of two range concatenation languages are trivially range concatenation languages: Possibly negative range concatenation languages are also closed under set complement.
A consequence of the above is that it is undecidable whether a (positive) range concatenation language is nonempty, because it is undecidable whether the intersection of two context-free languages is nonempty.
Hence range concatenation grammars are not generative.