Autonomization of monoidal categories Antonin Delpeuch March 28, 2019 SYCO 3 A. Delpeuch Autonomization March 28, 2019 SYCO 3 1 / 19
Outline 1 Pregroup grammars and compositional semantics 2 Free yourselves from the strings of tensors! 3 Examples of applications A. Delpeuch Autonomization March 28, 2019 SYCO 3 2 / 19
Pregroup grammars and compositional semantics Outline 1 Pregroup grammars and compositional semantics 2 Free yourselves from the strings of tensors! 3 Examples of applications A. Delpeuch Autonomization March 28, 2019 SYCO 3 3 / 19
Pregroup grammars and compositional semantics Context Pregroup grammars (Lambek, 1993, Lambek, 1999) the film that Emily directed n r · n · np ll · s l np r · s · np l np · n l n np A. Delpeuch Autonomization March 28, 2019 SYCO 3 4 / 19
Pregroup grammars and compositional semantics Context Pregroup grammars (Lambek, 1993, Lambek, 1999) the film that Emily directed n r · n · np ll · s l np r · s · np l np · n l n np A. Delpeuch Autonomization March 28, 2019 SYCO 3 4 / 19
Pregroup grammars and compositional semantics Context Pregroup grammars (Lambek, 1993, Lambek, 1999) the film that Emily directed n r · n · np ll · s l np r · s · np l np · n l n np A. Delpeuch Autonomization March 28, 2019 SYCO 3 4 / 19
Pregroup grammars and compositional semantics Context Pregroup grammars (Lambek, 1993, Lambek, 1999) the film that Emily directed n r · n · np ll · s l np r · s · np l np · n l n np A. Delpeuch Autonomization March 28, 2019 SYCO 3 4 / 19
Pregroup grammars and compositional semantics Autonomous (or rigid) categories Objects (= types) are closed under _ ⊗ _ (product of types), _ l and _ r (adjoints). contain basic types, and I , neutral for ⊗ . Arrows (= type reductions) between two objects can be composed with ◦ (sequential composition) and ⊗ (parallel composition) ; contain 1 A : A → A (identity of A ) and ǫ l : A l ⊗ A → I ǫ r : A ⊗ A r → I η l : I → A ⊗ A l η r : I → A r ⊗ A and such that some equations hold. A. Delpeuch Autonomization March 28, 2019 SYCO 3 5 / 19
Pregroup grammars and compositional semantics Representation A I A f f f B B I A A ⊗ A ⊗ C A C g = f ◦ g B = g f ⊗ g f f C ⊗ B ⊗ D B D C A. Delpeuch Autonomization March 28, 2019 SYCO 3 6 / 19
Pregroup grammars and compositional semantics ǫ and η A r A l A ⊗ ⊗ A I ǫ r = ǫ l = η r = A r I I ⊗ A I A η l = I A = A l A ⊗ A A. Delpeuch Autonomization March 28, 2019 SYCO 3 7 / 19
Pregroup grammars and compositional semantics Some equalities A I A I A A A r = = A l A A A A A A A A I I g f = = g f g f A. Delpeuch Autonomization March 28, 2019 SYCO 3 8 / 19
Pregroup grammars and compositional semantics Pregroup reductions as arrows Clouzot directed an Italian movie n r s n l d r d d r n n d A. Delpeuch Autonomization March 28, 2019 SYCO 3 9 / 19
Pregroup grammars and compositional semantics Pregroup reductions as arrows Clouzot directed an Italian movie n r s n l d r d d r n n d A. Delpeuch Autonomization March 28, 2019 SYCO 3 9 / 19
Pregroup grammars and compositional semantics Pregroup reductions as arrows Clouzot directed an Italian movie n r s n l d r d d r n n d s A. Delpeuch Autonomization March 28, 2019 SYCO 3 9 / 19
Pregroup grammars and compositional semantics Compositional semantics Clouzot directed an Italian movie I I I I I Word semantics d r d n r s s n l d r n s n s d Type reduction s Motto: Type reduction ◦ Word meanings = Sentence meaning A. Delpeuch Autonomization March 28, 2019 SYCO 3 10 / 19
Pregroup grammars and compositional semantics Distributional Compositional Categorial model DisCoCat (Coecke, Sadrzadeh, and Clark, 2011) : use ( Vect , ⊗ , I ), finite dimensional vector spaces over R and linear maps between them. I 0 . 73 − 0 . 3 3 . 9 − 2 . 1 0 . 4 I − 2 . 3 − 2 . 3 2 . 2 1 . 5 − 1 . 6 = = 0 . 1 0 . 1 0 . 3 − 3 . 8 1 . 2 n n r n 1 . 4 1 . 4 3 . 4 0 . 1 3 . 2 A. Delpeuch Autonomization March 28, 2019 SYCO 3 11 / 19
Pregroup grammars and compositional semantics Distributional Compositional Categorial model DisCoCat (Coecke, Sadrzadeh, and Clark, 2011) : use ( Vect , ⊗ , I ), finite dimensional vector spaces over R and linear maps between them. I 0 . 73 − 0 . 3 3 . 9 − 2 . 1 0 . 4 I − 2 . 3 − 2 . 3 2 . 2 1 . 5 − 1 . 6 = = 0 . 1 0 . 1 0 . 3 − 3 . 8 1 . 2 n n r n 1 . 4 1 . 4 3 . 4 0 . 1 3 . 2 The dimension of a word representation is exponential in the length of the grammatical type. A. Delpeuch Autonomization March 28, 2019 SYCO 3 11 / 19
Pregroup grammars and compositional semantics Why should we use the tensor product? The direct sum ⊕ is cartesian, so it cannot have cups and caps: = = � = A. Delpeuch Autonomization March 28, 2019 SYCO 3 12 / 19
Pregroup grammars and compositional semantics Why should we use the tensor product? The direct sum ⊕ is cartesian, so it cannot have cups and caps: = = � = General belief in the community: “sticking with the categorical framework [...] forces us to stay within the world of linear maps” (Wijnholds and Sadrzadeh, 2018). A. Delpeuch Autonomization March 28, 2019 SYCO 3 12 / 19
Free yourselves from the strings of tensors! Outline 1 Pregroup grammars and compositional semantics 2 Free yourselves from the strings of tensors! 3 Examples of applications A. Delpeuch Autonomization March 28, 2019 SYCO 3 13 / 19
Free yourselves from the strings of tensors! Just cheat and be free! Our semantic category does not need to have caps and cups: we can freely add them . Pat grows delicious kiwis f 1 + + n r s n l = n n l n n f 1 s s Trick: caps and cups can be eliminated in any sentence representation. A. Delpeuch Autonomization March 28, 2019 SYCO 3 14 / 19
Free yourselves from the strings of tensors! Constructing free autonomous categories Preller and Lambek (2007) construct the free autonomous category generated by a category. We need to start from a monoidal category instead. We factorize their construction: L ′ L Cat ⊥ Mon ⊥ Nom R ′ R A. Delpeuch Autonomization March 28, 2019 SYCO 3 15 / 19
Examples of applications Outline 1 Pregroup grammars and compositional semantics 2 Free yourselves from the strings of tensors! 3 Examples of applications A. Delpeuch Autonomization March 28, 2019 SYCO 3 16 / 19
Examples of applications Additive models Observation by Mikolov et al. (2013): − − − → queen + − − − − → woman + + − − → king + − − → man So, it tempting to define royal( x ) = x + − queen − − − − → − − − − → woman . A. Delpeuch Autonomization March 28, 2019 SYCO 3 17 / 19
Examples of applications Additive models Observation by Mikolov et al. (2013): − − − → queen + − − − − → woman + + − − → king + − − → man So, it tempting to define royal( x ) = x + − queen − − − − → − − − − → woman . That is forbidden in ( Vect , ⊗ , I )! A. Delpeuch Autonomization March 28, 2019 SYCO 3 17 / 19
Examples of applications Convolutional neural networks Socher et al. (2013) combine vectors following a Chomskyian tree: Lewis (2019) translates this approach to the categorical model, in ( Vect , ⊗ , I ). A. Delpeuch Autonomization March 28, 2019 SYCO 3 18 / 19
Examples of applications a man who ate a cake a b c d n r n s l n n r s n l d r n s d r n s d d a b c d a d = c b A. Delpeuch Autonomization March 28, 2019 SYCO 3 19 / 19
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