Multiple Context-Free Grammars (MCFG) have been introduced by Seki et al. [266] is similar to Linear Contex-Free Rewriting Systems (LCFRS) intro- duced by Joshi’s students, Vijay-Shanker and Weir [289]. We give here a pre- sentation that is slightly different from the one proposed in the original article but which is slightly more intuitive.
A Multiple Context Free Grammar (MCFG) G is a tuple (Θ, Σ, R, S) where Θ is a ranked alphabet, Σ is a finite set of letters, R is a set of rules and S is an element of Θ(1). The rules in R are of the form
A(α1, . . . , αn) ← B1(x11, . . . , x 1 l1), . . . , Bp(x p 1, . . . , x p lp)
where A is in Θ(n), B
jis in Θ(lj), the xkj are pairwise distinct variables and the
αjare elements of (Σ∪X)∗with X = {xkj | k ∈ [p]∧j ∈ [lk]} and the restriction
that each xkj may have at most one occurrence4 in the string α1· · · αn. Note
that p may be equal to 0 in which case the right part of the rule is empty, in such a case we will write the rule by omitting the symbol ←.
An MCFG such as G defines judgments of the form `GA(s1, . . . , sn) where
A is in Θ(n)and the si belongs to Σ∗. Such a judgment is said to be derivable
when there is a rule A(α1, . . . , αn) ← B1(x11, . . . , x1l1), . . . , Bp(x
p 1, . . . , x
p lp) and
there are derivable judgments `GBk(w1k, . . . , wklk) for all k in [p] such that sj is
equal to αj where the possible occurrences of the xkj are replaced by wkj. The
language defined by G is the set {w ∈ Σ∗| S(w) is derivable}. An MCFG is said well-nested when all its rules:
A(α1, . . . , αn) ← B1(x11, . . . , x 1 l1), . . . , Bp(x p 1, . . . , x p lp)
verify the following properties (where X = {xk
j | k ∈ [p] ∧ j ∈ [lk]}): • for i ∈ [p], if j < li then α1. . . αn∈ (Σ ∪ X)∗xij(Σ ∪ X)∗x i j+1(Σ ∪ X)∗, • if i 6= i0, j < l i and j0 < li0, then α1. . . αn∈ (Σ ∪ X)x/ ij(Σ ∪ X)∗xi 0 j0(Σ ∪ X)∗xi j+1(Σ ∪ X)∗xi 0 j0+1(Σ ∪ X)∗.
This means that the variables of introduced in the right-hand side of the rule appear in the same order in its left hand-side and that furthermore, whenever, for some i0 different from i, xij00 occurs in between xji and xij+1 in α1. . . αn,
then for all j00 in [li0] the variable xi 0
j00 occurs in between xij and xij+1. The
rules that satisfy these conditions are called well-nested rules and the class of languages that can be defined with well-nested MCFG is called well-nested Multiple Context Free Languages and written MCFLwn.
Even though this restriction may seem intricate, it decreases the expres- sive power of MCFGs significantly and MCFLwn is a very natural class of
languages that, as we mentioned earlier, coincides with many formalisms, like non-duplicating IO and OI grammars (so that MCFLwn are included in in-
dexed languages [126]), second order ACGs of complexity 3, coupled context- free grammars [170]. Furthermore, MCFLwn satisfy a strong form of pumping
lemma [171], but there is a 3-MCFL that does not satisfy such a lemma [S6]. An MCFG G = (Θ, Σ, R, S) is a k-MCFG(r) when the maximal arity of the elements of Θ is less than k and when the maximal number of non-terminal in the right hand side of a rule in R is r. A k-MCFG, is an MCFG that is a k-MCFG(r) for some r and similarly a MCFG(r) is an MCFG that is a k-MCFG(r) for some k. It is known [266] that for each k, k-MCFLs, the languages definable by k-MCFGs, form substitution-closed full Abstract Family of Languages [132]. In particular, this implies that k-MCFLs form a class of
4If we allow more than one occurrence, we obtain Parallel Multiple Context Free Gram-
languages that is closed under rational transduction for every k. Furthermore k- MCFLs form a strictly increasing hierarchy of languages. The two-dimensional hierarchy of k-MCFL(r) has been studied in detail by Rambow and Satta [252, 253]. Their results are summarized by the following theorem.
Theorem 36
• 1-MCFL is equal to the class of context-free languages, • 1-MCFL(r) = 1-MCFL(r + 1) when r > 1,
• 1-MCFL(1) is equal to the class of linear context-free languages, • 2-MCFL(2) = 2-MCFL(3)
• if k > 2 or r > 2, then k-MCFL(r) ( k-MCFL(r + 1).
In particular, this theorem implies that, in general, given a k-MCFG, there is no k-MCFG(2) defining the same language. Interestingly this is different when we consider MCFGwn. We proved [S9] the following theorem (a slightly
stronger form of that theorem has been independently obtained in [138]). Theorem 37 k-MCFLwn = k-MCFLwn(2).
This theorem gives a way of putting k-MCFGwn in a sort of Chomsky normal
form.
A result by Staudacher [277] can be exploited so as to prove the proper inclusion of MCFLwn into MCFL. This result gives an example of a language
that is an MCFL but that is not an Indexed language and thus not a MCFLwn.
But when separating classes of languages, rather than finding a language that is in one class and not in the other, it is better to characterize certain phenomena that are possible within one class and not within the other. In the case of MCFGs, one such phenomenon is copying: given a language L, the language L(p), the p-copying of L, is defined by:
L(p)= {wp| w ∈ L} .
It can easily be showed that MCFLs are closed under p-copying for every p, i.e. if L is an MCFL, then for every p, L(p) is an MCFL. Though k-MCFL is not
closed under 2-copying. An interesting result Engelfriet and Skyum [122] shows that Indexed languages and thus MCFLwn are not closed under 3-copying.
More precisely, they prove that, for every L, L(3) is indexed iff L is an EDT0L
(a restricted kind of indexed languages). In [S9], we refine slightly that result and we prove the following:
Theorem 38 L(2) is a MCFL
1-MCFLs are languages with a very simple structure: unary trees, and those languages are always captured by well-nested grammars. Thus, model- ing linguistic phenomena with MCFGwnimplies that copying can be performed
only on very simple structures. Such a hypothesis has to be confronted with linguistic models and linguistic data. So far, we have not observed complex structural copies, at least in configurational languages. This makes an inter- esting argument in favor of the well-nestedness constraint for the modelisation of natural language. Well-nestedness has also been studied from the point of view of dependency structures by Kuhlmann and Nivre [194] where they show that in the Pragues Dependency Treebank [150] and in the Danish Dependency Treebank [191], almost all the dependencies satisfy the well-nestedness prop- erty. Moreover and as we will see, it seems to us that MCFLwn are closer to
Joshi’s definition of mildly context sensitive languages proposed than MCFLs.