now the extensions…

NL: a pure logic of residuation Axiom Transitivity Residuation

Lambek calculus (sequents)

+ modalities If brings you an A, then when provided with the structure S, it gives you an A with the structure S If with S brings you an A, then without it it gives you an A which lacks S!

consequences [] A A ([] A) A A L [] L A [] A (A) A A [] R R

+ structural postulates Example :

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn npsss s P s ss s P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \ []/)\),/)\(,(( 2 )[]),\ /)\(,((( 1 )\),[]),/)\(,((( )\)),[],/)\((, )\)),,/)\((,.... nn\ nps),[]//(

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn npsss s P s ss s P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \ []/)\),/)\(,(( 2 )[]),\ /)\(,((( 1 )\),[]),/)\(,((( )\)),[],/)\((, )\)),,/)\((,.... nn\ nps),[]//(

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn P snpss s P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \,)ss\nps ),/)\((( 2 )[]),\ /)\(,((( 1 )\),[]),/)\(,((( )\)),[],/)\((, )\)),,/)\((,.... nn\ nps),[]//( s/ np[]

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn P sssnps P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \,)ss\nps ),/)\((( 2 )),\ /)\(,((( 1 )\),[]),/)\(,((( )\)),[],/)\((, )\)),,/)\((,.... nn\ nps),[]//( s/ np[] np[]

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn P sssnps P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \,)ss\nps ),/)\((( 2 )),\ /)\(,((( 1 )\), /)\(,((( )\)),[],/)\((, )\)),,/)\((,.... nn\ nps),[]//( s/ np[] np[] np[]

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn P sssnps P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \,)ss\nps ),/)\((( 2 )),\ /)\(,((( 1 )\),),,/)\(,(((((( )\)),)),[],/)\(,( )\)),,/)\((,.... nn\ nps),[]//( s/ np[] np[] np[] )

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn P sssnps P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \,)ss\nps ),/)\((( 2 )),\ /)\(,((( 1 )\),),,/)\(,(((((( )\)),)),[],/)\(,( )\)),,/)\((,.... nn\ nps),[]//( s/ np[] np[] np[] )

Example: non peripheral extraction (that I met _ yesterday) nnnn nnnn nnnn P sssnps P sss s L sss s sss s ss)\) s ),/\(,((nn)\( \\ \,)ss\nps ),/)\((( 2 )),\ /)\(,((( 1 )\),),,/)\(,(((((( )\)),)),[],/)\(,( )\)),,/)\((,.... nn\ nps),[]//( s/ np[] np[] np[] )

adapted proof-nets To take restructuring of resources into account To represent modalities see Richard Moots 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 a a

a a a a a a

a a a a restructuring-1

a/a a a restructuring-2

a/a a contraction step