+

X2|X3

X1|X2

X4|X5

X5|X6

X3|X4

X2|X3

X1|X2

X4|X5

X5|X6

X3|X4

(C) CCG derivation for non-canonical linearization.

(d) Separating tree that derives the same non-
canonical linearization.

数字 3
Relation between CCG and separating trees.

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We begin by proving three helpful lemmas. To prove the first of these lemmas we
use strong mathematical induction, sometimes also called complete induction. To prove
that some proposition P(n) holds for all natural numbers using strong induction it is
sufficient to prove the following statements:

(我)

(二)

base case P(1) 和
inductive step [磷(1) ∧ P(2) ∧ · · · ∧ P(k)] =⇒ P(k + 1) holds for all k ∈ N

Strong induction is different than the more typical weak induction in the strength of
the assumption that is taken in the inductive step: Weak induction assumes only P(k),
whereas strong induction assumes P(1) ∧ · · · ∧ P(k). Both are equally sound proving
策略, but strong induction is more convenient for proving propositions about trees.
A more detailed distinction between weak and strong induction is available in many
textbooks that cover proof methods (Eccles 1997; Liebeck 2010; Gunderson 2014).

3.4 The Proof

Lemma 1
For a fixed NOD of n elements with canonical linearization X1|X2 X2|X3 . . . Xn|Xn+1:

CCG derivations can form all possible contiguous binary branching trees;

Every node that spans from i to j in every tree derives a category Xi|Xj+1.

(我)

(二)

22

Stanojevi´c and Steedman

Formal Basis of a Language Universal

Proof. Base case:
For n = 1 we have a single category in the canonical linearization X1|X2. This is the only
possible tree and it is binary in a trivial sense. Its only node spans from 1 到 1 deriving
category X1|X1+1.
Inductive step: Assuming the strong inductive hypothesis holds for all 1 ≤ k ≤
n, we prove that it also holds for n + 1. We analyze the canonical linearization
. . . Xn+1|Xn+2 by splitting it into two contiguous parts where the first
X1|X2 X2|X3
. . . Xn+1|Xn+2 for any
. . . Xk|Xk+1 and the second one is Xk+1|Xk+2
one is X1|X2
k ∈ [1 . . . n]. Each of these two parts forms a smaller NOD with length ≤ n. 从
inductive assumption it follows that for both of these parts CCG can form all possible
binary branching trees, and that all nodes spanning i to j within these parts will be
labeled with Xi|Xj+1. That means that the root node of the left and right part will be
X1|Xk+1 and Xk+1|Xn+2, 分别. Now we can combine the left and right part into
the whole structure using forward function composition >B to give us a new node
X1|Xn+2. This node satisfies the labeling condition of the lemma. The binary branching
condition is also satisfied because we put no constraints on the size of the spans of the
left and the right part (as long as they are non-empty), and by the inductive assumption,
(西德:3)
each part can derive all possible binary trees.

Lemma 2
All word orders that can be derived by CCG for a given NOD can also be derived by
an isomorphic separating tree that reorders the same permutation into the canonical
linearization for that NOD.

Proof. It is sufficient to show that any CCG derivation tree corresponds to an isomorphic
separating tree over the same permutation.

Consider any arbitrary CCG derivation tree for a canonical linearization of a NOD

of length n. 例子: One such derivation for the NOD for n = 5 is in Figure 3a.

From Lemma 1, we know that all nodes with span i to j will have label Xi|Xj+1
and all binary nodes will use forward function composition >B. We can derive the
same order with an isomorphic separating tree where instead of using CCG forward
combination we have the + operator at every node, because the children are in the
canonical order. 例子: Figure 3b.

Non-canonical linearization in CCG requires backward combinations. For any node
in a CCG derivation for the canonical linearization whose left child has category
席|Xj+1 and right child category Xj+1|Xk+1 we can obtain the inverse order by reversing
the directionality of the combination and reversing the order of the child nodes, 制作
Xj+1|Xk+1 precede Xi|Xj+1, so that all words in the range j + 1 to k in the canonical order
precede words i to j. 例子: Figure 3c shows this reordering effect for the two nodes
where forward is replaced by backward combination.

Any CCG derivation reordered in this way corresponds to an isomorphic separating
tree in which the unrotated nodes corresponding to forward combination are relabeled
作为 + and the rotated nodes corresponding to backward combination are relabeled as −.
例子: Figure 3d.

By the result of Bose, Buss, and Lubiw discussed in Section 3.2, applying the −
operator to convert all such nodes to + and invert the linear order of their children
reverses the process to recover the canonical linearization of which this is a separable
(西德:3)
permutation.

23

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计算语言学

体积 47, 数字 1

Lemma 3
CCG can derive all possible separating trees for an NOD of arbitrary length.

Proof. By Lemma 1, CCG can derive all possible binary branching trees on the canonical
linearization. By Lemma 2, each node in each of these binary trees can be labeled either
+ or −, defining a separating tree isomorphic to the CCG derivation for the correspond-
ing linearization. Because we saw in Section 3.2 that a separating tree determines the
identities of its leaves, the set of all possible binary trees with all possible node labelings
(西德:3)
is the set of all possible separating trees under the definition (17) 那里.

最后, we can prove the main claims that relate CCG derivations of NOD and

separable permutations of canonical linearization of NOD.

Theorem 1
CCG derives only separable permutations.

Proof. By Lemma 2, all CCG derivations can be mapped to separating trees. All separat-
ing trees correspond to a separable permutation (Bose, Buss, and Lubiw 1998). 所以
(西德:3)
CCG derives only separable permutations.

Theorem 2
CCG derives all separable permutations over a NOD.

Proof. By Lemma 3, CCG can derive all possible separating trees. Every separable
permutation corresponds to a separating tree (Bose, Buss, and Lubiw 1998). 所以,
(西德:3)
CCG derives all separable permutations.

Theorem 3
The cardinality of the set of permutations derivable by CCG is bounded by Sn−1, 这
n − 1th Large Schr ¨oder Number.

Proof. By Theorems 1 和 2, CCG derives all and only separable permutations over an
NOD of length n. The number of separable permutations of length n is the n−1th Large
Schr ¨oder Number (Shapiro and Stephens 1991). 所以, the number of permutations
(西德:3)
derivable by CCG for that NOD is also n−1th Large Schr ¨oder Number.

The recurrent formula for this series is given below, the base case being S0 = 1:

(18)

Sn = Sn−1 +

n−1
(西德:88)

k=0

SkSn−k−1

3.5 Generalization to Full CCG

The above results have been proven using a restricted version of CCG called “pure
first-order” CCG (Koller and Kuhlmann 2009), in which all variables in categories
席|Xi+1 correspond to atomic types such as N, rather than function categories, 例如
(S\NP). The standard version of CCG is a second-order calculus, in which arguments
can have function types, a possibility that is central to the CCG analysis of relativization,
协调, and adjunction, including the phenomenon of parasitic gaps. 的确, 全部

24

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Stanojevi´c and Steedman

Formal Basis of a Language Universal

cases of so-called movement reduce in CCG to contiguous merger with a semantically
second-order functor.

Categories that select for multiple arguments are “Curried” unary functors in CCG,
with a function as their result, 例如 (席|Xj)|Xi+1. They do not define a NOD over the
arguments Xi+1, Xj, because those arguments do not stand in a dominance relation, 和
they potentially allow the second-level composition rules such as (12) to apply. Never-
theless, they have a canonical linearization in which all combinations are forward, 和
(by Lemma 2) dominate separating trees over separable permutations of that canonical
ordering, although they no longer allow all separable permutations. (The large Schr ¨oder
numbers thus constitute an upper bound on the allowable permutations of canonical
linearizations in CCG.)

The present results generalize immediately to full CCG. The semantics of the com-
binatory rules is invariant, so it makes no difference to the combinatorial possibilities
if the “canceling” variable Y in a rule such as application matches an atomic category
or a function category. (实际上, we have already exploited this possibility in writing the
atomic symbols VP and N for what are semantically predicate or functional types.)

Type-raising is a lexical operation that exchanges the role of argument and function
by making the former into a function over the latter and that does allow permutations
that would not be allowed by the original categories. It does so by changing the NOD
among the categories. 然而, because this is a lexical operation whose effects are
determined before syntactic derivation can apply, the main result of the present paper
仍然适用: Only separable permutations on the canonical linearization order defined
by the lexical categories can be captured.

4. Normal-Form Separating Trees and CCG Parsing

The results shown above depended on the fact that the canonical linearization on the
category sets of interest is binary tree-complete, and therefore separating tree-complete
under rotation, and therefore separable-permutation-complete. 然而, we noted ear-
lier that there are more separating trees than separable permutations, because many
separating trees yield the same permutation. 的确, for the canonical linearization of
NOD of length n and its inverse, the number of separating trees is Cn−1, the n−1th
Catalan Number (Church and Patil 1982), which is much larger than the corresponding
Large Schr ¨oder number.

例如, 数字 4 shows all eight separating trees derived by rotation from the
two binary trees on the canonical linearization of the canonical linearization CL3 of three
元素, 其中 + on each of the two nodes means that it is unrotated and − means
that it is rotated. These are indeed all separable permutations of three elements, 但是
two orders 1,2,3 和 3,2,1 are represented by two separating trees, where C3 = 2. 为了
the purpose of efficient parsing with CCG, we would like there to be only one CCG
derivation, and one separating tree, for each separable permutation.

4.1 Normal-Form Separating Trees

Shapiro and Stephens (1991) observed that the spurious ambiguity of separating
trees arises when two nodes that are in parent–child relation have the same label, as for
the trees in Figure 4a and 4b. They proposed a normal form that makes the mapping
between the set of normal form separating trees and the set of separable permutations
bijective. This normal form states that no node can be labeled with the same +/−
label as its right child. This excludes trees like those in Figure 4a and 4e and means

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计算语言学

体积 47, 数字 1

+

+

-

-

+

+

+

+

321

21

3

32

1

213

(A)

-

(乙)

-

(C)

+

(d)

+

-

-

-

-

123

23

1

12

3

231

(e)

(F)

(G)

(H)

数字 4
All possible separating trees for a three-word sentence.

that the number of normal-form separating trees is the same as that of the separable
permutations.

This normal form for separating trees is reminiscent of the CCG derivational
normal-form proposed by Eisner (1996) to enforce a one-to-one mapping between CCG
syntactic derivations and semantic interpretation that they generate (比照. Hockenmaier
and Bisk 2010). Eisner normal-form forbids particular sequences of combinatory rules
that are of the same type from occurring in a parent-child relationship in the derivation.
Using normal-form separating trees, we can distinguish the possible cases, 显示
图中 5. Figures 5a and 5b show the configurations where the right child is a
single word. Figures 5c and 5d show those where the right child is a non-single word
constituent that adheres to the normal form. The length of the examples is identified as
n + 1 in order to be consistent with the conventional numeration for the Large Schr ¨oder
numbers. The length of the constituent A is k. We are interested in the number of normal-
form separating-trees Sn, since that will also be the number of separable permutations
of length n + 1.

The number of trees that conform to the pattern in Figure 5a clearly depends on the
number of possible concrete sub-trees that can act as constituent A. Here A is of length
k = n, which means that the number of separating trees of the pattern in Figure 5a is
Sn−1. If we ignore the label of the root node, the other three patterns show all possible
ways of constructing the right child: having a single terminal node, or binary node
labeled + or −. This means that we can express the sum of trees confirming to these
three patterns as the number of trees that have their root unlabeled (since we are ignor-
ing the root label) and have the left and right child confirming to the normal form for
separating trees. For a given k, which plays a role of a split point in parsing, 号码
of those trees would be Sk−1Sn+1−k−1, but because k can be any number between 1 和
n, and is not fixed, we need to sum over all of those values (西德:80)n
k=1 Sk−1Sn−k. We can shift
this sum to start from zero to obtain the equivalent formula (西德:80)n−1
k=0 SkSn−k−1. 什么时候我们
put the formulas for all four patterns together we get the Large Schr ¨oder recurrence in
方程 (18). We can conclude that the number of normal-form separating trees for a
NOD of length n is also Sn−1.

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Formal Basis of a Language Universal

+

-

A

A

0

n

n+1

0

n

n+1

(A)

+

(乙)

-

-

+

A

乙

C

A

乙

C

0

k

(C)

n+1

0

k

(d)

n+1

数字 5
Normal form branching configurations.

4.2 The Relation of Large Schr ¨oder Numbers to Shift-Reduce Parsing

The Large Schr ¨oder numbers are often also presented in relation to the number of lattice
paths in the Cartesian plane such that each path starts at lower-left corner (0, 0), 结束
at upper right corner (n, n), contains no points above the diagonal y = x (那是, 能
only be in the lower right triangle), and is composed only of unit steps →, ↑, 和 (西德:37)
or more concretely (1, 0), (0, 1), 和 (1, 1) (Weisstein 2018). In this section we show how
this view of Large Schr ¨oder numbers can be interpreted as Shift-Reduce parsing with
CCG derivations corresponding to the normal-form separating trees defined above.

Shift-Reduce is one of the most widely used parsing algorithms in computational
linguistics and it is probably the simplest way to construct a parser. It is based on
having a parsing state with a stack (filled with constituents built so far) and a buffer
(filled with words in the sentence suffix that are yet to be processed). A shift transition
takes a word from the buffer and puts it on top of the stack. For binarized grammars
like CCG, a reduceX transition replaces the two topmost constituents on the stack with
a single new constituent labeled X. In our CCG derivations we will have two reduce
运营: reduce+ for forward combinators and reduce− for backward combinators.
We will again take an example sentence of length n + 1. The initial parsing state will be
a stack containing the first word and a buffer containing the other n words.

We can visually represent any valid successful shift-reduce parsing trace by starting
at the point (0, 0) and taking → to represent a shift transition and ↑ labeled + or − to
represent a reduceX transition. The Shift-Reduce transition sequence for the derivation
tree in Figure 6a is shown in Figure 6b. 清楚地, such paths will never cross the diagonal,

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+

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2

(A) separating tree

(乙) all reduce transitions are labeled

6

4

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1

3

2

6

4

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1

3

2

(C) only first reduce in the reduce
sequence is labeled

(d) no labels

数字 6
A separating tree and different representations of the shift-reduce transition sequence that
recognizes it.

because we can never have more reduce operations than shifts and every successful
parse has n shifts and n reduce steps, so that every path ends at (n, n).

The set of SR parse paths can be further restricted to those corresponding to the
the normal-form parses and separating trees defined in the preceding Section 4.1, 作为
如下. 第一的, the normal form says that a parent node cannot have the same label as
its right child node. What that means in SR parsing terms is that we cannot have two
consecutive transitions both of which are reduce+ or reduce−. 所以, when we have a
sequence of reduce operations, expressed as an uninterrupted vertical sequence of ↑,
we need only know the label of the first reduce transition in the sequence because the
other labels can be inferred from the normal form: They will alternate between + 和
-. This modified representation is shown in Figure 6c.

The final modification is to use a slightly different encoding of the initial reduce in
each of the reduce sequences. The first reduce step in the sequence always comes after
a shift, which means that it forms a shape → ↑. If that sequence is [转移, reduce−], 我们
will encode it with a diagonal (西德:37) and if it is [转移, reduce+] we will encode it with
existing→ ↑ but without any label, as shown in Figure 6d. This encoding defines a
one-to-one mapping between CCG normal-form shift-reduce parses and paths in the
Schr ¨oder lattice.

These observations support another way to think about the number of permutations
that CCG allows for a NOD of length n in terms of SR normal-form parsing. 什么时候

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Formal Basis of a Language Universal

+ 1,y)

(y

Sn−k−1

Sn−1

Sk

1

0
(A) Paths starting with (西德:37) 过渡

n

0

k+1

1
(乙) Paths starting with → transition

n

数字 7
Counting paths in the lattice for different initial transitions.

we start in the initial position (0, 0) we can take paths that start either with (西德:37) or →
(↑ is illegal because it would cross the upper diagonal). If we take (西德:37) (Figure 7a) 我们
find ourselves in a lattice of the same shape, smaller by one unit, 这意味着
number of normal-form paths through it is Sn−1. 如果, 另一方面, we take → we
will end up in position (1, 0). Any path that goes from (1, 0) 到 (n, n) must at some point
have a vertical arrow that crosses diagonal (y + 1, y), as shown in Figure 7b. 第一个
point at which the path crosses that diagonal will be some concrete point (k + 1, k). 这
number of paths that go from (1, 0) 到 (n, n) and contain arc ↑ at (k + 1, k) is SkSn−k−1.
再次, we need to sum over all possible paths that cross the diagonal (y + 1, y) 超过
each possible k, which gives (西德:80)n−1
k=0 SkSn−k−1. Summing results for initial (西德:37) and initial
→ again yields the Large Schr ¨oder number recurrence function from Equation (18).

5. Statistical Significance of the Global Result

The above results confirm that all 22 separable permutations are allowed, at least as al-
ternates, for any construction defined by four elements of the form A|乙, 乙|C, C|D, D, 和
这 2 non-separable permutations are forbidden. The fact that in both the constructions
被调查的, the distribution of the frequencies with which the separable permutation
orders are actually observed are so skewed means that some orders are so vanishingly
rare as to only be attested in one construction, and/or even by only one or two lan-
guages. The possibility that this skewness arises from soft constraints related to ease of
processing or learnability is interesting in their own right, and is the subject of ongoing
linguistic research (Culbertson, 斯摩棱斯基, and Legendre 2012; Merlo 2015; Dryer 2018;
Merlo and Ouwayda 2018).13

尽管如此, according to the present theory, the two permutations (G) 和 (j) 是
excluded for a different reason, 即, by a hard constraint that is a consequence of
the theory of grammar itself. The fact that the two disallowed orders are the only two
that are totally unattested in the union of Cinque’s, Nchare’s, and Abels’s surveys is a

13 This possibility is also acknowledged by Abels and Neeleman (2012).

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strong result that is unlikely to have arisen by chance. The strength of this result can be
assessed quantitatively, as follows.

有 21 permutations that are attested for NP. We will assume that all 21 是
sampled without replacement from a uniform distribution of 24 permutations. 没有
knowing which 21 are attested, we could form 24 theories that exclude one permutation,
24 × 23 that exclude two, 和 24 × 23 × 22 that exclude 3 permutations. For each one
of them, there is a different number of possible observations (IE。, different sets of 21
permutations) that would not falsify them. 例如, a theory that forbids 3 permu-
tations will be consistent only with one possible set of 21 observed permutations, 尽管
a theory that forbids 2 (like the one presented here) will be consistent with 22 不同的
sets of 21 permutations. The probability of our predictions being not falsified by chance
is the number of observations (permutation sets) that would be consistent with our
theory divided by the total number of possible observations—that is, 大约 1
在 100:

(19)

(西德:19)

(西德:19) = 22
2024

≈ 0.0109

p =

(西德:18) 22
21
(西德:18) 24
21

If on the other hand we accept (H), as tentatively attested on the basis of Abels’ and
Wurmbrand’s accounts, the probability of getting this stronger result by chance drops
to around 1 在 250:

我

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哦
w
n
哦
A
d
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F
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米
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d
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t
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米

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F
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(20)

(西德:19)

(西德:19) = 1
276

≈ 0.0036

p =

(西德:18) 22
22
(西德:18) 24
22

Both results are evidently unlikely to come up by chance.

6. 讨论

The prediction of CCG concerning possible word orders is significant from both sci-
entific and engineering perspectives. The Large Schr ¨oder series Sn grows much more
slowly with n than the total factorial number of permutations n!, so that the proportion
n!−Sn−1
of non-separable permutations that are disallowed by CCG grows rapidly with
n!
Sn−1
n! of separable

n, 作为 0%, 0%, 0%, 8%, 25%, 45%, 64%, 79%, 89%, . . . . The proportion
permutations that are allowed shrinks correspondingly rapidly with n.

From a practical NLP perspective there are many areas where this saving can
be achieved, with machine translation and parsing, including “sequence-to-sequence”
语义解析, being the obvious use-cases. If we know the limited number of word-
order variations among languages, alignment algorithms (or attention as its neural
equivalent [Bahdanau, 给, and Bengio 2015]) can be constrained (or biased) 在里面
right direction (Zhang and Gildea 2005; Dong and Lapata 2016).14

14 This observation is also implicit in Whitelock’s (1992) early proposal for “shake and bake” translation for

Japanese.

30

Stanojevi´c and Steedman

Formal Basis of a Language Universal

Unsupervised parser induction is another application where introducing an induc-
tive bias (or hard constraint) for contiguous derivation trees with a normal form sig-
nificantly reduces the number of alternatives that learner needs to consider (Clark and
Lappin 2010). These savings are especially significant in comparison to other theories
of grammar that use discontinuous constituents, empty strings, or very powerful, 经常
Turing-complete, unification mechanisms.

We saw earlier that the number and type of permutations allowed by CCG appears
to correspond to the crosslinguistic variation in word order that is actually attested.
CCG also provides a strong and easily testable prediction of what word orders will be
found in the future in human languages that have not so far been examined (斯蒂德曼
2020). These properties stem from the combinatory projection principle (13), 并在
particular from the restriction to binary combinatory rules.

迄今为止, the empirical studies of crosslinguistic variation cited above have been
confined to no more than four core elements of NP and VP. 然而, we have shown
that the results proved here are completely general to sets of categories of any size that
define a natural order of dominance. The verb group in particular can be naturally
extended to serial verbs of various kinds. The cartographic spines of functional pro-
jections identified by Cinque and Rizzi (2008) offer several tens of functional heads in
a supposedly universal order of dominance, although there is more to say here about
the relation of syntax and morphology and the degrees of crosslinguistic variation they
allow. Any firm evidence for any non-separable permutation of these elements in any
language for which the natural order of dominance can be unequivocally determined
would immediately falsify the present theory. As we noted earlier, the number of for-
bidden non-separable falsifying cases rises very rapidly as the size of such sets increases
超过 4.

然而, the power-law distributions that we observe on all such dimensions of
variation, and the fact that the performance factors that lie behind them favor continuity
and semantic homomorphism, mean that those falsifying cases will always lie out in the
long tail. We have seen that the total number of languages that have ever been studied in
the requisite detail is in the case of the nominal construction only just enough to take us
far enough out into that long tail to achieve a confidence level against ever seeing a non-
separable permutation of even 1 在 100. That means our chances of reaching a definitive
answer to the question for even a five-element construction are quite daunting.

Naturally, our work is not the first work to try to find a formal constraint of this

种类. Here we review some results from other formalisms.

ITG Inversion Transduction Grammars (吴 1996) are a type of syntax transduction
grammar developed for the task of machine translation. ITG can generate only separa-
ble permutations of the words in the source language, so it is natural to ask how this
property of ITG differs from the one proposed here. The main difference lies in what is
considered the reference word order that is being permuted. 这里, CCG gives a clear
definition: The reference word order is the CL that reflects the NOD, and ultimately the
语义学. To say that CCG generates only separable permutations of the NOD is to
make a clear prediction about what permutations can be found in natural languages.
然而, the same is not the case with ITG, which does not identify any NOD. For ITG
the reference word order is that of the source language in the MT task. Taking a different
language as a source language gives a different prediction of what will be found in the
world’s languages. 拿, 例如, the previous example “these five young lads.” If
ITG takes English as a source language then it will make the same prediction as CCG.
But if we had, 例如, taken Spanish as a source language, with the word order
“these five lads young,” ITG would be able to derive “five lads these young,” which has

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not been attested and that CCG predicts to be impossible. 更重要的是, starting
from Spanish ITG would be unable to derive the word order “lads these young five,”
which was attested by both Cinque and Nchare for NP and by Abels and Neeleman for
the equivalent VP permutation. Extending ITG to a more powerful formalism such as
Permutation Trees (Zhang and Gildea 2007; Stanojevi´c and Sima’an 2015) does not solve
this problem as long as the NOD is ignored.

TAG Tree-Adjoining Grammars (Joshi 1985) are often compared to CCG because
they are weakly equivalent, in the sense that they generate the same set of languages
(Joshi, Vijay-Shanker, and Weir 1991). The problem discussed in our article is about the
set of permutations allowed for a fixed natural order of dominance. 这意味着
results from our paper would transfer to TAG only if TAG was strongly equivalent to
CCG, which Koller and Kuhlmann (2009) have shown not to be the case: There are some
dependency structures that are derivable by CCG that are not derivable by TAG and vice
versa. 尤其, Koller and Kuhlmann’s (2009, Figure 8b) example of a dependency
structure derivable by TAG but not by CCG is actually the dependency structure of the
non-separable permutation 2413, 或者 (G) under the alphanumeric convention used in this
文章.

Previous work on the permutations derivable by TAG have looked at the different
scrambled argument orders allowed in German. This was motivated by the work of
Becker, Rambow, and Niv (1992), who claim that scrambling requires even higher
generative power than LCFRS. Joshi, Becker, and Rambow (2000) show that that is
indeed the case if we want unbounded embedding levels of scrambling, but note that
it is not clear that this is needed in modeling attestable natural language data. The tree-
local multi-component version of TAG (TL-MC-TAG) does not increase weak generative
power but does increase strong generative power, allowing complete scrambling of
three arguments but not four.

Scrambling in CCG is handled by the assumption that all arguments are lexically
type-raised as functions over the verb or verb complex, which allows less oblique
论据 (关键地, the subject) to compose with the German verb complex in ad-
vance of more oblique arguments. 如前所述, lexical type raising alters the NOD,
making the subject take position 1 and the verb position n in the order of dominance.
If the generalization of composition rules in CCG is limited to the second-order case
illustrated in (12), CCG is subject to the same limitation as TL-MC-TAG: three arguments
can scramble freely, but four cannot. For more discussion on German scrambling and
CCG, see Hockenmaier and Young (2008).

Chen-Main and Joshi (2008) take the problem of scrambling of noun arguments
of four verbs: N1N2N3N4V1V2V3V4. They show that there are 22 permutations of N1
to N4 that can be derived with the extended version of TL-MC-TAG that supports
flexible composition and multiple adjoining. 然而, these permutations cannot be
directly compared with the NP and VP constructions considered in the empirical studies
discussed earlier.15

MG Minimalist Grammars (Stabler 1996) are a rigorously defined formalism for
Chomskyan Minimalism (Chomsky 1995b) that has higher generative capacity than
CCG. Stabler (2011) has used MG to derive possible permutations for the four core ele-
ments of a noun phrase and found that, under some assumptions, MG can generate only

15 When Chen-Main and Joshi (2008) report that their variation of TAG can capture permutation (G) 2413 的
verbal arguments, they are actually referring to a permutation 24135678 of eight elements, of which the
last four (the verbs) are polyvalent, defining a quite different (部分的) NOD from the 1 > 2 > 3 > 4
linearization.

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Formal Basis of a Language Universal

16 out of the complete 24 possible permutations. 这些 16 include the 14 permutations
claimed by Cinque (2005), but exclude some of those claimed by Nchare (2012) 为了
free word-order language Shupamem. The assumption that Stabler makes is that the
grammar uses neither empty strings nor head-movement, both of which are prevalent
in most linguistic work on minimalist syntax. 有趣的是, Stabler’s 16 include permu-
站 (j), which is one of the two non-separable permutations that are excluded under
the present theory.

The limitation of combinatory reduction to the separable permutations is also
implicit in Svenonius’s (2007) *1-3-X-2 restriction on adjunct placement (除其他外
constructions), where X is an adjunct to 1, and in Williams’s (2003, 页面 203-211)
proposal for his categorial calculus CAT. CAT has a standard directional categorial
lexicon and rule of application, with a combinatory operation REASSOCIATE equiv-
alent to composition, and an operation FLIP, unavailable in CCG, which reverses the
directionality on the argument of a functor category.

Williams’s CAT variety of Minimalism is closely related to CCG. 然而, 没有
the addition of morpho-lexical type-raising, permitting arguments to compose, CAT
is unable to express the variety of constructions that CCG makes available, 哪个
include relativization, coordination-reduction, Germanic scrambling, and Hungarian
VM fronting (斯蒂德曼 2020). In the absence of lexical type-raising these constructions
require the extension of CAT by the addition of powerful rules of “action at a distance”
unavailable in CCG, such as movement, copying, and/or deletion, together with atten-
dant constraints to limit their considerable expressive power.

A related proposal to ours by Medeiros (2018) hypothesizes that the set of allowed
permutations is the one that could be sorted with a single stack, so-called stack-sortable
permutations or 231-avoiding permutations (Knuth 1968). This set of permutations
exactly covers Cinque’s 14, but does not fit the other empirical data considered here:
7 permutations out of 21 that are observed are predicted not to be possible by Medeiros.
As with Williams (2003), it is also unclear that Medeiros’s mechanism will also han-
dle phenomena of unbounded or wh movement, as CCG does. 然而, there is an
interesting relation between the sorting automata for 231-avoiding permutations and
the automata that sort separable permutations: 231-avoiding permutations are sortable
with a single pop-stack while separable permutations are sortable with a sequence of
n - 1 pop-stacks (Avis and Newborn 1981).

7. 结论

This article has argued that the facts of attested word order in the constructions consid-
ered exploits all and only the degrees of freedom that CCG allows, despite the fact that
some of the orders that CCG allows for some constructions are vanishingly rare due
to unrelated performance soft constraints that induce Zipfian (power-law) 分布
discussed by Culbertson, 斯摩棱斯基, and Legendre (2012) and Culbertson and Adger
(2014).

Explanatory adequacy in a theory depends on being able to explain why the data
has just the degrees of freedom it does, and why other things don’t happen. If the latter
have been prevented by hard constraints, as in various ways they have been for these
constructions by Cinque, Svenonius, Abels, Neeleman, Nchare, and Stabler, then those
constraints themselves have in turn to be explained.

No hard constraints are needed at the level of competence grammar in CCG. 这
limitation on the grammatical permutation of n elements in natural grammars to the
Large Schr ¨oder number Sn−1 of separable permutations is a formal universal stemming

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from the limited expressivity of CCG itself—specifically, its restriction of its operators
to strictly adjacent pairs of non-empty constituent types via the Combinatory Projec-
tion Principle (13). This result is a corollary of the Combinatory Categorial theory of
grammar, and the formally explicit reduction that it affords of what in Minimalist terms
would be called MOVE or “internal merge” to contiguous (那是, “external”) MERGE.

附录A: The Ensemble of Attested Word Orders Over 4 Elements

In this appendix we review the attested orders of four core elements of NP and VP as
reported by Cinque, Nchare, and Abels and Neeleman. We also give CCG derivation
trees for these orders from Steedman (2020).

If our lexicon consists of four entirely unrestricted categories of the form A|乙, 乙|C,
C|D, D, then for both the nominal and verbal constructions all 22 separable permuta-
tions are allowed, and only the two non-separable permutations are unanalyzable, 作为一个
consequence of the combinatory projection principle (13), as follows.

The NP
For the NP, 这 22 orders are allowed under the following derivations (“×” marks
the two sequences (G) 和 (j) that are unanalyzable). Only essential compositions are
表明的: all other combinations are application. For non-basic orders, the annotation
“from z” indicates the basic pure-applicative order among those in (8) on whose lexicon
a particular derived order is based:16

两个都 (basic: v.many)

两个都 (basic: 许多)

Cinque (from b: v.few)

Cinque (from b: 很少)

Nchare (from r)

Nchare (from s)

(21) A.

乙.

C.

这些

这些

five
young lads
NP/NumP NumP/N(西德:48) 氮(西德:48)/N N
lads young
five
NP/NumP NumP/N(西德:48) N N(西德:48)\氮
young
NP/NumP N NumP/N(西德:48) 氮(西德:48)\氮
>B×

这些

lads

five

NumP\N

d. Lads

five

这些

young
N NP/NumP NumP/N(西德:48) 氮(西德:48)\氮
NP/N(西德:48)

>B

NP\N

>B×

e.

F.

Five

这些
young lads
NumP/N(西德:48) NP\ NumP N(西德:48)/N N

B×

我. Young

这些

lads
氮(西德:48)/N NP/NumP NumP\N(西德:48) 氮

five

NP\N(西德:48)

NP/N

>B×

B×

five

NP\N(西德:48)

我. Lads young

这些
N N(西德:48)\N NP/NumP NumP\N(西德:48)
>B×

five

NP\N(西德:48)
five

米.

n.

哦.

这些

young

lads
NP/NumP N(西德:48)/N NumP\N(西德:48) 氮

B×

NP\N

q.

Five

young

lads
NumP/N(西德:48) 氮(西德:48)/N NP\NumP N
>B

这些

NumP/N

NP/N

B×

NumP\N

你. Young

five

这些

lads
氮(西德:48)/N NumP\N(西德:48) NP\NumP N
NP\N(西德:48)

B×