Glue Semantics Ash Asudeh March 5, 2011

1 Introduction Glue Semantics is a theory of the syntax–semantics interface and semantic composition (Dalrymple et al. 1993, Dalrymple 1999, 2001, Asudeh 2011), developed principally in light of the non-transformational, constraint-based syntactic framework of Lexical-Functional Grammar (Kaplan and Bresnan 1982, Bresnan 2001, Bresnan et al. 2011, Dalrymple 2001). LFG is committed to the separation of variable aspects of syntax — surface syntax — from more invariant, universal aspects of syntax — abstract syntax. Surface syntax is represented as a constituent structure (c-structure) tree, which is mapped to a representation of abstract syntax — a functional structure (f-structure). F-structure is the level of syntax that feeds semantic interpretation. Glue Semantics was developed as a theory of compositionality that could deal with f-structure syntactic representations, particularly the fact that f-structures are much more abstract than surface syntax representations. This is accomplished by using the commutative resource logic, linear logic (Girard 1987), as a language that ‘glues’ lexical meanings together based on an f-structure parse.

2 Substructural Logics (1)

Three key structural rules a.

Weakening: Premises can be freely added.

b.

Contraction: Additional occurrences of a premise can be freely discarded.

c.

Commutativity: Premises can be freely reordered. Weakening

Contraction

Commutativity

Γ⊢B

Γ, A, A ⊢ B

Γ, A, B ⊢ C

Γ, A ⊢ B

Γ, A ⊢ B

Γ, B, A ⊢ C

Figure 1: Three key structural rules Classical/Intuitionistic Logic A; A → B ⊢ B A; A → B ⊢ B ∧ A Premise A reused, conjoined with conclusion B

Linear Logic A; A ⊸ B ⊢ B A; A ⊸ B 6⊢ B ⊗ A Premise A is consumed to produce conclusion B, no longer available for conjunction with B

Figure 2: Resource Sensitivity: No reuse of premises/resources Classical/Intuitionistic Logic A; B ⊢ A Can ignore premise B

Linear Logic A; B 6⊢ A Cannot ignore premise B

Figure 3: Resource Sensitivity: No discarding premises/resources

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3 Core Natural Deduction Proof Rules (2)

Application : Implication Elimination · · · · · · f : A⊸B a:A ⊸E

f (a) : B

(3)

Abstraction : Implication Introduction [x : A]1 · · · f :B λx .f : A ⊸ B

(4)

⊸I,1

Pairwise substitution : Conjunction Elimination · · · a : A⊗B

[x : A]1 [y : B ]2 · · · f :C

⊗E,1,2

let a be x × y in f : C

4 Examples 4.1 Basic Composition (5) (6)

(7)

(8)

(9)

(10)

Alfie chuckled.   PRED ‘chucklehSUBJi’ h i f SUBJ g PRED ‘Alfie’ a.

alfie : ↑σe

b.

chuckle : (↑

a.

alfie : gσe

b.

chuckle : gσe ⊸ fσt

SUBJ )σe

1. alfie : a 2. chuckle : a ⊸ c alfie : a

Lex. Alfie Lex. chuckled

chuckle : a ⊸ c

chuckle(alfie) : c (11)

⊸ ↑σt

1. alfie : a 2. chuckle : a ⊸ c 3. chuckle(alfie) : c

⊸E

Lex. Alfie Lex. chuckled E ⊸, 1, 2

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4.2 Anaphora (12)



 ‘sayhSUBJ , COMPi’ h i   t PRED ‘Thora’    PRED ‘gigglehSUBJi’  h i  g SUBJ p PRED ‘pro’

PRED

 SUBJ  s   COMP (13)

λz.z × z : (↑σ ANTECEDENT )e ⊸ ((↑σ

(14)

λz .z × z : A ⊸ (A ⊗ P )

(15)

Thora said she giggled.

(16)

ANTECEDENT )e

λuλq.say(u, q) : t⊸g⊸s

1

[x : t] thora : t

λz.z × z : t ⊸ (t ⊗ p)

⊗ ↑σe )

[y : p]

λq.say(x, q) : g⊸s

thora × thora : t ⊗ p

λx.giggle(x) : p⊸g

2

giggle(y) : g

say(x , giggle(y)) : s

let thora × thora be x × y in say(x , giggle(y)) : s say(thora, giggle(thora)) : s

⊗E,1,2

⇒β

4.3 Control (17)



PRED

 SUBJ  t  XCOMP (18)

 ‘tryhSUBJ , XCOMPi’ h i   g PRED ‘Gonzo’  " #  PRED ‘leavehSUBJi’  l SUBJ

Composition with a complement property, property denotation for complement gonzo : g

λx λP .try(x , P ) : g ⊸ (g ⊸ l) ⊸ t

λP .try(gonzo, P ) : (g ⊸ l) ⊸ t

⊸E

leave : g ⊸ l ⊸E

try(gonzo, leave) : t

(19)

Composition with a complement property, propositional denotation for complement gonzo : g

λx λP .try(x , P (x )) : g ⊸ (g ⊸ l) ⊸ t

λP .try(gonzo, P (gonzo)) : (g ⊸ l) ⊸ t try(gonzo, leave(gonzo)) : t

⊸E

leave : g ⊸ l ⊸E

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4.4 Scope (20)



PRED

  s SUBJ (21)



PRED

   SUBJ  s    OBJ (22)

most:

 ‘speakhSUBJi’    PRED ‘president’ h i  p SPEC PRED ‘most’  ‘speakhSUBJ , OBJi’    PRED ‘president’ h i p  SPEC PRED ‘most’      PRED ‘language’  h i  l SPEC PRED ‘some’ D0

(↑ SPEC PRED ) = ‘most’ λRλS .most(R, S ) : [(↑σ VAR)e ⊸ (↑σ RESTR )t ] ⊸ ∀X .[(↑σ e ⊸ Xt ) ⊸ Xt ]

(23)

president : (↑σ

VAR )e

(24)

Most presidents speak.

⊸ (↑σ

RESTR )t

(25)

1. λRλS .most(R, S ) : (v ⊸ r ) ⊸ ∀X .[(p ⊸ X ) ⊸ X ] 2. president ∗ : v ⊸ r 3. speak : p ⊸ s

(26)

λRλS.most(R, S) : (v ⊸ r) ⊸ ∀X.[(p ⊸ X) ⊸ X]

president∗ : v⊸r

λS.most(president∗ , S) : ∀X.[(p ⊸ X) ⊸ X]

speak : p⊸s

most(president ∗ , speak ) : s (27) (28)

Lex. most Lex. presidents Lex. speak

⊸E , [s/X]

Most presidents speak some language. 1. λRλS .most(R, S ) : (v1 ⊸ r1 ) ⊸ ∀X .[(p ⊸ X ) ⊸ X ] 2. president ∗ : v1 ⊸ r1 3. λuλv .speak (u, v ) : p ⊸ l ⊸ s 4. λP λQ .some(P , Q ) : (v2 ⊸ r2 ) ⊸ ∀Y .[(l ⊸ Y ) ⊸ Y ] 5. language : v2 ⊸ r2

Lex. most Lex. presidents Lex. speak Lex. some Lex. language

Based on these premises, we can construct two valid linear logic proofs. Both proofs share the same initial sub-proof, shown in (31). The proofs then diverge, depending on which quantifier is scoped first. The proof in figure 4 provides the surface scope reading and the proof in figure 5 provides the inverse scope reading. For presentational purposes, I have left implicit in figures 4 and 5 the sub-proofs that show the composition of the quantificational determiners with their nominal restrictions; these are presented separately in (29) and (30).

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(29)

λRλS .most(R, S ) : (v1 ⊸ r1 ) ⊸ ∀X .[(p ⊸ X ) ⊸ X ]

president ∗ : v1 ⊸ r1 ⊸E

λS .most(president ∗ , S ) : ∀X .[(p ⊸ X ) ⊸ X ]

(30)

λP λQ .some(P , Q ) : (v2 ⊸ r2 ) ⊸ ∀Y .[(l ⊸ Y ) ⊸ Y ]

language : v2 ⊸ r2 ⊸E

λQ .some(language, Q ) : ∀Y .[(l ⊸ Y ) ⊸ Y ] (31)

[x : p]1

λuλv .speak (u, v ) : p ⊸ l ⊸ s

⊸E

λv .speak (x , v ) : l ⊸ s

[y : l]2

speak (x , y) : s

· · · λS.most(president∗ , S) : ∀X.[(p ⊸ X) ⊸ X]

· · · λQ.some(language, Q) : ∀Y.[(l ⊸ Y ) ⊸ Y ]

⊸E

· · · speak (x , y) : s λy.speak (x , y) : l ⊸ s

some(language, λy.speak (x , y)) : s λx .some(language, λy.speak (x , y)) : p ⊸ s

most(president ∗ , λx .some(language, λy.speak (x , y)))

⊸I,1 ⊸E , [s/Y]

⊸I,2 ⊸E , [s/X]

Figure 4: Surface scope proof

· · · λQ.some(language, Q) : ∀Y.[(l ⊸ Y ) ⊸ Y ]

· · · λS.most(president∗ , S) : ∀X.[(p ⊸ X) ⊸ X]

· · · speak (x , y) : s λx .speak (x , y) : p ⊸ s

most(president ∗ , λx .speak (x , y)) : s λy.most(president ∗ , λx .speak (x , y)) : l ⊸ s

some(language, λy.most(president ∗ , λx .speak (x , y))) Figure 5: Inverse scope proof

⊸I,1 ⊸E , [s/X]

⊸I,2 ⊸E , [s/Y]

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Suggested Further Reading There are two draft chapters from Asudeh (2011) available from the workshop website. Much of the material in these notes is taken from these chapters. The introduction to the Glue chapter contains several references to other relevant work. The best in-depth general introduction is Dalrymple (2001).

Selected References Asudeh, Ash. 2011. The Logic of Pronominal Resumption. Oxford: Oxford University Press. To appear. Bresnan, Joan. 2001. Lexical-Functional Syntax. Oxford: Blackwell. Bresnan, Joan, Ash Asudeh, Ida Toivonen, and Stephen Wechsler. 2011. Lexical-Functional Syntax. Oxford: Wiley-Blackwell, 2nd edn. Forthcoming. Dalrymple, Mary, ed. 1999. Semantics and Syntax in Lexical Functional Grammar: The Resource Logic Approach. Cambridge, MA: MIT Press. Dalrymple, Mary. 2001. Lexical Functional Grammar. San Diego, CA: Academic Press. Dalrymple, Mary, Ronald M. Kaplan, John T. Maxwell III, and Annie Zaenen, eds. 1995. Formal Issues in LexicalFunctional Grammar. Stanford, CA: CSLI Publications. Dalrymple, Mary, John Lamping, and Vijay Saraswat. 1993. LFG Semantics via Constraints. In Proceedings of the Sixth Meeting of the European ACL, 97–105. European Chapter of the Association for Computational Linguistics, University of Utrecht. Girard, Jean-Yves. 1987. Linear Logic. Theoretical Computer Science 50(1): 1–102. Kaplan, Ronald M., and Joan Bresnan. 1982. Lexical-Functional Grammar: A Formal System for Grammatical Representation. In Joan Bresnan, ed., The Mental Representation of Grammatical Relations, 173–281. Cambridge, MA: MIT Press. Reprinted in Dalrymple et al. (1995: 29–135).