Structure and Global Ambiguity
First, we will present some well-known results, which are important for our further
Theorem 1 RTGs are weakly equivalent to
CFGs (that is, the set of strings of leaves generated by CFGs is equivalent to the
strings of leaves generated by RTGs).
RTGs are strongly equivalent
to graphs generated by a CFGs, modulo a homomorphism of node labels (i.e., a homomorphism
which maps various node labels in a given tree onto a single one), provided the RTG
LTGs are strongly equivalent
to graphs generated by CFGs, provided finitary branching.
Proof is trivial: as in LTGs every node label is generated by exactly one nonterminal;
in CFGs, nodes which are not leaves are labeled by nonterminals, there is a one-to-one
correspondence. This has some importance for the relation of LTGs and RTGs. It follows,
LTGs and RTGs are equivalent up to homomorphism, as well as all grammar types in between.
Every discussion we have about generative capacity concerns only non-equivalence in
isomorphism. Since also the strings of leaves for all grammar types are identical,
we can only
be interested in the sets of trees. In the sequel, if we speak of the weak generative
of a tree grammar, we refer to the sets of trees it generates, not the strings of
Since to our knowledge, only the strong generative capacity of the grammar types of
Murata et al., 2005 has been in the focus of research, we will now scrutinize their weak
Theorem 4 The
sets of trees generated by STGs form a proper subset of the sets of trees generated
For proof, consider the trees generated by the following grammar:
S → a(AB)
A → b(C)
B → b(D)
C → c(D)
D → c(ε)
This is an RCG, since A and B do not occur in similar contexts. In order to see that
is no way to generate this tree with an STG, consider the following fact: a governs two identical labels, which however govern different
subtrees. It is therefore impossible to introduce them with identical rules, and (by
definition of STGs) forbidden to have two competing rules in the content model of
rule. This is sufficient for the proof of weak inclusion, since all STG rules are
rules, and therefore the languages generated form a proper subset.
Theorem 5 The
sets of trees generated by LTGs form a proper subset of the sets of trees generated
Consider the trees generated by the following grammar:
A → a(B)
B → b(C)
C → a(D)
D → c(ε)
This is a single type tree grammar, and no LTG is able to generate such a tree (remember
that LTGs are strongly equivalent to graphs generated by CFG, provided finitary
Restrained Competition Grammars and Variants
In this section we will scrutinize formal properties of the different types of
restrained competition grammars. We will show which kind of languages cannot be generated
RCGs; we will prove that there are GRCGs which are ambiguous, and that for every language
which can be generated by an RTG, there is a GRCG which generates the same language.
Finally, we will show that URCGs do not have these properties, are properly included
GRCGs and properly include RCGs.
The type of languages we cannot generate with RCGs is quickly described as follows:
grammars, where a single content model contains competing nonterminals, which can
uniquely distinguished only from their left (right, respectively) context, are not
Consequently, we cannot generate sets of trees, where a certain node has different
depending on its right siblings. If we want to get rid if this asymmetry, and allow
GRCGs, where competing nonterminals in a single content model are uniquely determined
their left or right context, we run into problems:
Theorem 6 There
are GRCGs which are ambiguous.
For proof, consider the following rule: S →
a(AB|BA). Suppose, A and B are competing nonterminals; suppose furthermore, that there is some
overlapping between A and B; i.e., the nonterminals generate overlapping sets of trees. In particular,
we may assume that the trees generated by A form a subset
of the trees generated by B. For example, A and B generate identical trees
up to depth n; B in
addition generates a tree of depth n+1. In this case, the
trees of the language have the root a, with two symmetrical
sets of subtrees up to depth n, and possibly one subtree
with depth n+1. It is easily seen that now it is impossible
to merge A and B, for then
we would be incapable of expressing the condition that at most one subtree has depth
n+1. However, for the trees, where the subtrees
introduced by A and B have
depth at most n, there is necessarily more than one
analysis. The grammar we have described so far is, however, a GRCG, because neither
A nor B occur in
identical contexts (though in similar contexts, remember the preceding section).
Theorem 7 For
every language which can be generated by an RTG, there is a GRCG which generates the
To proof this theorem, we describe a simple procedure to convert any RTG into a GRCG,
which generates the same language. We define competing sequences of length n of nonterminals as follows: two sequences of nonterminals
compete, if for all n, the nth nonterminal of one sequence competes with the nth nonterminal of the other sequence. We have to assume a content model
r which is not GRCG conform. Therefore, there have to be
two competing nonterminals or competing sequences of nonterminals A and B in r, such that for possibly empty sequences of nonterminals Γ and Δ, (Γ A Δ) and (Γ B Δ) match
Given this, we can be sure, that in the instantiations of r, which violate the GRCG condition, A and
B occur in exactly the same global tree contexts. By
global tree context we here mean that a tree with a governing the subtrees generated
A is part of the language iff a also governs the set of
subtrees generated by B. Since this is the case, we can
simply merge the two nonterminals to a new one, C, which is
the union of the former two. This new nonterminal substitutes all instantiations of
A and B, which occur in
the same global tree context. This, by definition, are the instantiations which violate
GRCG condition. This we can apply to all nonterminals which violate the GRCG condition.
only thing we have to take care of is that we apply this only to those instantiations
content models where two competing nonterminals match equally (this might force us
some regular expressions). We do not show an exact algorithm at this point, since
clear that an equivalent GRCG exist, and the details of the construction are of no
interest at this point.
We now show that there is a hierarchy of proper inclusion RCG ⊂ URCG ⊂ GRCG.
To show that RCG ⊂ URCG, consider the following: every rule which is admitted by an
RCG is also admitted by a URCG, because if competing nonterminals in the same content
have a unique prefix, a fortiori they also have a unique context (we have already
a unique prefix of nonterminals is paramount to a unique prefix of labels/siblings,
induction). Above, we have already shown that for an RCG it is impossible to generate
languages as the following, which is a URCG.
S → a(AC|BD)
A → b(C)
B → b(D)
C → c(ε)
D → d(ε)
compete, but are
determined uniquely by their context.
This concludes the first part; the second part will be a corollary of the next section:
We will show that some languages are inherently ambiguous, that is, there is no unambiguous
grammar for them. By Theorem 7 we know that we can generate these languages with GRCGs,
Proposition 1 A URCG cannot be ambiguous.
This is easy to see: an ambiguous grammar assigns two different sequences of
nonterminals to the daughters of one node (since root nodes are unambiguous): Then
there must be at least two competing nonterminals which occur in the same content
similar contexts, which, by definition, is impossible.
Inherently Ambiguous Languages
As a corollary, we can show that there are regular tree languages, for which there
unambiguous grammar. There are sets of trees, which are generated by an ambiguous
by no URCG. We will call these languages inherently
Theorem 8 Some
regular tree languages are inherently ambiguous.
This can be seen easily, if we spell out a grammar which we described in the above
subsection. We will then show that there is now way to write an unambiguous grammar
generates the same language.
S → r(AB|BA)
A → a(C)
B → a(D)
C → b(ε)
D → b(ε|E)
E → c(ε)
There is no way to merge A and B, since they generate different sets of subtrees (we can write L(A)≠L(B)); but since they overlap (L(A)∩ L(B)≠ ∅), there is no way to have a unique
interpretation in the cases where the subtrees generated by the nonterminals are identical.
There will always be two ways to generate trees in this case.
We can, furthermore, precisely state the conditions, under which a regular tree language
is ambiguous. To this end, however, we need to introduce some notation. We now for
simplicity write trees as terms: a tree with root a and
daughters b and c is
denoted as a(b,c), etc. As a next step, we define a context
as the position, where certain subtrees occur within trees of a language.
A context C is a tree-term with exactly one variable. We say that a
set of subtrees α occurs in a context C in a language L, if the following holds: We
can instantiate the variable of C with any tree from α, and the resulting tree is
Note that sets of subtrees correspond to nonterminals, when we speak of languages
than grammars. In the sequel, for simplicity we will use lower case Greek variables
for sets of subtrees as for ordered sequences of sets of subtrees. The definition
context is easily accommodated to sequences. A set of sequences of trees of length
n consists of ordered tuples of trees of length n, of the form (t0,...,tn-1). Sets of subtrees
are then simply sets of one-tuples. Importantly, we will not provide a proof for the
following proposition, and leave it open as a conjecture. However, we will sketch
argument. We now make the following conjecture:
A tree-language is inherently ambiguous iff at least one node fulfills
all of the following conditions:
We need to have one node with an arbitrary label a, with at least
two (sequences of) sets of subtrees α and β, such that
α ∩ β ≠ ∅;
α ≠ β;
There is at least one context C in L, such that both a(Γ,
(u1,...,un), Θ) and
(t1,...,tn), Θ) occur
in C, for all (t1,...,tn) ∈ α
and all (u1,...,un)
∈ β, where uppercase Greek letters designate possibly empty
sequences of daughter sub-trees; note that the sequences need to have equal length
order to meet condition 1.
Due to space restrictions, we leave the prove for this conjecture open here; this
reminds however of a theorem in Odgen, 1968 for string languages. But we
will give some rather informal discussion of the points in the next section. It is
to see that this is merely a generalization of the cases we have been described above.
will see, we can derive some useful facts from these properties of ambiguous languages,
without a general proof: we can show that we can construe grammars for languages which
not fulfill one of the conditions, and, moreover, which type of grammars we can
From the grammar types sketched so far, there is no type which
generates all and only the RTLs that are not inherently ambiguous.
We will demonstrate this going through the three conditions mentioned in the preceding
section, and look which unambiguous grammar we can construe if one condition is not
This is to be read as follows: if one condition is not met, then it means, that from
nodes of the tree language, there might be any one which meets the ones not in question,
none which meets the one currently under consideration.
If there is no intersection between the subtrees of a given node, the grammar is of
course not ambiguous. We can, however not necessarily construe a URCG for this grammar,
since in the content model of the mother node there are competing nonterminals in
similar contexts (recall the example given above).
We can, however, construe a USG for such a language, since subtrees are uniquely
This means that there are no two sets of subtrees governed by the same node which
are not identical. We can thus introduce them by the same nonterminals, and have a
tree grammar (having no different sets of subtrees governed by the same node amounts
say we need no competing rules in the grammar, as nonterminals correspond to sets
If the third condition is not met, then we can construe nonterminals (corresponding
to the sets of subtrees) such that for all of them the following holds: assuming they
compete (introduce identical labels), they either occur in different contexts, in
case they are distinguishable thereby and no ambiguity arises; or they occur in
identical contexts, in which case we can use a unique nonterminal which is the merge
both (this also holds for root nodes). The critical case, where the content of one
(sequence of) set(s) of subtrees depends on the other one, which makes them occur
similar contexts, while making it impossible to merge them, however, we have excluded
Since this argument holds inductively from the root to all subtrees, we can construe
a URCG for the language were condition 3 is not met, but we cannot use any strictly
weaker type. The only thing we have provided is that if two sets of subtrees occur
similar contexts (for the grammar we construe), then they actually occur in identical
contexts. It follows that we do not need competing nonterminals in similar
This shows that we still have not solved the problem to define a canonical grammar
which generates all and only the unambiguous languages, since there are languages
generated by USGs, but no other canonical class which does not allow any ambiguity
languages which are generated by URCGs and no other such type. So far, we are still
a characterization of the unambiguous languages in terms of grammar rules.
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