LDG-basic-slides

Logical Description Grammar
for sentence and discourse interpretation
Introduction to the grammar system
Noor van Leusen
Radboud University
2012
Noor van Leusen (Radboud University) Logical Description Grammar for sentence and discourse interpretation 2012 1 / 42

Content
Content
Logical Description Grammar as a method for the description of
sentence syntax and semantics.
Muskens 1996, 2001. /cf. ‘Saarbruecken school’.
Underspeciﬁed representations. Syntactic and semantic properties of
linguistic trees. Parsing by deduction. CDRT and local contexts in the
semantic dimension.
vLeusen&Muskens 2003
Discourse processing and the computation of coherence in LDG.
Anaphora and Presupposition.
vLeusen 2007 /cf. Gardent&Webber 1998, Egg&Redeker 2006.
Implicit discourse relations. Inferring discourse meaning.

LDG framework for Sentence Analysis Introducing Logical Description Grammar
Description Grammar for sentence analysis
LDG combines description theory (Vijay-Shanker c.s.) lexicalised TAG
(Joshi, Webber c.s.) and compositional DRT (Kamp, Muskens).
With a view to ambiguity and implicitness of meaning it generates
‘underspeciﬁed representations’.
It describes syntactic, semantic and pragmatic constraints in parallel,
and is speciﬁcally tuned to model the interaction of these constraints.
The semantics integrates a context parameter, employed in the
treatment of anaphora, presupposition and accommodation.

Discourse contexts and discourse descriptions
Discourse contexts are represented by discourse descriptions.
Discourse descriptions are built up incrementally from sentence descriptions.
r
k′
+
relation
k
Sentence representations are linked to their discourse context by explicit
connectives or implicit discourse relations.

Underspecified representations
There are two levels of linguistic analysis:
sentence description (underspecified representation)
?
set of verifying trees (fully specified linguistic repr.)
Sentence descriptions are statements in classical type logic.
They partially describe properties of linguistic trees, constraining
syntactic, semantic and pragmatic features in parallel.
In case of ambiguity, more than one tree class verifies a description.

LDG framework for Sentence Analysis Linguistic Representations
Linguistic Tree Representations
Sentence and discourse representations are tree structures, decorated with
lexemes, syntactic labels, semantic values, and local contexts.

Example tree, showing just syntax and lexemes:
S
S
DP
D
Max
VP
V
opened
DP
D
a
NP
N
can
In situ analysis of quantiﬁers (’a can’): lexeme in surface position,
quantiﬁcation potential is scoped out (at top S).

Fine-grained compositional DRT
[..|...] is a DRS, to the left of the | sign is the universe, to the right
the conditions.
uk, wk, ok , . . . are discourse markers;
uk , ok store individuals.
uk represent new referents (generated in the discourse);
ok represent given referents; they are globally available and interpreted
referentially rather than existentially;
wk store worlds.
wr is the marker for the actual world.
⊕ merges DRSs, ⊑ denotes inclusion, |≍ entailment.
‘proper(K)’ deﬁnes K as a DRS in which no free discourse markers
occur.

Context Parameter
Each node in a tree structure carries a DRS which is its local context.
(cf. Karttunen 1974)
The compositional semantics refers to this context parameter in the
resolution of anaphora, presupposition satisfaction and projection, and
accommodation in general.

Local contexts and context-sensitivity
Local contexts ‘collect’ information on the semantic tier of the
discourse context.
They are constructed going top-down and from left to right through a
discourse or sentence tree.
The local context of the root node is B, the implicit background.
Anaphora and projective elements such as presupposition triggers
introduce conditions on local contexts.
Accommodation/projection results from satisfying these conditions in
partially underspeciﬁed local backgrounds.

LDG framework for Sentence Analysis Description Grammar as a reasoning system
Description Grammar, a constraint-based reasoning system
A description grammar G consists of lexical descriptions and some
general axioms.
It generates a sentence description ∆u per processed utterance u.
The grammar is itself a description, a set of statements in type logic.
Given the theory G + ∆u, a language user may reason about and
obtain the possible verifying trees of the sentence description M(∆u).

LDG framework for Sentence Analysis Description Grammar as a reasoning system
While G represents a language user’s general linguistic knowledge, a
sentence description ∆u represents his speciﬁc linguistic knowledge of
the sentence.
Linguistic knowledge means knowledge in any of the relevant
domains, such as phonology, syntax, semantic and pragmatics.
Sentence interpretation is a reasoning process.
Since descriptions can be partial, G + ∆u can have more than a single
verifying tree, and the syntax or semantics of the sentence can remain
underspeciﬁed.
Each verifying tree comes with a possible reading of the sentence.

LDG framework for Sentence Analysis Lexicon and General Axioms
G consists of lexical descriptions and general axioms
General axioms constrain the general conﬁgurational and semantic
properties of linguistic representations.
∀k ¬[k ≺ k] No node precedes itself
∀k [Γk |≍⊥ ∧ Γr ⊕ σr |≍⊥] Local contexts are consistent

LDG framework for Sentence Analysis Lexicon and General Axioms
Lexical descriptions
characterise the syntactic, semantic, and pragmatic properties of
lexemes. They describe ‘little trees’ similar to elementary structures in
TAG.
Lexical description of the noun can:
∀k [can(k) → (cn(k) ∧ σ(k) = λv [ |wr : can v]) ]
∀k [cn(k) → ∃k2(ℓ(k) = n ∧ ℓ(k2) = np ∧ k2 ¡ k∧
σ(k2) = σ(k) ∧ k
+
←֓ {k2} ∧ k
−
←֓ ∅)]
Graphic representation:
NP+
k2
λv[ | wr : can v]
Nk
can

LDG framework for Sentence Analysis Example description
Sentence description of ‘Max opened a can’
Sentence descriptions consist of the lexical descriptions of the lexemes
occurring in them, taking into account their surface order.
Graphic representation:
S−
r
Γr = B
DP+
10
o0
[o0 | wr : Max o0] ⊑ B
D0
Max
S+
11
σ13(σ12)
DP−
12 VP−
13
VP+
14
λv[ | wr : v opened σ15]
V1
opened
DP−
15
S+
16
[u2 | ] ⊕ σ19(u2) ⊕ σ17
S−
17
Γ17 = Γ16 ⊕ [u2 | ] ⊕ σ19(u2)
DP+
18
u2
D2
a
NP−
19
NP+
20
λv[ | wr : can v]
N3
can

Syntactic properties
Each lexeme enters its own lexical description.
S, VP, DP, NP, D, V are syntactic labels.
Dashed lines stand for underspecified dominance relations, straight
lines for immediate dominance.
The general axioms enforce that the described structures are tree
structures.
+ and - represent anchoring relations. Each node must be both
positively and negatively anchored to some lexical element. The
general axioms enforce that + and - marked nodes are paired off
one-to-one and the pairs identified.
Parsing comes down to reasoning about node identifications.

Semantic properties
σ associates semantic values with nodes.
σα
k indicates a semantic value of type α.
Semantic values are in ‘finegrained compositional DRT’.
Γ associates Karttunen-style local contexts with nodes.
B, the local context of the root node, represents the implicit, general
background to the discourse.
Indefinite descriptions introduce fresh discourse referents.
Names are presuppositional elements. They introduce a condition on
B which has the effect of resolving or accommodating them in the
main DRS of the resulting discourse meaning.

LDG framework for Sentence Analysis Reasoning about the description
Parsing as deduction I
S−
r
Γr = B
DP+
10
o0
[o0 | wr : Max o0] ⊑ B
D0
Max
S+
11
σ13(σ12)
DP−
12 VP−
13
VP+
14
V1
opened
DP−
15
S+
16
[u2 | ] ⊕ σ19(u2) ⊕ σ17
S−
17
Γ17 = Γ16 ⊕ [u2 | ] ⊕ σ19(u2)
DP+
18
u2
D2
a
NP−
19
NP+
20
λv[ | wr : can v]
N3
can
What tree structures ﬁt this description?

Parsing as deduction II
On the assumption that there are currently no other lexemes than the ones
processed, the language user may reason about verifying trees of the
description.
Parsing The only way to pair of + and - marked nodes that is in
accordance with the grammar is
r = 16 ∧ 11 = 17 ∧ 10 = 12 ∧ 13 = 14 ∧ 15 = 18 ∧ 19 = 20

Parsing as deduction III
S−
r
Γr = B
DP+
10
o0
[o0 | wr : Max o0] ⊑ B
D0
Max
S+
11
σ13(σ12)
DP−
12 VP−
13
VP+
14
V1
opened
DP−
15
S+
16
[u2 | ] ⊕ σ19(u2) ⊕ σ17
S−
17
Γ17 = Γ16 ⊕ [u2 | ] ⊕ σ19(u2)
DP+
18
u2
D2
a
NP−
19
NP+
20
λv[ | wr : can v]
N3
can
r = 16 ∧ 11 = 17 ∧ 10 = 12 ∧ 13 = 14 ∧ 15 = 18 ∧ 19 = 20

If the language user adds this inferred information to the discourse
description, we get
S16
[u2 | wr : o0 opened u2, wr : can u2]
B
[o0 | wr : Max o0] ⊑ B
S11
DP10
D0
Max
VP14
V1
opened
DP18
D2
a
NP20
N3
can
The linguistic tree we saw before is the single verifying tree of this
description; there is no ambiguity.

Inferring Discourse Meaning
The meaning of a discourse given a general background B is
identiﬁed with B ⊕ σr
what was ‘taken for granted’ updated with ‘what was said’, while at
the same time all constraints collected in the description are satisﬁed.
The discourse meaning of ‘Max opened a can’ is
B ⊕ [u2 | wr : o0 opened u2, wr : can u2],
where [wr | ] ⊑ B and [o0 | wr : Max o0] ⊑ B hold.

Sentence meaning
Sentences ﬁgure as minimal discourses.
Sentence meaning cannot be computed independently of computing
discourse meaning, unless an out of the blue context is assumed.
Given a discourse description and a sentence whose root node is k,
the sentence meaning ‘in context’ corresponds to Γk ⊕ σk, where all
constraints contributed by elements dominated by k must be satisﬁed.

Discourse Processing An example Discourse
Discourse processing: an incrementation step
‘Max opened a can. It contained the money.’
S−
r
B
S+
16
[o0 | wr : Max o0] ⊑ B
Max opened a can
DP+
24
σ5
[σk | ] ⊑ Γ24
σ5 ∈ Tms
24
D5
it
S+
25
σ26(σ27)
DP−
26
VP−
27
VP+
28
λv[ | wr : v contained σ29]
V6
contained
DP−
29
DP+
30
σ7
[σ7 | ] ⊑ Γ30
Γ30 |≍[ | wr : money σ7]
D7
the
NP9
money
‘it’ and ‘the money’ are context dependent items.

Discourse Processing An example Discourse
Anaphora resolution and presupposition
is the topic of another slide-show.
In the example, the hearer will infer σ5 = u2,
and accommodate/project
[ u7′ | wr : money u7′ ] ⊑ B and [o0 | wr : Max o0] ⊑ B

Discourse Processing Further reasoning about the discourse description
Further reasoning about node identiﬁcations...
(1) S−
r
B
S+
16
[o0 | wr : Max o0] ⊑ B
Max opened a can
S+
25
[ | wr : σ5 contained σ7 ]
[σ5 | ] ⊑ Γ24, [σ7 | ] ⊑ Γ30
Γ30 |≍[ | wr : money σ7]
σ5 ∈ Tmostsalient
24
It contained the money
Heuristics must guide the reasoning process: ‘safe’ conclusions drawn
earlier must not be lost through the incrementation step.

Discourse Processing Selecting an appropriate discourse relation
.. and implicit discourse relations
(2) S−
r
B
S+
16
[o0 | wr : Max o0] ⊑ B
Max opened a can
S+
25
[σ5 | ] ⊑ Γ24, [σ7 | ] ⊑ Γ30
Γ30 |≍[ | wr : money σ7]
24
The grammar enforces coherence by requiring that every discourse
description describes a discourse parse tree.
So the sentence structure must be attached to the discourse context
structure.

There must be an implicit anchor
Additional structure can only be contributed by a lexical anchor (due
to lexicalisation).
Since no overt lexeme is available, the anchor must be implicit.
The only implicit elements in the grammar that can contribute the
link are discourse relations.
It must be one of the discourse relations in the lexicon.

Lexical description for discourse relation Elaboration
S+
k1
σk2 ⊕ σk4 ⊕ [ | wr :εk4 ⊆ εk2 ]
topicsubord(k2, k4)
S−
k2
Γk2 = Γk1
Sk3
Rel⋄
k
elaboration
S−
k4
Γk4 = Γk2 ⊕ σk2
Elaboration expresses that there is a part-whole relation linking the main
eventualities introduced by the two arguments.
There is an additional constraint topicsubord(k2, k4) on the information
structural tier.

Discourse topical constraints
This presupposes a discourse topical hierarchy can be constructed on
the basis of the information structure of input clauses and
characteristic constraints contributed by individual discourse relations.
Cf. Asher & Lascarides (2003), B¨uring, Roberts, van Kuppevelt.
topicsubord(k, k′) means the discourse unit headed by node k ‘topic
subordinates’ the discourse unit headed by node k′.
c.commontopic(k, k′) conveys that the discourse units share a
‘contingent common topic’.

Lexical description for discourse relation Narration
S+
k1
σk2 ⊕ σk3 ⊕ [ | wr :εk2 < εk3 ]
c.commontopic(k2, k3)
S−
k2
Γk2 = Γk1
Rel⋄
k
narration
S−
k3
Narration expresses a temporal sequence: the main eventuality of the
contextual argument temporally precedes the main eventuality of the right
argument.

Lexical description for discourse relation Background
S+
k1
σk2 ⊕ σk4 ⊕ [ | wr :εk2 εk4 ]
topicsubord(k1, k4)
S−
k2
Γk2 = Γk1
Sk3
Rel⋄
k
background
S−
k4
Background expresses that the main eventualities introduced by the two
arguments are overlapping.
There is a subordinating topic relation (very unclear! Cf. A&L 2003).

There must be a discourse relation (underspeciﬁed as yet)
Sr
σ26(σ16)(σ25)
B
S16
B
[o0 | wr : Max o0] ⊑ B
Max opened a can
S
Rel26
S25
Γ25
[σ5 | ], [σ7 | ] ⊑ Γ25
Γ25 |≍[ | wr : money σ7]
24

The hearer selects a discourse relation from his lexicon, on the basis of his
linguistic knowledge of this discourse, world knowledge, and common sense
expectations, e.g. Background:
Sr
[u2 | wr : o0 opened u2, wr : can u2] ⊕ [ | wr : σ5 contained σ7] ⊕ [ | wr :εk2
εk4
]
B
S16
B
[o0 | wr : Max o0] ⊑ B
Max opened a can
S
Rel4
background
topicsubord(k1, k4)
S25
B ⊕ [u2 | wr : o0 opened u2, wr : can u2]
[σ5 | ], [σ7 | ] ⊑ B ⊕ [u2 | wr : o0 opened u2, wr : can u2]
B ⊕ [u2 | wr : o0 opened u2, wr : can u2]|≍[ | wr : money σ7]
24

Multidimensionality/interacting grammatical levels
The selection of a suitable discourse relation and the resolution of
context dependent elements in the new sentence is interdependent.
This is modelled direcly in that the choice of a suitable relation is the
outcome of constraint satisfaction in all grammatical levels (syntax,
semantics, pragmatics) in parallel.

Preferences
A full-fledged discourse grammar must contain means to compute
preferences over interpretations, some form of soft constraints, to
obtain full disambiguation at any stage in an ongoing discourse. cf.
Asher and Lasc. 2003, vLeusen 2007
Given the choice of a specific discourse relation the example
description describes a single verifying tree structure.
Underspecification/ambiguity of interpretation is the case when a
hearer derives two or more equally preferred discourse relations.

Discourse Processing Computing a Discourse Meaning
Computing a Discourse Meaning
The discourse meaning that can be inferred from the description is an
underspeciﬁed DRS:
B ⊕ [ u2 | wr : o0 opened u2, wr : can u2, wr : σ5 contained σ7] ⊕ [ | wr :εk2 εk4 ],
where, among others, the following conditions must be satisﬁed:
(3) a. [o0 | wr : Max o0] ⊑ B
b. [σ5 | ] ⊑ B ⊕ [u2 | wr : o0 opened u2, wr : can u2]
c. σ5 ∈ Tmostsalient
24
d. B ⊕ [u2 | wr : o0 opened u2, wr : can u2]|≍[ | wr : money σ7]
e. [σ7 | ] ⊑ B ⊕ [u2 | wr : o0 opened u2, wr : can u2]

Wrapping up Wrapping up
Some issues w.r.t. current LDG formalism
Processing units are sentences or clauses. What changes if we
implement incrementation per lexeme?
Decidability issue.
Lexical descriptions can contain feature structures, but how do
features interact with semantic composition and local contexts?
Do discourse tree structures represent update history, d-topic
structure, speech acts, or RST-like coherence relations?
How to put a preference system in place?
Can goals, beliefs, intentions, commitments, attitudes of the
discussion participants, dynamics of the utterance situation be
modelled?

Wrapping up Wrapping up
Pointer to further topics
Another slide-show explains in more detail
The integration of the context parameter (local contexts) in the
compositional semantics of the grammar.
The speciﬁcs of anaphora resolution and presupposition ’by constraint
satisfaction’.
and of accommodation and projection by abduction of underspeciﬁed
content or global contextual information.

Wrapping up References
References
Asher & Lascarides 2003. Logics of Conversation.
Buering 2003. On D-Trees, Beans, and B-Accents.
Gardent & Webber 1998. Describing Discourse Semantics.
Gazdar 1979. Pragmatics. Implicature, presupposition and logical form.
Heim 1982. The Semantics of Deﬁnite and Indeﬁnite Noun Phrases.
Karttunen 1974. Presupposition and Linguistic Context.
van Kuppevelt 1995 Discourse Structure, Topicality and Questioning
van Leusen 2007. Description Grammar for Discourse.
van Leusen & Muskens 2003.Construction by Description in Discourse Representation.
Lewis 1979. Score-keeping in a language game.
Muskens 1996. Combining Montague Semantics and Discourse Representation.
Muskens 2001. Talking about Trees and Truth-conditions.
van der Sandt 1992. Presupposition Projection as Anaphora Resolution.
Vijay-Shanker 1992. Using descriptions of trees in a tree-adjoining grammar.
Webber & Joshi 1998. Anchoring a lexicalised tree-adjoining grammar for discourse.

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