Problems of syntax-semantics interface ESSLLI 02 Trento.

Problems of syntax-semanticsinterface

ESSLLI 02

Trento

summary

• The need for lambda calculus• From Montague grammar to categorial

grammar• Lambek calculus• Curry-Howard isomorphism• Proof-nets• Extensions (and restrictions) of L• Extended proof-nets

Jackendoff

• Where (narrow) syntax has structural relations such as head-to-complement, head-to-specifier, and head-to-adjunct, conceptual structure has structural relations such as predicate-to-argument, category-to-modifier, and quantifier-to-bound variable. Thus, although conceptual structure undoubtedly constitutes a syntax in the generic sense, its units are not NPs, VPs, etc. […] In particular, unlike syntactic and phonological structures, conceptual structures are purely relational, in the sense that linear order plays no role.

recall:Montague grammars

• Truth-conditional approach:– sentence logical formula (true or false)– noun phrase term (constant, variable,

complex term)

• But what for other linguistic expressions?– verb open atomic formula?– but how to combine?

• kiss(x,y) composed with p and m gives:

kiss(p,m) or kiss(m,p)?

fortunately : lambda calculus

• constants, variables : -terms

• If M and N are -terms, then (M N) [or M(N)] is a -term,(application)

• If M is a -term and if x is a variable, then x.M is a -term (abstraction)

• + -reduction : (x.M, N) M[N/x]

• Goal : x (child(x) play(x))• Identical to :(P.[x (enfant(x) P(x))] u.play(u))• therefore :every child = P.[x (child(x) P(x))]• Identical to :(Q.P.[x (Q(x) P(x))] v.child(v))• therefore:every = Q.P.[x (Q(x) P(x))]

Example : how to extract the « meaning » of quantifiers?

other quantifiers

• a, an = Q.P.[x (Q(x)P(x))]

• no = Q.P.[x (Q(x)P(x))]

But we cannot apply anything to anything…

• x is a -term• (x x) is a -term x.(x x) is a -term • (x.(x x) x.(x x)) is a -term

• But• (x.(x x) x.(x x)) (x.(x x) x.(x x))• (no end to the reduction : the normalisation

process does not stop)

• « Intransitive verbs » apply to nominal entities (and they give propositions)

• « Transitive verbs » apply to nominal entities (and they give intransitive verbs…)

• « Propositional verbs » apply to propositions (and they give propositions)

• « Adjectives » apply to nominal entities (and they give nominal entities)

Typed -calculus

• Constants and variables of type a are -terms of type a

• if M is a -term of type <a, b> and N a -term of type a, then (M N) is a -term of type b

• If M is a -term of type b and if x is a variable of type a, then x. M is a -term of type <a, b>

• In other words:

b

aba

ba

b

a

...

][

Correspondance syntactic categories – semantic types

• sentences• VP, IV• NP, PN• TT• verbal adverbs VI/VI• CN (common noun)• sentential adverbs• preposition• propositional verb• intentional verb• article

• t• <e, t>• e ou bien <<e,t>, t>• <<<e,t>,t>,<e,t>>• <<e,t>, <e, t>>• <e, t>• <t, t>• <<<e,t>,t>, <<e,t>, <e, t>>>

• <t, <e, t>>• <<e,t>, <e, t>>• <<e, t>, <<e,t>, t>>

syntax

• For each syntactic category A, the set PA of all expressions of category A contains at least the set BA of the « dictionary words » of category A,

• If PA and if PB, then, in some cases to enumerate, F(,) for some function F belongs to some set PC.

Example of rule

• S2 : if PT/CN and if PCN, then, F2(,) PT, where F2(,) = *, where * = except if is equal to a and if the first word of begins by a vowel, in which case * = an

• Remark : T is the category of terms, example : a man, an aristocrat

Example of rule

• S4 : if PT and if PVI, then F4(, ) Pt,

• where F4(, ) = *, where * is obtained from by replacing the first verb by its 3rd person singular form

• Example : = John, = walk,

• F4(, ) = John walks

Montagovian analysis

• John seeks a unicorn• S1 : aT/CN, unicornCN

• S2 : F2(a, unicorn) = a unicorn T

• S1 : seek VI/T

• S5 : F5(seek, a unicorn) = seek a unicorn VI

• S1 : John T

• S4 : F4(John, seek a unicorn) = John seeks a unicorn t

John seeks a unicorn

John seek a unicorn

a unicornseek

a unicorn

Second analysis !

• John seeks a unicorn

• S1 : seek VI/T, he1 T

• S5 : F5(seek, he1) = seek him1 VI

• S4 : F4(John, seek him1) = John seeks him1 t

• S2 : F2(a, unicorn) = a unicorn T

• S14 : F14,1(a unicorn, John seeks him1) = John seeks a unicorn t

John seeks a unicorn

John

John seeks him1

seek

a unicorn

a unicorn seek him1

him1

remark

• In a « modern» grammar (cf. GPSG in the eighties), syntagmatic rules are put in correspondance with some semantic counterpart,

• In a « logical » grammar (eg. Lambek grammars), the correspondance automatically follows from a known isomorphism between logical derivations and -terms (Curry-Howard)

Syntagmatic grammar

• S SN SV• SN Det N• SN Np• SV Vi• SV Vt SN• SV Vp que S• SV Vint SV

(S) = ((SN) (SV)) (SN) = ((Det) (N)) (SN) = (Np) (SV) = (Vi) (SV) = (SN) o (Vt) (SV) = ((Vp) (S)) (SV) = (SV)o(Vint)

• Det chaque | tout• Det un • N enfant | ballon• Np stéphane• Vi joue• Vt cherche• Vp dit• Vint essaie

(tout) = Q.P.[x (Q(x) P(x))] (un) = Q.P.[x (Q(x)P(x))] (enfant) = x.enfant(x) (stéphane) = P.P(stéphane) (joue) = x.joue(x) (cherche) = x. y.cherche(x, y) (dit) = P. x. dit(x,P) (essaie) = x. P.essaie(x, P)

lexical rules

Example : stéphane cherche un ballon

SN

Det N

un ballon

x. ballon(x)Q.P.x[Q(x)P(x)]

(Q.P.x[Q(x)P(x)] x. ballon(x))P.x[(x. ballon(x) x)P(x)] P.x[ballon(x)P(x)]

Example : stéphane cherche un ballon

SN

Det N

un ballon

P.x[ballon(x)P(x)]

SV

Vt

x.y. chercher(x,y)

Example : stéphane cherche un ballon

SN

Det N

un ballon

P.x[ballon(x)P(x)]

SV

Vt

x.y. chercher(x,y)

Composition : (x.f(x)) o (y.g(y)) = z. (x.f(x), (y.g(y), z))

z. (P.x[ballon(x)P(x)],(x.y. chercher(x,y)z)) z. (P.x[ballon(x)P(x)], y. chercher(z,y)) z. x[ballon(x) (y. chercher(z,y), x)], z. x[ballon(x) chercher(z,x)]

Example : stéphane cherche un ballon

SN

Det N

un ballon

P.x[ballon(x)P(x)]

SV

Vt

x.y. chercher(x,y)

z. x[ballon(x) chercher(z,x)]

S

SN

NpStéphaneP. P(stéphane)

Example : stéphane cherche un ballon

SN

Det N

un ballon

P.x[ballon(x)P(x)]

SV

Vt

x.y. chercher(x,y)

z. x[ballon(x) chercher(z,x)]

S

SN

NpStéphaneP. P(stéphane)

(P. P(stéphane) z. x[ballon(x) chercher(z,x)])(z. x[ballon(x) chercher(z,x)] stéphane) x[ballon(x) chercher(stéphane,x)]

Example : stéphane cherche un ballon

SN

Det N

un ballon

P.x[ballon(x)P(x)]

SV

Vt

x.y. chercher(x,y)

z. x[ballon(x) chercher(z,x)]

S

SN

NpStéphaneP. P(stéphane)

x[ballon(x) chercher(stéphane,x)]