Proof-nets and semantic applications

1
Proof-nets and semantic applications

• Alain Lecomte
• ESSLLI2002

2
Semantic proof nets

• child

xe, child e?t - child(x) t hence child
e?t - ?x.child(x)e?t
3

• run

e
t-
x
e-
t
?x
child
4

• run

e
t-
x
e-
t
?x
child
5

• run

e
t-
x
e-
t
?x
child
6

• run

e
t-
x
e-
t
x
?x
child
?x.child(x)
7

• run

e
t-
x
e-
t
x
?x
child
?x.child(x)
8
each, every

• A determiner like every, each decomposes into
• A quantifier, for instance ?
• type (e?t)?t
• A connective, for instance ?
• type t?(t?t)

9

• needs two predicates (e ? t) for obtaining one
  proposition (t)
• A determiner is therefore of type
• (e?t)?((e?t)?t)

10

• A determiner is therefore associated with a
  sequent
• Its semantic is represented by its proof

11
deduction
12
remark

• With a very remarkable step an application of
  the contraction rule!
• ? necessity of working inside Intuitionistic
  linear logic with exponentials
• The exact sequent which encodes the determiner is
  

!e
!e
)
)
((
)
(
)
(
),
(
t
t
t
t
t
!e
t
t
t
--
--o
--o
--o
--o
--o
--o
--o
--o
13
Exponentials
14
Exponentials (one-sided)
15
Representation of the proof
c
?
?
(!e o t) o ((!e o t) o t)
16
every child
c
child
(!e o t) o t
?
?
17
every child likes to play
c
likes to play
t
child
?
?
18
Application

-

A
A o B
19
Application
A -
B
20
Abstraction
B -
21
Abstraction
B -
B o A
22
Syntactic proof-nets

• Proof-nets for Lambek calculus
• Like PN for MILL
• condition on semi-planarity

23
every child plays

• 1) unfolding

np -
s
s -
np\s
np
(s/(np\s)) -
n
s -
n - child
np\s - plays
s
(s/(np\s))/n - every
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every child plays

• 2) links

np -
s
s -
np\s
np
(s/(np\s)) -
n
s -
n - child
np\s - plays
s
(s/(np\s))/n - every
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Attention!

• 2) links

WRONG !
np -
s
s -
np\s
np
(s/(np\s)) -
n
s -
n - child
np\s - plays
s
(s/(np\s))/n - every
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Parsing

• through homomorphism
• H(s) t
• H(np) !e
• H(n) !e o t
• H(A/B) H(B\A) H(B) o H(A)

27
every child plays

np -
s
s -
np\s
np
(s/(np\s)) -
n
s -
n - child
np\s - plays
s
(s/(np\s))/n - every
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every child plays

• 3) homomorphism

!e
t
t
!e o t
!e
(!e o t) o t
!e o t
t
!e o t child
!e o t plays
t
(!e o t) o ((!e o t) ot)) every
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• semantic recipes
• child ?x.child(x)
• every ?P.?Q.?(?x.(P(x)?Q(x))
• plays ?x.play(x)

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• represented by proof-nets

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• represented by proof-nets

d
32
every
c
?
?
(!e o t) o ((!e o t) o t)
33
plugging lexical semantic types to the
homomorphic PN by cut
34

!e
t
t
!e o t
!e
(!e o t) o t
!e o t
t
!e o t child
!e o t plays
t
(!e o t) o ((!e o t) ot)) every
CUT
35

!e
t
t
!e o t
!e
t
!e
(!e o t) o t
t
!e o t plays
t
(!e o t) o ((!e o t) ot)) every
d
child
36

!e
t
t
!e o t
!e
t
!e
(!e o t) o t
t
!e o t plays
t
(!e o t) o ((!e o t) ot)) every
d
child
CUT
37

!e
t
d
t
!e o t
!e
t
!e
(!e o t) o t
t
plays
t
(!e o t) o ((!e o t) ot)) every
d
child
38

PNevery
!e
t
d
t
!e o t
!e
t
!e
(!e o t) o t
t
plays
t
(!e o t) o ((!e o t) ot)) every
d
child
CUT
39

!e
t
c
plays
t
d
?
?
child
40

!e
t
c
plays
t
d
?
?
child
41

!e
t
c
plays
t
d
?
?
?
child
42

!e
t
c
plays
?x
t
d
?
?
?
child
43

!e
t
c
plays
?x
t
d
?
?
?
?
child
44

!e
t
c
plays
?x
plays
t
d
?
?
?
?
child
45

!e
t
c
plays
?x
plays
t
d
?
?
child
?
?
child
46

child(x)
!e
t
c
plays
?x
plays
t
d
?
?
child
?
?
child
47

plays(x)
child(x)
!e
t
c
plays
?x
plays
t
d
?
?
child
?(?x.(?(child(x),plays(x))))
?
?
child
48
Logical synthesisfrom a formula to a sentence

• the reverse story
• Start
• a semantic formula ?
• semantic recipes for lexical entries ?1, ?2,
  ?n
• Goal
• A sentence using all these recipes the meaning of
  which is ?

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Usual solutions?-term unification
? skiss(p,m)
Peter np p
kisses (np\s)/np ?x.?y.kiss(y,x)
Mary np m
GOAL
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s-
np
?
kiss(?,?)
np
?y.kiss(y, ?)
?
? skiss(p,m)
Peter np- p
kisses (np\s)/np ?x.?y.kiss(y,x)
Mary np- m
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s-
np
kiss(?,?)
?
np
?y.kiss(y, ?)
?
? skiss(p,m)
Peter np- p
kisses (np\s)/np ?x.?y.kiss(y,x)
Mary np- m
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kiss(?,?) kiss(p,m) ? ? m ? p
s-
np
?
kiss(?,?)
np
?y.kiss(y, ?)
?
? skiss(p,m)
Peter np- p
kisses (np\s)/np ?x.?y.kiss(y,x)
Mary np- m
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kiss(?,?) kiss(p,m) ? ? m ? p
? m
s-
np
?
kiss(?,?)
np
?y.kiss(y, ?)
?
? skiss(p,m)
Peter np- p
kisses (np\s)/np ?x.?y.kiss(y,x)
Mary np- m
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kiss(?,?) kiss(p,m) ? ? m ? p
? m
? p
s-
np
?
kiss(?,?)
np
?y.kiss(y, ?)
?
? skiss(p,m)
Peter np- p
kisses (np\s)/np ?x.?y.kiss(y,x)
Mary np- m
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The net is a proof-net for L, therefore the
correct sentence is Peter kisses Mary
kiss(?,?) kiss(p,m) ? ? m ? p
? m
? p
s-
np
?
kiss(?,?)
np
?y.kiss(y, ?)
?
? skiss(p,m)
Peter np- p
kisses (np\s)/np ?x.?y.kiss(y,x)
Mary np- m
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But

• Non decidability for second order
• xf f(fa) ?
• no mgu
• two incomparable solutions
• ?z.z(fa)
• ?z.f(za)

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the use of PN

• Generation (or synthesis )
• research of a proof in L (which is decidable!),
  helped by a semantic form

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The problem

• Find PNL such that
• plug by cuts lexical PNs PN?1, PN?2, PN?n
• cut-elimination
• Given PN?

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?
APNL
T1
T2
Tn
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?
AH(PNL)
HT1
HT2
HTn
CUT
CUT
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PN?
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the execution formula

• Let P be a PN, U the set of its axiom links, ?
  the set of its cut links
• u incidence matrix of U
• ? incidence matrix of ?

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Cut elimination between axioms
e1
e2
e3
e4
e1
e2
e3
e4
e1
e2
e3
e4
- X, X?
- X, X?
CUT
- X, X?
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• after a first step of cut-elimination on axioms
  u replaced by u?u
• links coming from cut-elimination of level 1
• To suppress all the links the premisses of which
  are premisses of a new cut and all the links
  which had no incident cut
• u?u - ?2u?u? - u?u?2 ?2u?2
• (1- ?2)u?u(1- ?2)

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• Idem for links coming from elimination of level
  2, 3, , n, cuts
• Resulting graph after cut-elimination

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?
U
HT1
HT2
HTn
CUT
?
CUT
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Res(U,?)
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• U can be calculated from Res(U,?) and ?
• Cf. PhD thesis by Sylvain POGODALLA
• http//www.xrce.xerox.com/pogodalla
• Condition that each lexical semantics contain
  at least one constant which intervenes in the
  global semantic representation.