Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.
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Transcript of Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.
![Page 1: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/1.jpg)
now the extensions…
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NL: a pure logic of residuation
• Axiom
• Transitivity
• Residuation
AA
CAthenCBandBAIf
BCACBACAB /\
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Lambek calculus(sequents)
AA
CBACBA
)]\,[(][
CABCBA
)],/[(][
BAAB/
),(
ABAB\
),(
CBACBA
][)],[(
BABA
),(
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+ 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!
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consequences
[]A A([]A) AA A
L
[]L
A [] A(A) AA A
[]RR
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+ structural postulates
• Example :
1)]),,[((
))],(,[(P
DCBA
DCBA
2)]),,[((
)]),,[((P
DBCA
DCBA
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Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnn
npsssnpsnpnp
Psnpssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
[]/)\),/)\(,((
2)[]),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//(
![Page 8: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/8.jpg)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnn
npsssnpsnpnp
Psnpssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
[]/)\),/)\(,((
2)[]),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//(
![Page 9: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/9.jpg)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
snpssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)[]),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]
![Page 10: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/10.jpg)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),[]),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
![Page 11: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/11.jpg)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),),/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
![Page 12: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/12.jpg)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),,/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
)
![Page 13: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/13.jpg)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),,/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
)
![Page 14: Now the extensions…. NL: a pure logic of residuation Axiom Transitivity Residuation.](https://reader036.fdocuments.us/reader036/viewer/2022062417/55155bca55034691418b48ce/html5/thumbnails/14.jpg)
Example: non peripheral extraction(that I met _ yesterday)
nnnnnnnnnnnnP
sssnpsnpnp
Psssnpsnpnp
Lsssnpnpsnpnp
sssnpnpsnpnp
ss )\) npsnpnp ),/\(,((nn )\(
\\\
, )ss \npsnpnp ),/)\(((
2)),\),/)\(,(((
1)\),,/)\(,(((
)\)),[],/)\((,((
)\)),,/)\((,((....
nn \
nps ),[]//( s / np[]np[]
np[]
)
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adapted proof-nets
• To take restructuring of resources into account
• To represent modalities
• see Richard Moot’s thesis– « Proof Nets for Linguistic Analysis »
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• Uses a contraction criterion (graph-rewriting)
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a/a a/a a/a a/a
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a/a a/a a/a a/a
a/a aa
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a/a a/a a/a a/a
a/a aa
a/a aa
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a/a a/a a/a a/a
a/a aa
a/a aa
a/a aa
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a/a a/a a/a a/a
a/a aa
a/a aa
a/a aa
a/a
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a/a a/a a/a a/a
a/a aa
a/a aa
a/a aa
a/a
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a/a aa
a/aa
a
a/a
a/a
restructuring-1
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a/aa
a
a/aa/a
a/a
restructuring-2
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a/a
a
a/aa/a
contraction step