now the extensions…
Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.
Mar 27, 2015
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now the extensions…
NL: a pure logic of residuation
• Axiom
• Transitivity
• Residuation
AA
CAthenCBandBAIf
BCACBACAB /\
Lambek calculus(sequents)
AA
CBACBA
)]\,[(][
CABCBA
)],/[(][
BAAB/
),(
ABAB\
),(
CBACBA
][)],[(
BABA
),(
+ modalities
RAA
)( L
BABA
][])[(
RAA
[][])( L
BABA
[]
])[([]][
If brings you an A, then whenprovided with the structure S, it gives youan A with the structure S
If with S brings you an A, then without itit gives you an A which lacks S!
consequences
[]A A([]A) AA A
L
[]L
A [] A(A) AA A
[]RR
+ structural postulates
• Example :
1)]),,[((
))],(,[(P
DCBA
DCBA
2)]),,[((
)]),,[((P
DBCA
DCBA
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnn
npsssnpsnpnp
Psnpssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
[]/)\),/)\(,((
2)[]),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//(
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnn
npsssnpsnpnp
Psnpssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
[]/)\),/)\(,((
2)[]),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//(
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
snpssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)[]),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),,/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),,/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),,/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
)
adapted proof-nets
• To take restructuring of resources into account
• To represent modalities
• see Richard Moot’s thesis– « Proof Nets for Linguistic Analysis »
• Uses a contraction criterion (graph-rewriting)
a/a a/a a/a a/a
a/a a/a a/a a/a
a/a aa
a/a a/a a/a a/a
a/a aa
a/a aa
a/a a/a a/a a/a
a/a aa
a/a aa
a/a aa
a/a a/a a/a a/a
a/a aa
a/a aa
a/a aa
a/a
a/a a/a a/a a/a
a/a aa
a/a aa
a/a aa
a/a
a/a aa
a/aa
a
a/a
a/a
restructuring-1
a/aa
a
a/aa/a
a/a
restructuring-2
a/a
a
a/aa/a
contraction step
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