Problems of syntaxsemantics interface

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Problems of syntax-semanticsinterface

• ESSLLI 02
• Trento

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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

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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.

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recallMontague 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)?

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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) ? MN/x

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Example how to extract the  meaning  of
quantifiers?

• 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))

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other quantifiers

• a, an ?Q.?P.?x (Q(x)?P(x))
• no ?Q.?P.??x (Q(x)?P(x))

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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)

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•  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)

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Typed ?-calculus

• Constants and variables of type a are ?-terms of
  type a
• if M is a ?-term of type lta, bgt 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 lta, bgt

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• In other words

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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
• lte, tgt
• e ou bien ltlte,tgt, tgt
• ltltlte,tgt,tgt,lte,tgtgt
• ltlte,tgt, lte, tgtgt
• lte, tgt
• ltt, tgt
• ltltlte,tgt,tgt, ltlte,tgt, lte, tgtgtgt
• ltt, lte, tgtgt
• ltlte,tgt, lte, tgtgt
• ltlte, tgt, ltlte,tgt, tgtgt

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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.

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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

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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

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Montagovian analysis

• John seeks a unicorn
• S1 a?T/CN, unicorn?CN
• 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

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John seeks a unicorn
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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

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John seeks a unicorn
John seeks him1
John
seek him1
seek
him1
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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)

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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)

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lexical rules

• 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)

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Example stéphane cherche un ballon

• (?Q.?P.?xQ(x)?P(x) ?x. ballon(x))
• ?P.?x(?x. ballon(x) x)?P(x)
• ?P.?xballon(x)?P(x)

SN
N
Det
un
ballon
?Q.?P.?xQ(x)?P(x)
?x. ballon(x)
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Example stéphane cherche un ballon
SV
?P.?xballon(x)?P(x)
Vt
SN
?x.?y. chercher(x,y)
N
Det
un
ballon
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Example stéphane cherche un ballon

• ?z. (?P.?xballon(x)?P(x),(?x.?y. chercher(x,y)
• z)) ? ?z. (?P.?xballon(x)?P(x), ?y.
  chercher(z,y))
• ?z. ?xballon(x)? (?y. chercher(z,y), x),
• ?z. ?xballon(x)? chercher(z,x)

Composition (?x.f(x)) o (?y.g(y)) ?z.
(?x.f(x), (?y.g(y), z))
SV
?P.?xballon(x)?P(x)
Vt
SN
?x.?y. chercher(x,y)
N
Det
un
ballon
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Example stéphane cherche un ballon
S

• ?z. ?xballon(x)? chercher(z,x)

SV
SN
?P.?xballon(x)?P(x)
Vt
SN
?x.?y. chercher(x,y)
N
Det
Np Stéphane ?P. P(stéphane)
un
ballon
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Example stéphane cherche un ballon
S

• (?P. P(stéphane) ?z. ?xballon(x)?
  chercher(z,x))
• (?z. ?xballon(x)? chercher(z,x) stéphane)
• ?xballon(x)? chercher(stéphane,x)

• ?z. ?xballon(x)? chercher(z,x)

SV
SN
?P.?xballon(x)?P(x)
Vt
SN
?x.?y. chercher(x,y)
N
Det
Np Stéphane ?P. P(stéphane)
un
ballon
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Example stéphane cherche un ballon
?xballon(x)? chercher(stéphane,x)
S

• ?z. ?xballon(x)? chercher(z,x)

SV
SN
?P.?xballon(x)?P(x)
Vt
SN
?x.?y. chercher(x,y)
N
Det
Np Stéphane ?P. P(stéphane)
un
ballon