Title: Problems of syntaxsemantics interface
1Problems of syntax-semanticsinterface
2summary
- 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
3Jackendoff
- 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.
4recallMontague 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)?
5fortunately 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
6Example 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))
7other quantifiers
- a, an ?Q.?P.?x (Q(x)?P(x))
- no ?Q.?P.??x (Q(x)?P(x))
8But 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)
9 - 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)
10Typed ?-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
11 12Correspondance 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
13syntax
- 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.
14Example 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
15Example 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
16Montagovian 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
17John seeks a unicorn
18Second 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
19John seeks a unicorn
John seeks him1
John
seek him1
seek
him1
20remark
- 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)
21Syntagmatic 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)
22lexical 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)
23Example 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)
24Example stéphane cherche un ballon
SV
?P.?xballon(x)?P(x)
Vt
SN
?x.?y. chercher(x,y)
N
Det
un
ballon
25Example 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
26Example 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
27Example 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
28Example 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