Underspecified Beta Reduction
Manuel Bodirsky
Katrin Erk
Joachim Niehren
Programming Systems Lab
Saarland University
D-66041 Saarbr¨ucken, Germany
{bodirsky|erk|niehren}@ps.uni-sb.de
Alexander Koller
Department of Computational Linguistics
Saarland University
D-66041 Saarbr¨ucken, Germany
koller@coli.uni-sb.de
Abstract
For ambiguous sentences, traditional
semantics construction produces large
numbers of higher-order formulas,
which must then be a0 -reduced individ-
ually. Underspecified versions can pro-
duce compact descriptions of all read-
ings, but it is not known how to perform
a0 -reduction on these descriptions. We
show how to do this using a0 -reduction
constraints in the constraint language
for a1 -structures (CLLS).
1 Introduction
Traditional approaches to semantics construction
(Montague, 1974; Cooper, 1983) employ formu-
las of higher-order logic to derive semantic rep-
resentations compositionally; then a0 -reduction is
applied to simplify these representations. When
the input sentence is ambiguous, these approaches
require all readings to be enumerated and a0 -
reduced individually. For large numbers of read-
ings, this is both inefficient and unelegant.
Existing underspecification approaches (Reyle,
1993; van Deemter and Peters, 1996; Pinkal,
1996; Bos, 1996) provide a partial solution to this
problem. They delay the enumeration of the read-
ings and represent them all at once in a single,
compact description. An underspecification for-
malism that is particularly well suited for describ-
ing higher-order formulas is the Constraint Lan-
guage for Lambda Structures, CLLS (Egg et al.,
2001; Erk et al., 2001). CLLS descriptions can
be derived compositionally and have been used
to deal with a rich class of linguistic phenomena
(Koller et al., 2000; Koller and Niehren, 2000).
They are based on dominance constraints (Mar-
cus et al., 1983; Rambow et al., 1995) and extend
them with parallelism (Erk and Niehren, 2000)
and binding constraints.
However, lifting a0 -reduction to an operation on
underspecified descriptions is not trivial, and to
our knowledge it is not known how this can be
done. Such an operation – which we will call un-
derspecified a0 -reduction – would essentially a0 -
reduce all described formulas at once by deriv-
ing a description of the reduced formulas. In this
paper, we show how underspecified a0 -reductions
can be performed in the framework of CLLS.
Our approach extends the work presented in
(Bodirsky et al., 2001), which defines a0 -reduction
constraints and shows how to obtain a complete
solution procedure by reducing them to paral-
lelism constraints in CLLS. The problem with
this previous work is that it is often necessary to
perform local disambiguations. Here we add a
new mechanism which, for a large class of de-
scriptions, permits us to perform underspecified
a0 -reduction steps without disambiguating, and is
still complete for the general problem.
Plan. We start with a few examples to show
what underspecified a0 -reduction should do, and
why it is not trivial. We then introduce CLLS
and a0 -reduction constraints. In the core of the
paper we present a procedure for underspecified
a0 -reduction and apply it to illustrative examples.
2 Examples
In this section, we show what underspecified a0 -
reduction should do, and why the task is nontriv-
ial. Consider first the ambiguous sentence Every
student didn’t pay attention. In first-order logic,
the two readings can be represented as
a2
a3a4a6a5
a7
a8
a2
a9a11a10a13a12 a14
a4a16a15
a2
a14
a4a16a15
a14
a4a16a15
a3a4a6a5
a2
a17
a4a19a18a20a4
a10a13a10 a14
a4a16a15
a21
a22a24a23
a25a27a26
a25a27a28
a25a27a29
a25a31a30a25a32a23
a25a27a33
a25a35a34
a25a37a36
a7
a8
a2
a9a11a10a38a12 a14
a4a16a15
a2
a3a4a6a5
a2
a17
a4a19a18a20a4
a10a39a10 a14
a4a16a15
a14
a4a16a15
a21
a22a40a33
a41a43a42
a41a43a26
a41 a28
a41a43a36
a41a43a29
a41a44a30
a7
a8
a2
a9a11a10a13a12 a14
a4a16a15
a2
a17
a4a19a18a20a4
a10a13a10 a14
a4a45a15
a21
a22a40a28
Figure 1: Underspecified a0 -reduction steps for ‘Every student did not pay attention’
a2
a2
a46
a14a48a47
a15a49a18
a9a11a10a13a12
a3a4a6a5
a2
a17
a4a19a18a20a4
a10a39a10 a14
a4a16a15
a21
a22a50a26
a51a52a26
Figure 2: Description of ‘Every student did not
pay attention’
a7a27a53a55a54
a9a11a10a38a12
a53
a8
a21
a54
a17
a4a19a18a20a4
a10a39a10
a53a57a56a11a56
a21
a54a58a7a27a53a55a54
a9a11a10a38a12
a53
a8 a17
a4a19a18a20a4
a10a39a10
a53a57a56a11a56
A classical compositional semantics construction
first derives these two readings in the form of two
HOL-formulas:
a54a38a46
a14a48a47
a15a49a18
a9a11a10a13a12
a56
a1
a53a59a54
a21
a17
a4a60a18a61a4
a10a13a10
a53a31a56
a21
a54a11a54a38a46
a14a48a47
a15a62a18
a9a11a10a38a12
a56
a1
a53a59a54
a17
a4a60a18a61a4
a10a13a10
a53a31a56a11a56
where a46 a14a48a47 a15a49a18 is an abbreviation for the term
a46
a14a16a47
a15a49a18a64a63
a1a66a65a67a1a27a68
a54a58a7a27a53a55a54
a65
a53
a8
a68
a53a31a56a11a56
An underspecified description of both readings is
shown in Figure 2. For now, notice that the graph
has all the symbols of the two HOL formulas as
node labels, that variable binding is indicated by
dashed arrows, and that there are dotted lines indi-
cating an “outscopes” relation; we will fill in the
details in Section 3.
Now we want to reduce the description in Fig-
ure 2 as far as possible. The first a0 -reduction step,
with the redex at a51a52a26 is straightforward. Even
though the description is underspecified, the re-
ducing part is a completely known a1 -term. The
result is shown on the left-hand side of Figure 1.
Here we have just one redex, starting at a25 a26 , which
binds a single variable. The next reduction step
is less obvious: The a21 operator could either be-
long to the context (the part between a22a64a23 and a25a37a26 )
a2
a3a4a6a5
a69
a2
a70
a14
a4a16a15
a71
a21
a69
a2
a70
a71
a21
a51
a25 a72
a73
Figure 3: Problems with rewriting of descriptions
or to the argument (below a25a35a34 ). Still, it is not dif-
ficult to give a correct description of the result:
it is shown in the middle of Fig. 1. For the final
step, which takes us to the rightmost description,
the redex starts at a41a44a42 . Note that now the a21 might
be part of the body or part of the context of this
redex. The end result is precisely a description of
the two readings as first-order formulas.
So far, the problem does not look too difficult.
Twice, we did not know what exactly the parts of
the redex were, but it was still easy to derive cor-
rect descriptions of the reducts. But this is not
always the case. Consider Figure 3, an abstract
but simple example. In the left description, there
are two possible positions for the a21 : above a51 or
below a25 . Proceeding na¨ıvely as above, we arrive
at the right-hand description in Fig. 3. But this de-
scription is also satisfied by the term a69 a54 a21 a54a74a70a16a54 a71 a56a11a56a11a56 ,
which cannot be obtained by reducing any of the
terms described on the left-hand side. More gen-
erally, the na¨ıve “graph rewriting” approach is
unsound; the resulting descriptions can have too
many readings. Similar problems arise in (more
complicated) examples from semantics, such as
the coordination in Fig. 8.
The underspecified a0 -reduction operation we
propose here does not rewrite descriptions. In-
stead, we describe the result of the step using a
“a0 -reduction constraint” that ensures that the re-
duced terms are captured correctly. Then we use a
saturation calculus to make the description more
explicit.
3 Tree descriptions in CLLS
In this section, we briefly recall the definition of
the constraint language for a1 -structures (CLLS).
A more thorough and complete introduction can
be found in (Egg et al., 2001).
We assume a signature a75 a63 a76 a69a37a77a38a78a35a77a80a79a80a79a80a79a82a81 of
function symbols, each equipped with an arity
a4a16a15a13a54 a69 a56a84a83a86a85 . A tree
a87 consists of a finite set of
nodes a88a90a89a59a91a93a92 , each of which is labeled by a sym-
bol a94a95a92 a54 a88 a56 a89a96a75 . Each node a88 has a sequence of
children a88a43a97 a77a80a79a80a79a80a79a55a77 a88a31a98a84a89a99a91a93a92 where a98 a63a100a4a16a15a38a54 a94a101a92 a54 a88 a56a11a56
is the arity of the label of a88 . A single node a102 , the
root of a87 , is not the child of any other node.
3.1 Lambda structures
The idea behind a1 -structures is that a a1 -term can
be considered as a pair of a tree which represents
the structure of the term and a binding function
encoding variable binding. We assume a75 contains
symbols a14 a4a45a15 (arity 0, for variables), a3a4a103a5 (arity 1,
for abstraction), a2 (arity 2, for application), and
analogous labels for the logical connectives.
Definition 1. A a1 -structure a104 is a pair a54 a87 a77 a1 a56 of
a tree a87 and a binding function a1 that maps every
node a88 with label a14 a4a16a15 to a node with label a3a4a6a5 , a7 ,
or a105 dominating a88 .
a3a4a6a5
a2
a69
a14
a4a16a15
The binding function a1 explicitly
maps nodes representing variables to
the nodes representing their binders.
When we draw a1 -structures, we rep-
resent the binding function using dashed arrows,
as in the picture to the right, which represents the
a1 -term a1
a53 a79a106a69 a54a107a53a57a56 .
A a1 -structure corresponds uniquely to a closed
a1 -term modulo a108 -renaming. We will freely
consider a1 -structures as first-order model struc-
tures with domain a91a93a92 . This structure defines
the following relations. The labeling relation
a88a110a109
a69 a54
a88
a23 a77a80a79a80a79a80a79a32a77
a88a35a111
a56 holds in
a87 if a94a95a92
a54
a88
a56a112a63 a69 and
a88a66a113
a63
a88a31a114 for all a97a116a115a117a114a93a115a117a98 . The dominance re-
lation a88a57a118a120a119a60a88a31a121 holds iff there is a path a88a31a121a121 such that
a88a57a88 a121a121
a63
a88 a121 . Inequality a122
a63 is simply inequality of
nodes; disjointness a88a32a123a50a88a31a121 holds iff neither a88a57a118a120a119a60a88a31a121
nor a88 a121a118 a119 a88 .
3.2 Basic constraints
Now we define the constraint language for a1 -
structures (CLLS) to talk about these relations.
a51 a77 a25 a77 a41 are variables that will denote nodes of a
a1 -structure.
a124
a109a49a109
a63 a51
a118 a119
a25a126a125a61a51
a122
a63 a25a126a125a61a51
a123
a25a126a125
a124a116a127a116a124
a121
a125 a51
a109
a69 a54a51a128a23 a77a80a79a80a79a80a79a32a77 a51
a111
a56 a54a74a4a16a15a11a54 a69 a56a43a63
a98
a56
a125
a1
a54a51 a56a11a63 a25a129a125
a1a57a130
a23
a54a51a52a26 a56a11a63a64a76 a51a128a23 a77a80a79a80a79a80a79a82a77 a51
a111
a81
A constraint a124 is a conjunction of literals (for
dominance, labeling, etc). We use the abbrevi-
ations a51 a118a120a131 a25 for a51 a118a120a119 a25 a127 a51 a122a63 a25 and a51 a63
a25 for a51
a118 a119
a25
a127
a25
a118 a119
a51 . The
a1 -binding literal
a1
a54a51 a56a11a63 a25 expresses that a25 denotes a node which
the binding function maps to a51 . The inverse
a1 -binding literal a1a57a130
a23
a54a51a52a26 a56a11a63a64a76 a51a128a23 a77a80a79a80a79a80a79a32a77 a51
a111
a81 states
that a51 a23 a77a80a79a80a79a80a79a32a77 a51 a111 denote the entire set of vari-
able nodes bound by a51a52a26 . A pair a54 a104 a77a133a132 a56 of a a1 -
structure a104 and a variable assignment a132 satisfies a
a1 -structure iff it satisfies each literal, in the obvi-
ous way.
a3a4a6a5
a14
a4a45a15
a14
a4a16a15
a51
a51 a23 a51 a33
Figure 4: The constraint graph of
a1 a130
a23
a54a51 a56a11a63a67a76 a51a128a23 a77 a51a52a33 a81
a127
a51
a118a120a119
a51a128a23
a127
a51
a118a120a119
a51a52a33
We draw constraints as graphs (Fig. 4) in which
nodes represent variables. Labels and solid lines
indicate labeling literals, while dotted lines repre-
sent dominance. Dashed arrows indicate the bind-
ing relation; disjointness and inequality literals
are not represented. The informal diagrams from
Section 2 can thus be read as constraint graphs,
which gives them a precise formal meaning.
3.3 Segments and Correspondences
Finally, we define segments of a1 -structures and
correspondences between segments. This allows
us to define parallelism and a0 -reduction con-
straints.
A segment is a contiguous part of a a1 -structure
that is delineated by several nodes of the structure.
Intuitively, it is a tree from which some subtrees
have been cut out, leaving behind holes.
Definition 2 (Segments). A segment a108 of a a1 -
structure a54 a87 a77 a1 a56 is a tuple a88 a26a74a134 a88 a23 a79a80a79a80a79a32a77 a88a35a111 of nodes
in a91a93a92 such that a88 a26 a118 a119 a88a35a113 and a88a35a113a74a123a50a88a136a135 hold in a87 for
a97a129a115a137a114a138a122
a63a137a139
a115a137a98 . The root a140
a54
a108
a56 is
a88
a26 , and
a141a35a142a136a54
a108
a56a143a63
a88
a23 a77a80a79a80a79a80a79a32a77
a88a66a111 is its (possibly empty) se-
quence of holes. The set a144
a54
a108
a56 of nodes of
a108 is
a144
a54
a108
a56a95a63a100a76
a88a145a89a59a91a147a146
a125
a140
a54
a108
a56
a118a120a119a60a88
a77 and not
a88 a113 a118a120a131a110a88
for all a97a148a115a149a114a95a115a150a98 a81
To exempt the holes of the segment, we define
a144 a130
a54
a108
a56a148a63
a144
a54
a108
a56a152a151a150a141a27a142a6a54
a108
a56 . If a141a27a142a6a54
a108
a56 is a singleton
sequence then we write a141a32a54 a108 a56 for the unique hole
of a108 , i.e. the unique node with a141a153a54 a108 a56 a89 a141a35a142a136a54 a108 a56 .
For instance, a108 a63 a88 a23a154a134 a88 a33 a77 a88 a28 is a segment in
Fig. 5; its root is a88 a23 , its holes are a88 a33 and a88 a28 , and
it contains the nodes a144
a54
a108
a56a95a63a155a76
a88
a23 a77
a88
a36 a77
a88
a33 a77
a88
a28 a81 .
Two tree segments a108 a77a11a0 overlap properly iff
a144a120a130
a54
a108
a56a157a156
a144a120a130
a54a0 a56
a122
a63a159a158 . The syntactic equivalent
of a segment is a segment term a51a52a26a80a134a82a51a128a23 a77a80a79a80a79a80a79 a51 a111 .
We use the letters a160 a77a133a161a162a77a164a163a50a77 a91 for them and extend
a140
a54
a160
a56 , a141a27a142a6a54
a160
a56 , and a141a32a54
a160
a56 correspondingly.
A correspondence function is intuitively an iso-
morphism between segments, mapping holes to
holes and roots to roots and respecting the struc-
tures of the trees:
Definition 3. A correspondence function be-
tween the segments a108 a77a11a0 is a bijective mapping
a165
a109a166a144
a54
a108
a56
a8
a144
a54a0 a56 such that
a165 maps the
a114 -th hole
of a108 to the a114 -th hole of a0 for each a114 , and for every
a88a90a89a90a144a120a130
a54
a108
a56 and every label a69 ,
a88a110a109
a69 a54
a88a43a97
a77a80a79a80a79a80a79a82a77
a88a31a98
a56a110a167
a165
a54
a88
a56
a109
a69 a54
a165
a54
a88a43a97
a56 a77a80a79a80a79a80a79
a165
a54
a88a31a98
a56a11a56 a79
There is at most one correspondence function
between any two given segments. The correspon-
dence literal coa54 a163a40a77 a91 a56a60a54 a51 a56a11a63 a25 expresses that a
correspondence function a165 between the segments
denoted by a163 and a91 exists, that a51 and a25 denote
nodes within these segment, and that these nodes
are related by a165 .
Together, these constructs allow us to define
parallelism, which was originally introduced for
the analysis of ellipsis (Egg et al., 2001). The par-
allelism relation a108a149a168 a0 holds iff there is a corre-
spondence function between a108 and a0 that satis-
fies some natural conditions on a1 -binding which
we cannot go into here. To model parallelism in
the presence of global a1 -binders relating multiple
parallel segments, Bodirsky et al. (2001) general-
ize parallelism to group parallelism. Group par-
allelism a54 a108 a23 a77a80a79a80a79a80a79a32a77 a108a32a111 a56 a168 a54a0 a23 a77a80a79a80a79a80a79a32a77a11a0 a111 a56 is entailed
a3a4a6a5
a78
a69
a2
a3a4a103a5
a2
a14
a4a16a15
a14
a4a16a15
a71
a69
a2
a14
a4a16a15
a71
a88
a26
a88a31a121
a26
a88
a23
a88
a36
a88 a121
a23
a88
a33
a88
a34
a88
a28
a88a31a121
a28
Figure 5: a69 a54a11a54 a1 a53 a79a170a169 a54a107a53a57a56a11a56a60a54 a71 a56a11a56 a8a172a171 a69 a54 a169 a54 a71 a56a11a56
by the conjunction a127 a111
a113a62a173
a23
a108 a113 a168
a0
a113 of ordinary par-
allelisms, but imposes slightly weaker restrictions
on a1 -binding. By way of example, consider the a1 -
structure in Fig. 5, where a54 a88 a26a20a134 a88 a23 a77 a88 a33a61a134 a88 a34 a77 a88 a28a20a134 a56 a168
a54
a88a31a121
a26
a134
a88a31a121
a23
a77
a88a31a121
a23
a134
a88a31a121
a34
a77
a88a31a121
a34
a134 a56 holds.
On the syntactic side, CLLS provides
group parallelism literals a54 a160 a23 a77a80a79a80a79a80a79a32a77 a160 a111 a56 a168
a54 a161 a23 a77a80a79a80a79a80a79a174a77a133a161
a111
a56 to talk about (group) parallelism.
4 Beta reduction constraints
Correspondences are also used in the definition of
a0 -reduction constraints (Bodirsky et al., 2001).
A a0 -reduction constraint describes a single a0 -
reduction step between two a1 -terms; it enforces
correct reduction even if the two terms are only
partially known.
Standard a0 -reduction has the form
a163 a54a11a54
a1
a53 a79a170a161 a56
a160
a56
a8a172a171
a163 a54 a161a176a175a53 a134
a160a178a177
a56a179a53 free for
a160
a79
The reducing a1 -term consists of context a163 which
contains a redex a54 a1 a53 a79a170a161 a56 a160 . The redex itself is an
occurrence of an application of a a1 -abstraction
a1
a53 a79a170a161 with body a161 to argument
a160 .
a0 -reduction
then replaces all occurrences of the bound vari-
able a53 in the body by the argument while preserv-
ing the context.
We can partition both redex and reduct into ar-
gument, body, and context segments. Consider
Fig. 5. The a1 -structure contains the reducing a1 -
term a69 a54a11a54 a1 a53 a79a170a169 a54a107a53a31a56a11a56a60a54 a71 a56a11a56 starting at a88 a26 . The reduced
term can be found at a88 a121a26 . Writing a180 a77 a180 a121 for the
context, a0a44a77a11a0 a121 for the body and a108
a77
a108a32a121 for the ar-
gument tree segments of the reducing and the re-
duced term, respectively, we find
a180
a63
a88
a26a181a134
a88
a23 a0 a63
a88
a33a181a134
a88
a34
a108
a63
a88
a28a181a134
a180a27a121
a63
a88a31a121
a26
a134
a88a31a121
a23
a0
a121
a63
a88a31a121
a23
a134
a88a31a121
a28
a108a32a121
a63
a88a31a121
a28
a134
Because we have both the reducing term and the
reduced term as parts of the same a1 -structure, we
can express the fact that the structure below a88 a121a26
can be obtained by a0 -reducing the structure be-
low a88 a26 by requiring that a108 corresponds to a108a32a121 , a0
to a0 a121 , and a180 to a180 a121 , again modulo binding. This is
indeed true in the given a1 -structure, as we have
seen above.
More generally, we define the a0 -reduction re-
lation
a54
a180
a77a11a0a44a77
a108
a56
a171
a151
a8
a54
a180 a121
a77a11a0
a121
a77
a108 a121
a23
a77a80a79a80a79a80a79a153a77
a108 a121
a111
a56
for a body a0 with a98 holes (for the variables bound
in the redex). The a0 -reduction relation holds iff
two conditions are met: a54a180 a77a11a0a44a77 a108 a56 must form a re-
ducing term, and the structural equalities that we
have noted above must hold between the tree seg-
ments. The latter can be stated by the following
group parallelism relation, which also represents
the correct binding behaviour:
a54
a180
a77a11a0a44a77
a108
a77a80a79a80a79a80a79a110a77
a108
a56
a168
a54
a180a37a121
a77a11a0
a121
a77
a108a32a121
a23
a77a80a79a80a79a80a79a174a77
a108a32a121a111
a56
Note that any a1 -structure satisfying this relation
must contain both the reducing and the reduced
term as substructures. Incidentally, this allows us
to accommodate for global variables in a1 -terms;
Fig. 5 shows this for the global variable a169 .
We now extend CLLS with a0 -reduction con-
straints
a54 a163a40a77a133a161a182a77
a160
a56
a171
a151
a8
a54 a163
a121
a77a133a161
a121
a77
a160 a121
a23
a77a80a79a80a79a80a79a174a77
a160 a121
a111
a56 a77
which are interpreted by thea0 -reduction relation.
The reduction steps in Section 2 can all be
represented correctly by a0 -reduction constraints.
Consider e.g. the first step in Fig. 1. This is repre-
sented by the constraint a54 a22a64a23a19a134a136a25a27a26 a77 a25a37a33a20a134a136a25a27a28 a77 a25a27a34a20a134 a56
a171
a151
a8
a54 a22a50a33a20a134a80a41a43a26 a77 a41a43a26a20a134a80a41a44a28 a77 a41a43a28a20a134 a56 . The entire middle con-
straint in Fig. 1 is entailed by the a0 -reduction lit-
eral. If we learn in addition that e.g. a25a31a30 a118a120a119
a25a27a26 ,
the a0 -reduction literal will entail a41 a30 a118a120a119 a41 a26 because
the segments must correspond. This correlation
between parallel segments is the exact same ef-
fect (quantifier parallelism) that is exploited in
the CLLS analysis of “Hirschb¨uhler sentences”,
where ellipses and scope interact (Egg et al.,
2001).
a0 -reduction constraints also represent the prob-
lematic example in Fig. 3 correctly. The spuri-
ous solution of the right-hand constraint does not
usb(a124 , X) =
if all syntactic redexes in a124 below a51
are reduced then return a54 a124 a77 a51 a56
else
pick a formula redexa183 a54 a163a40a77a133a161a182a77 a160 a56 in a124
that is unreduced, with a51 a63 a140
a54 a163 a56 in
a124
add a54 a163a40a77a133a161a182a77 a160 a56
a171
a151
a8
a54 a163
a121
a77a133a161
a121
a77
a160 a121
a23
a77a80a79a80a79a80a79a174a77
a160 a121
a111
a56
to a124 where a163 a121a77a133a161 a121a77 a160a184a121a23 a77a80a79a80a79a80a79a32a77 a160a184a121
a111
are new
segment terms with fresh variables
add a51 a123a50a140
a54 a163
a121
a56 to
a124
for all a124 a121a57a89 solvea54 a124 a56 do usba54 a124 a121a77 a140 a54 a163 a121a56a11a56
end
Figure 6: Underspecified a0 -reduction
satisfy the a0 -reduction constraint, as the bodies
would not correspond.
5 Underspecified Beta Reduction
Having introduced a0 -reduction constraints, we
now show how to process them. In this section,
we present the procedure usb, which performs a
sequence of underspecified a0 -reduction steps on
CLLS descriptions. This procedure is parameter-
ized by another procedure solve for solving a0 -
reduction constraints, which we discuss in the fol-
lowing section.
A syntactic redex in a constraint a124 is a subfor-
mula of the following form:
redexa183 a54 a163a40a77a133a161a182a77 a160 a56a95a63 df a141a32a54 a163 a56 a109a2 a54 a25 a77 a140 a54 a160 a56a11a56
a127
a25
a109
a3a4a6a5a64a54
a140
a54 a161 a56a11a56
a127
a1 a130
a23
a54 a25 a56a43a63a185a141a35a142a136a54 a161 a56
A context a163 of a redex must have a unique hole
a141a153a54 a163 a56 . An
a98 -ary redex has a98 occurrences of the
bound variable, i.e. the length of a141a35a142a136a54 a161 a56 is a98 . We
call a redex linear if a98 a63 a97 .
The algorithm a186a57a187a20a188 is shown in Figure 6. It
starts with a constraint a124 and a variable a51 , which
denotes the root of the current a1 -term to be re-
duced. (For example, for the redex in Fig. 2,
this root would be a22 a26 .) The procedure then se-
lects an unreduced syntactic redex and adds a de-
scription of its reduct at a disjoint position. Then
the solve procedure is applied to resolve the a0 -
reduction constraint, at least partially. If it has
to disambiguate, it returns one constraint for each
reading it finds. Finally, usb is called recursively
with the new constraint and the root variable of
the new a1 -term.
Intuitively, the solve procedure adds entailed
literals to a124 , making the new a0 -reduction literal
more explicit. When presented with the left-hand
constraint in Fig. 1 and the root variable a22a64a23 , usb
will add a a0 -reduction constraint for the redex at
a25a174a23 ; then solve will derive the middle constraint.
Finally, usb will call itself recursively with the
new root variable a22a50a33 and try to resolve the redex
at a41a44a28 , etc. The partial solving steps do essentially
the same as the na¨ıve graph rewriting approach
in this case; but the new algorithm will behave
differently on problematic constraints as in Fig. 3.
6 A single reduction step
In this section we present a procedure solve for
solving a0 -reduction constraints. We go through
several examples to illustrate how it works. We
have to omit some details for lack of space; they
can be found in (Bodirsky et al., 2001).
The aim of the procedure is to make explicit
information that is implicit in a0 -reduction con-
straints: it introduces new corresponding vari-
ables and copies constraints from the reducing
term to the reduced term.
We build upon the solver for a0 -reduction con-
straints from (Bodirsky et al., 2001). This solver
is complete, i.e. it can enumerate all solutions of
a constraint; but it disambiguates a lot, which we
want to avoid in underspecified a0 -reduction. We
obtain an alternative procedure solve by dis-
abling all rules which disambiguate and adding
some new non-disambiguating rules. This al-
lows us to perform a complete underspecified a0 -
reduction for many examples from underspecified
semantics without disambiguating at all. In those
cases where the new rules alone are not sufficient,
we can still fall back on the complete solver.
6.1 Saturation
Our constraint solver is based on saturation with
a given set of saturation rules. Very briefly, this
means that a constraint is seen as the set of its lit-
erals, to which more and more literals are added
according to saturation rules. A saturation rule
of the form a124 a26 a8 a189a55a111
a113a62a173
a23
a124
a113 says that we can add
one of the a124 a113 to any constraint that contains at
least the literals in a124 a26 . We only apply rules where
each possible choice adds new literals to the set; a
constraint is saturated under a set a190 of saturation
rules if no rule in a190 can add anything else. solve
returns the set of all possible saturations of its in-
put. If the rule system contains nondeterminis-
tic distribution rules, with a98a192a191a193a97 , this set can be
non-singleton; but the rules we are going to intro-
duce are all deterministic propagation rules (with
a98
a63
a97 ).
6.2 Solving Beta Reduction Constraints
The main problem in doing underspecified a0 -
reduction is that we may not know to which part
of a redex a certain node belongs (as in Fig. 1).
We address this problem by introducing under-
specified correspondence literals of the form
coa54a39a76a6a54 a163 a23 a77 a91 a23 a56 a77a80a79a80a79a80a79a153a77 a54 a163 a111 a77 a91a147a111 a56 a81 a56a60a54a51 a56a11a63 a25 a79
Such a literal is satisfied if the tree segments
denoted by the a163 ’s and by the a91 ’s do not
overlap properly, and there is an a114 for which
coa54 a163 a113 a77 a91a147a113 a56a60a54a51 a56a101a63 a25 is satisfied.
In Fig. 7 we present the rules UB for under-
specified a0 -reduction; the first five rules are the
core of the algorithm. To keep the rules short, we
use the following abbreviations (with a97a24a115a149a114a95a115a149a98 ):
beta a63a50a194a19a195a197a196a198a54 a163a50a77a133a161a162a77 a160 a56
a171
a151
a8
a54 a163
a121
a77a133a161
a121
a77
a160a184a121
a23
a77a80a79a80a79a80a79a32a77
a160a184a121
a111
a56
coa113 a63a50a194a19a195a197a196 coa54a39a76a6a54 a163a40a77a164a163 a121a56 a77 a54 a161a182a77a133a161 a121a56 a77 a54 a160 a77 a160 a121
a113
a56 a81 a56
The procedure solve consists of UB together
with the propagation rules from (Bodirsky et al.,
2001). The rest of this section shows how this
procedure operates and what it can and cannot do.
First, we discuss the five core rules. Rule
(Beta) states that whenever the a0 -reduction rela-
tion holds, group parallelism holds, too. (This al-
lows us to fall back on a complete solver for group
parallelism.) Rule (Var) introduces a new variable
as a correspondent of a redex variable, and (Lab)
and (Dom) copy labeling and dominance literals
from the redex to the reduct. To understand the
exceptions they make, consider e.g. Fig. 5. Every
node below a88 a26 has a correspondent in the reduct,
except for a88 a28 . Every labeling relation in the redex
also holds in the reduct, except for the labelings of
the a2 -node a88 a23 , the a3a4a6a5 -node a88 a28 , and the a14 a4a16a15 -node
a88
a34 . For the variables that possess a correspon-
dent, all dominance relations in the redex hold in
the reduct too. The rule (a1 .Inv) copies inverse a1 -
binding literals, i.e. the information that all vari-
ables bound by a a1 -binder are known. For now,
(Beta) a199a49a200a37a201a203a202a55a201a203a204a37a205a101a206a207 a199a208a200a37a209a210a201a107a202a31a209a210a201a107a204a27a209a211a38a201a197a212a181a212a197a212a48a201a107a204a27a209a213a82a205 a207 a199a208a200a37a201a74a202a55a201a107a204a32a201a197a212a197a212a39a212a82a201a210a204a31a205a103a214a162a199a49a200a37a209a58a201a203a202a31a209a58a201a107a204a35a209a211a11a201a197a212a181a212a181a212a48a201a210a204a35a209a213a16a205
(Var) beta a215 redexa216a32a199a208a200a37a201a74a202a55a201a107a204a31a205a20a215a95a217a19a199a208a200a57a205a49a218a164a219a154a220a184a215a166a220a110a221a173 a183 a207a112a222 a220 a209a212coa223a74a199a49a220a27a205a173 a220 a209
(Lab) beta a215 redexa216 a199a208a200a37a201a74a202a55a201a107a204a31a205a20a215a166a220a103a224a19a225a226a48a199a49a220 a211 a201a39a212a197a212a197a212a16a201a107a220a120a227a39a205a82a215a166a228 a227a229a13a230
a224
coa223a203a199a208a220 a229 a205a173 a220a35a209a229 a215a101a220a103a224a20a221a173a31a231 a199a208a200a57a205a48a215a101a220a6a224a82a232a233 a231a20a234 a199a208a202a174a205 a207 a220a66a209
a224
a225a226a48a199a49a220a66a209a211 a201a181a212a197a212a181a212a48a201a107a220a35a209
a227
a205
(Dom) beta a215 a228a110a235a229a38a230 a211 coa223a203a199a208a220 a229 a205a173 a220a35a209a229 a215a166a220 a211 a218 a219 a220
a235
a207
a220a35a209a211 a218 a219 a220a66a209
a235
(a236 .Inv) betaa215 redexa216 a199a49a200a37a201a203a202a110a201a210a204a37a205a39a215a66a236a16a237 a211 a199a49a220a103a224a133a205a173a174a238 a220 a211 a201a197a212a181a212a181a212a48a201a107a220a103a239a44a240a60a215 a228 a239a229a38a230
a224
coa211 a199a49a220 a229 a205a173 a220 a209a229 a207 a236a16a237 a211 a199a208a220 a209
a224
a205
a173a174a238
a220 a209a211 a201a197a212a39a212a197a212a82a201a203a220 a209a239 a240 redex linear
(Par.part) betaa213 a215 coa223a203a199a49a220a27a205a173 a220 a209 a215 a183 a233a152a241 a199a208a204a37a205a16a215a166a220a66a218a164a219 a183 a207 a220 a209 a232a233a152a242a199a208a202 a209a205
a23a57a243
a113
a243
a111
(Par.all) coa199
a238
a199a49a244a101a201a203a244 a209a205a203a201a197a212a39a212a197a212a74a240a38a205a74a199a208a220a37a205
a173
a220 a209 a215a166a220
a233a245a241
a199a208a244a43a205
a207
a220 a209
a233a245a241
a199a208a244 a209a205a48a215 coa199a208a244a101a201a203a244 a209a205a203a199a49a220a27a205
a173
a220 a209
Figure 7: New saturation rules UB for constraint solving during underspecified a0 -reduction.
it is restricted to linear redexes; for the nonlinear
case, we have to take recourse to disambiguation.
It can be shown that the rules in UB are sound
in the sense that they are valid implications when
interpreted over a1 -structures.
6.3 Some Examples
To see what the rules do, we go through the first
reduction step in Fig. 1. The a0 -reduction con-
straint that belongs to this reduction is
a54 a163a40a77a133a161a182a77
a160
a56
a171
a151
a8
a54 a163
a121
a77a133a161
a121
a77
a160a178a121
a23
a56 with
a163 a63 a22a64a23a164a134a136a25a37a26 a77 a161 a63 a25a174a23a164a134a136a25a37a28 a77
a160
a63 a25a35a34a20a134 a77
a163
a121
a63 a22a50a33a61a134a80a41a43a26 a77a246a161
a121
a63 a41a44a26a61a134a80a41a43a28 a77
a160a178a121
a23
a63 a41a44a28a20a134
Now saturation can add more constraints, for
example the following:
a199
a23
a205
a183a61a247
a221
a173 a183 a211
a199
a36
a205
a183a61a247
a221
a173 a183a61a248
a199
a33
a205
a183a20a249
a221
a173 a183 a211
a199
a29
a205a250a220
a247
a225a251a252a199a208a220
a249
a205 (Lab)
a199
a28
a205
a222
a220
a247
a212co
a211
a199
a183a61a247
a205
a173
a220
a247 (Var)
a199
a30
a205a250a253
a235
a218 a219 a220
a247 (Dom)
a199
a34
a205
a222
a220
a249
a212co
a211
a199
a183a20a249
a205
a173
a220
a249 (Var)
We get (1), (2), (5) by propagation rules from
(Bodirsky et al., 2001): variables bearing differ-
ent labels must be different. Now we can apply
(Var) to get (3) and (4), then (Lab) to get (6). Fi-
nally, (7) shows one of the dominances added by
(Dom). Copies of all other variables and literals
can be computed in a completely analogous fash-
ion. In particular, copying gives us another redex
starting at a41a44a42 , and we can continue with the algo-
rithm usb in Figure 6.
Note what happens in case of a nonlinear redex,
as in the left picture of Fig. 8: as the redex is a254 -
ary, the rules produce two copies of the a21 labeling
constraint, one via coa23 and one via coa33 . The result
is shown on the right-hand side of the figure. We
will return to this example in a minute.
6.4 More Complex Examples
The last two rules in Fig. 7 enforce consistency
between scoping in the redex and scoping in the
reduct. The rules use literals that were introduced
in (Bodirsky et al., 2001), of the forms a51 a89a145a144 a54 a160 a56 ,
a51 a134
a89a155a255
a54 a161 a56 , etc., where
a160 ,
a161 are segment terms.
We take a51 a89a90a144
a54
a160
a56 to mean that a51 must be inside
the tree segment denoted by a160 , and we take a51 a89
a255
a54 a161 a56 (i for ’interior’) to mean that a51
a89a145a144
a54 a161 a56 and
a51 denotes neither the root nor a hole of a161 .
As an example, reconsider Fig. 3: by rule
(Par.part), the reduct (right-hand picture of Fig.
3) cannot represent the term a69 a54 a21 a54a74a70a48a54 a71 a56a11a56a11a56 because
that would require the a21 operator to be in a255a54 a161 a121a56 .
Similarly in Fig. 8, where we have introduced
two copies of the a21 label. If the a21 in the redex
on the left ends up as part of the context, there
should be only one copy in the reduct. This is
brought about by the rule (Par.all) and the fact that
correspondence is a function (which is enforced
by rules from (Erk et al., 2001) which are part of
the solver in (Bodirsky et al., 2001)). Together,
they can be used to infer that a41a44a26 can have only
one correspondent in the reduct context.
7 Conclusion
In this paper, we have shown how to perform an
underspecifieda0 -reduction operation in the CLLS
framework. This operation transforms underspec-
ified descriptions of higher-order formulas into
descriptions of their a0 -reducts. It can be used to
essentially a0 -reduce all readings of an ambiguous
sentence at once.
It is interesting to observe how our under-
specified a0 -reduction interacts with parallelism
constraints that were introduced to model el-
lipses. Consider the elliptical three-reading ex-
ample “Peter sees a loophole. Every lawyer does
too.” Under the standard analysis of ellipsis in
CLLS (Egg et al., 2001), “Peter” must be rep-
resented as a generalized quantifier to obtain all
three readings. This leads to a spurious ambigu-
a2
a3a4a6a5
a127
a2
a14
a4a16a15
a17 a47a60a10a38a47
a15
a2
a14
a4a16a15 a5a40a4a45a15a49a18
a3a4a6a5
a3a4a1a0a3a2a5a4
a14
a4a45a15
a21
a41a44a26
a41 a23
a127
a2
a3a4a6a5
a3a4a1a0a6a2a5a4
a14
a4a16a15
a17 a47a60a10a38a47
a15
a2
a3a4a6a5
a3a4a7a0a3a2a5a4
a14
a4a16a15
a5a40a4a16a15a62a18
a21 a21
a41
a121
a26
a41
a121
a23
a41
a121a121
a26
a41
a121a121
a23
Figure 8: “Peter and Mary do not laugh.”
ity in the source sentence, which one would like
to get rid of by a0 -reducing the source sentence.
Our approach can achieve this goal: Adding
a0 -reduction constraints for the source sentence
leaves the original copy intact, and the target sen-
tence still contains the ambiguity.
Under the simplifying assumption that all re-
dexes are linear, we can show that it takes time
a8 a54a10a9
a98
a28
a56 to perform a9 steps of underspecified a0 -
reduction on a constraint with a98 variables. This
is feasible for large a9 as long as a98a12a11a14a13 a85 , which
should be sufficient for most reasonable sen-
tences. If there are non-linear redexes, the present
algorithm can take exponential time because sub-
terms are duplicated. The same problem is known
in ordinary a1 -calculus; an interesting question to
pursue is whether the sharing techniques devel-
oped there (Lamping, 1990) carry over to the un-
derspecification setting.
In Sec. 6, we only employ propagation rules;
that is, we never disambiguate. This is concep-
tually very nice, but on more complex examples
(e.g. in many cases with nonlinear redexes) dis-
ambiguation is still needed.
This raises both theoretical and practical issues.
On the theoretical level, the questions of com-
pleteness (elimination of all redexes) and conflu-
ence still have to be resolved. To that end, we
first have to find suitable notions of completeness
and confluence in our setting. Also we would like
to handle larger classes of examples without dis-
ambiguation. On the practical side, we intend to
implement the procedure and disambiguate in a
controlled fashion so we can reduce completely
and still disambiguate as little as possible.
References
M. Bodirsky, K. Erk, A. Koller, and J. Niehren. 2001.
Beta reduction constraints. In Proc. 12th Rewriting
Techniques and Applications, Utrecht.
J. Bos. 1996. Predicate logic unplugged. In Proceed-
ings of the 10th Amsterdam Colloquium.
R. Cooper. 1983. Quantification and Syntactic The-
ory. Reidel, Dordrecht.
M. Egg, A. Koller, and J. Niehren. 2001. The con-
straint language for lambda structures. Journal of
Logic, Language, and Information. To appear.
K. Erk and J. Niehren. 2000. Parallelism constraints.
In Proc. 11th RTA, LNCS 1833.
K. Erk, A. Koller, and J. Niehren. 2001. Processing
underspecified semantic representations in the Con-
straint Language for Lambda Structures. Journal of
Language and Computation. To appear.
A. Koller and J. Niehren. 2000. On underspecified
processing of dynamic semantics. In Proc. 18th
COLING, Saarbr¨ucken.
A. Koller, J. Niehren, and K. Striegnitz. 2000. Re-
laxing underspecified semantic representations for
reinterpretation. Grammars, 3(2/3). Special Issue
on MOL’99. To appear.
J. Lamping. 1990. An algorithm for optimal lambda
calculus reduction. In ACM Symp. on Principles of
Programming Languages.
M. P. Marcus, D. Hindle, and M. M. Fleck. 1983. D-
theory: Talking about talking about trees. In Proc.
21st ACL.
R. Montague. 1974. The proper treatment of quantifi-
cation in ordinary English. In Formal Philosophy.
Selected Papers of Richard Montague. Yale UP.
M. Pinkal. 1996. Radical underspecification. In Proc.
10th Amsterdam Colloquium.
O. Rambow, K. Vijay-Shanker, and D. Weir. 1995.
D-Tree Grammars. In Proceedings of ACL’95.
U. Reyle. 1993. Dealing with ambiguities by under-
specification: construction, representation, and de-
duction. Journal of Semantics, 10.
K. van Deemter and S. Peters. 1996. Semantic Am-
biguity and Underspecification. CSLI Press, Stan-
ford.
