Towards Glue Semantics for Minimalist Syntax
Towards Glue Semantics for Minimalist Syntax
Matthew Gotham
Department of Linguistics
University College London
Cambridge Syntax Cluster event:
‘Interactions between syntax and semantics across frameworks’
8 January 2015
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Towards Glue Semantics for Minimalist Syntax
Plan for today
Glue semantics
Minimalist syntax
Putting the two together
So what?
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
Glue
An approach to the syntax-semantics interface
I
Compatible with different syntactic theories: LFG (Dalrymple, 1999),
HPSG (Asudeh and Crouch, 2002), LTAG (Frank and Genabith,
2001), CG. . .
I
Compatible with different meaning representations: IL, DRT,
situation semantics. . .
So this talk should really be called ‘towards glue semantics for minimalist
syntax and the lambda calculus’
A good introduction is provided by Lev (2007, Ch. 3).
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
Basic points
I
Lexicon+syntax produces premises in a fragment of linear logic (the
glue language), each of which is paired with a lambda term
I
Semantic interpretation uses deduction to assemble final meaning
from these premises
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
Linear logic
Classical logic proof rules:
conclusion follows from some
subset of the set of premises
P → Q, P → (Q → R) ` P → R
P, Q ` Q
Linear logic proof rules:
conclusion follows from the
multiset of premises
P ( Q, P ( (Q ( R) 0 P ( R
P, Q 0 Q
Linear logic keeps a strict accounting of the number of times a premise is
used in a proof. But it doesn’t care about how they are ordered or
grouped.
P, P ( Q ` Q
P ( Q, P ` Q
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
The Curry-Howard Correspondence
There’s a mapping between (intuitionistic) linear logic inference rules and
operations on meaning terms.
(-elimination (linear modus ponens) corresponds to function application.
f :A(B
x :A (
E
f (x) : B
And (-introduction (linear conditional proof) corresponds to
λ-abstraction
[x : A]n
..
..
f :B
(I n
λx.f : A ( B
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
A simple example
(1) Sam trains Rover
Sam trains Rover
s0 : B
t
* 0
r :A
λy .λx.train0 (x, y ) : A ( (B ( C )
λy .λx.train0 (x, y ) : A ( (B ( C ) r 0 : A
(E
λx.train0 (x, r 0 ) : B ( C
train0 (s 0 , r 0 ) : C
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s0 : B
Towards Glue Semantics for Minimalist Syntax
(E
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
An ambiguous sentence
(2) A boy trains every dog
A boy trains every dog
λP.some0 (boy0 , P) :
(B ( C ) ( C
M Gotham (UCL)
v
λy .λx.train0 (x, y ) :
A ( (B ( C )
(
λQ.every0 (dog0 , Q) :
(A ( C ) ( C
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
Surface scope
λy .λx.train0 (x, y )
: A ( (B ( C ) [y : A]1
(E
B(C
[x : B]2
(E
λQ.every0 (dog0 , Q)
C
(I 1
A(C
: (A ( C ) ( C
(E
every0 (dog0 , (λy .train0 (x, y ))) : C
λP.some0 (boy0 , P)
2
(I
: (B ( C ) ( C
B(C
(E
0
0
0
0
0
some (boy , λx.every (dog , λy .train (x, y ))) : C
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Towards Glue Semantics for Minimalist Syntax
Glue semantics
Inverse scope
λy .λx.train0 (x, y )
1
λP.some0 (boy0 , P) : A ( (B ( C ) [A] (
E
B(C
: (B ( C ) ( C
(E
some0 (boy0 , λx.train0 (x, y )) : C
λQ.every0 (dog0 , Q)
(I 1
A(C
: (A ( C ) ( C
(E
0
0
0
0
every (dog , λy .some (boy , λx.train0 (x, y ))) : C
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
Lexical items
I
Lexical items (LIs) are bundles of features.
I
Some features describe what an LI is.
I
Some features describe what an LI needs (uninterpretable features).
Those can be strong(*) or weak.
V
huD, uDi
|
train
T
huD*i
(I’m going to ignore morphosyntactic features and agreement.)
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
Structure-building operation(s)
Merge.
I
I
Hierarchy of Projections-driven.
Selectional features-driven.
I
I
External.
Internal.
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
Hierarchy of projections
Adger (2003)
Clausal:
Nominal:
Adjectival:
We’ll use:
Clausal:
Nominal:
has:
C i T i (Neg) i (Perf) i (Prog) i (Pass) i v i V
D i (Poss) i n i N
(Deg) i A
TiV
DiN
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
HoPs merge
A
h. . .i
A
h. . .i
+
B
hi
⇒
A B
hi hi
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Where A and B are in the
same hierarchy of projections
(HoPs) and A is higher on that
HoPs than B
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
Select merge
External
A
h. . .i
A
huB, . . .i
+
B
hi
⇒
B
A
huBi h i
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
Select merge
Internal
A
h. . .i
A
huB*i
A
huB*, . . .i
⇒
... B ...
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... B ...
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
External merge
An example
V
V
huDi
D
⇒
+
Bill
D
V
huDi
smiles
Bill smiles
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
HoPs merge
An example
T
huD*i
V
T
huD*i
V
T
+
D
V
huDi
⇒
D
V
huDi
Bill smiles
Bill smiles
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Towards Glue Semantics for Minimalist Syntax
Minimalist syntax
Internal merge
An example
T
T
huD*i
T
huD*i
V
T
⇒
D
V
huDi
V
T
D
V
huDi
Bill smiles
Bill smiles
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Indices on features
Every feature (interpretatble or uninterpretable) bears a numerical index
subject to the following contstraints:
I
I
The indices assigned to features within a single lexical item must all
be distinct.
E.g. in this:
Vi
huDj , uDk i
We must have i 6= j 6= k
|
train
Structure-building operations are sensitive to indices in the following
ways:
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
HoPs merge
with indices
Ai
h. . .i
+
Bi
hi
Ai
h. . .i
⇒
Ai Bi
hi hi
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Where A and B are in the
same hierarchy of projections
(HoPs) and A is higher on that
HoPs than B
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
External merge
with indices
Ai
huBj , . . .i
+
Bj
hi
Ai
h. . .i
⇒
Bj
Ai
huBj i h i
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Internal merge
with indices
Ai
h. . .i
Ai
huBj *i
Ai
huBj *, . . .i
⇒
. . . Bj . . .
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. . . Bj . . .
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Meaning constructors
Following Kokkonidis (2008), we’ll use a fragment of (monadic) first-order
linear logic as the glue language.
I
Predicates: e, eN and t.
I
Constants: 1, 2, 3 . . .
I
Variables: X , Y , Z . . .
I
Connectives: ( and ∀
We need a new rule of inference:
f : ∀X (P)
∀E
f : P[a/X ]
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Lexical items
Some examples
Sam
Syntax
Semantics
Ni
s 0 : e(i)
trains
Syntax
Semantics
Vi
huDj , uDk i
λx.λy .train0 (y , x) : e(j) ( (e(k) ( t(i))
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Lexical items
Some more examples
every
Syntax
Semantics
Di
λP.λQ.every0 (P, Q) :
(eN (i) ( t(i)) ( ∀X ((e(i) ( t(X )) ( t(X ))
dog
Syntax
Semantics
Ni
λx.dog0 (x) : eN (i) ( t(i)
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
a
Syntax
Semantics
Di
λF .λG .some0 (F , G ) :
(eN (i) ( t(i)) ( ∀X ((e(i) ( t(X )) ( t(X ))
boy
Syntax
Semantics
Ni
λx.boy0 (x) : eN (i) ( t(i)
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
T1
T1
huD2 *i
V1
T1
V1
huD2 i
D2
D2
N2
V1
huD3 , uD2 i
a
boy
trains
D3
D3
N3
every dog
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
The mapping to interpretation
The multiset of premises:
I
λP.λQ.some0 (P, Q) :
(eN (2) ( t(2)) ( ∀X ((e(2) ( t(X )) ( t(X ))
I
boy0 : eN (2) ( t(2)
I
λx.λy .train0 (y , x) : e(3) ( (e(2) ( t(1))
I
λF .λG .every0 (F , G ) :
(eN (3) ( t(3)) ( ∀Y ((e(3) ( t(Y )) ( t(Y ))
I
dog0 : eN (3) ( t(3)
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Solving from the multiset of premises
λP.λQ.some0 (P, Q) :
boy0 :
(eN (2) ( t(2)) ( ∀X ((e(2) ( t(X )) ( t(X )) eN (2) ( t(2)
(E
λQ.some0 (boy0 , Q) : ∀X ((e(2) ( t(X )) ( t(X ))
∀E
λQ.some0 (boy0 , Q) : (e(2) ( t(1)) ( t(1)
λF .λG .every0 (F , G ) :
dog0 :
(eN (3) ( t(3)) ( ∀X ((e(3) ( t(X )) ( t(X )) eN (2) ( t(2)
(E
λG .every0 (dog0 , G ) : ∀X ((e(3) ( t(X )) ( t(X ))
∀E
λG .every0 (dog0 , G ) : (e(3) ( t(1)) ( t(1)
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Surface scope
λy .λx.train0 (x, y )
: e(3) ( (e(2) ( t(1)) [y : e(3)]1
(E
e(2) ( t(1)
[x : e(2)]2
(E
t(1)
λQ.every0 (dog0 , Q)
(I 1
e(3) ( t(1)
: (e(3) ( t(1)) ( t(1)
(E
every0 (dog0 , (λy .train0 (x, y ))) : t(1)
λP.some0 (boy0 , P)
(I 2
: (e(2) ( t(1)) ( t(1)
e(2) ( t(1)
(E
some0 (boy0 , λx.every0 (dog0 , λy .train0 (x, y ))) : t(1)
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Towards Glue Semantics for Minimalist Syntax
Putting the two together
Inverse scope
λy .λx.train0 (x, y )
: e(3) ( (e(2) ( t(1)) [y : e(3)]1
λP.some0 (boy0 , P)
(E
: (e(2) ( t(1)) ( t(1)
e(2) ( t(1)
(
E
some0 (boy0 , λx.train0 (x, y )) : t(1)
λQ.every0 (dog0 , Q)
(I 1
e(3) ( t(1)
: (e(3) ( t(1)) ( t(1)
(E
every0 (dog0 , λy .some0 (boy0 , λx.train0 (x, y ))) : t(1)
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Towards Glue Semantics for Minimalist Syntax
So what?
Embedded QNPs
(3) No owner of a coffee machine drinks tea.
N1
P2
N1
huP2 i
owner
λx.λy .own0 (y , x) :
e(2) ( (eN (1) ( t(1))
λv .v : e(3) ( e(2)
P2
huD3 i
D3
of
a coffee
machine
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λP.some0 (cof-mach0 , P) :
∀X ((e(3) ( t(X )) ( t(X ))
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Towards Glue Semantics for Minimalist Syntax
So what?
λx.λy .own0 (y , x) :
λv .v :
e(2) ( (eN (1) ( t(1)) e(3) ( e(2)
HS
∀X ((e(3) ( t(X )) ( t(X ))
λv .λy .own0 (y , v ) :
λP.some0 (cof-mach0 , P) :
e(3) ( (eN (1) ( t(1))
∀E
Perm
(e(3) ( t(1)) ( t(1)
λy .λv .own0 (y , v ) :
λP.some0 (cof-mach0 , P) :
eN (1) ( (e(3) ( t(1))
HS
λy .some0 (cof-mach0 , λv .own0 (y , v )) : eN (1) ( t(1)
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Towards Glue Semantics for Minimalist Syntax
So what?
Phases
I
General idea: at certain points in the tree you must use the multiset
of premises you have in a proof with a conclusion of a particular type.
I
Scope islands: impossible interpretation would involve failling to
compute proof at one of those points.
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Towards Glue Semantics for Minimalist Syntax
So what?
References
Adger, David (2003). Core Syntax: A Minimalist Approach. Core Linguistics.
Oxford: Oxford University Press.
Asudeh, Ash and Richard Crouch (2002). “Glue Semantics for HPSG”. In:
Proceedings of the 8th International HPSG Conference. (Norwegian University of
Science and Technology, Trondheim). Ed. by Frank van Eynde, Lars Hellan, and
Dorothee Beermann. Stanford, CA: CSLI Publications.
Dalrymple, Mary, ed. (1999). Semantics and Syntax in Lexical Functional
Grammar: The Resource Logic Approach. Cambridge, MA: MIT Press.
Frank, Anette and Josef van Genabith (2001). “GlueTag: Linear Logic-based
Semantics for LTAG—and what it teaches us about LFG and LTAG”. In:
Proceedings of the LFG01 Conference. (University of Hong Kong). Ed. by
Miriam Butt and Tracy Holloway King. Stanford, CA: CSLI Publications.
Kokkonidis, Miltiadis (2008). “First-Order Glue”. In: Journal of Logic, Language
and Information 17 (1), pp. 43–68.
Lev, Iddo (2007). “Packed Computation of Exact Meaning Representations”.
PhD thesis. Stanford University.
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