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