Dialogue categories and Frobenius monoids Paul-Andr Mellis CNRS - PowerPoint PPT Presentation

Dialogue categories and Frobenius monoids Paul-André Melliès CNRS & Université Paris Diderot SamsonFest Oxford 28 May 2013

Two [ academic ] lifes entangled ⇐⇒ Dialogue games Frobenius algebras 2

Living on both sides of the Channel 3

The Australian connection A Frobenius monoid F is a monoid and a comonoid satisfying m m d = = d m d A deep relationship with ∗ -autonomous categories discovered by Brian Day and Ross Street. 4

Original purpose of tensorial logic To provide a clear type-theoretic foundation to game semantics Propositions as types ⇔ Propositions as games based on the idea that game semantics is a diagrammatic syntax of continuations 5

Continuations Captures the difference between addition as a function nat × nat ⇒ nat and addition as a sequential algorithm ( nat ⇒ ⊥ ) ⇒ ⊥ × ( nat ⇒ ⊥ ) ⇒ ⊥ × ( nat ⇒ ⊥ ) ⇒ ⊥ This enables to distinguish the left-to-right implementation λϕ. λψ. λ k . ϕ ( λ x . ψ ( λ y . k ( x + y )) ) lradd = from the right-to-left implementation = λϕ. λψ. λ k . ψ ( λ y . ϕ ( λ x . k ( x + y )) ) rladd 6

The left-to-right addition ¬ ¬ nat × ¬ ¬ nat ⇒ ¬ ¬ nat question question 12 question 5 17 λϕ. λψ. λ k . ϕ ( λ x . ψ ( λ y . k ( x + y )) ) lradd = 7

The right-to-left addition ¬ ¬ nat × ¬ ¬ nat ⇒ ¬ ¬ nat question question 5 question 12 17 λϕ. λψ. λ k . ψ ( λ y . ϕ ( λ x . k ( x + y )) ) rladd = 8

Tensorial logic tensorial logic = a logic of tensor and negation = linear logic without A � ¬¬ A = the syntax of linear continuations = the syntax of dialogue games A synthesis between linear logic and game semantics 9

Tensorial logic Every sequent of the logic is of the form: ⊲ A 1 , · · · , A n B ⊢ Main rules of the logic: ⊲ Γ , A , B , ∆ ⊢ C Γ ⊢ A ∆ ⊢ B Γ , ∆ ⊢ A ⊗ B Γ , A ⊗ B , ∆ ⊢ C Γ , A ⊢ ⊥ Γ ⊢ A Γ , ¬ A ⊢ ⊥ Γ ⊢ ¬ A The primitive kernel of logic 10

A different way to think of polarities Tensorial logic Linear logic Motto: linear logic is a depolarized tensorial logic 11

A different way to think of polarities Tensorial logic Linear logic Motto: linear logic is a depolarized tensorial logic 12

The left-to-right scheduler A ⊢ A B ⊢ B Right ⊗ A , B ⊢ A ⊗ B Left ¬ B , ¬ ( A ⊗ B ) , A ⊢ Right ¬ ¬ ( A ⊗ B ) , A ¬ B ⊢ Left ¬ A , ¬¬ B , ¬ ( A ⊗ B ) ⊢ Right ¬ ¬¬ B , ¬ ( A ⊗ B ) ⊢ ¬ A Left ¬ ¬ ( A ⊗ B ) , ¬¬ A , ¬¬ B ⊢ Right ¬ ¬¬ A , ¬¬ B ⊢ ¬¬ ( A ⊗ B ) Left ⊗ ¬¬ A ⊗ ¬¬ B ⊢ ¬¬ ( A ⊗ B ) λϕ. λψ. λ k . ϕ ( λ x . ψ ( λ y . k ( x , y )) ) lrsched = 13

The left-to-right scheduler ¬ ¬ A × ¬ ¬ B ⇒ ¬ ¬ A ⊗ B question question answer question answer answer λϕ. λψ. λ k . ϕ ( λ x . ψ ( λ y . k ( x , y )) ) lrsched = 14

The right-to-left scheduler A ⊢ A B ⊢ B Right ⊗ A , B ⊢ A ⊗ B Left ¬ A , B , ¬ ( A ⊗ B ) ⊢ Right ¬ B , ¬ ( A ⊗ B ) ¬ A ⊢ Left ¬ B , ¬ ( A ⊗ B ) , ¬¬ A ⊢ Right ¬ ¬ ( A ⊗ B ) , ¬¬ A ⊢ ¬ B Left ¬ ¬ ( A ⊗ B ) , ¬¬ A , ¬¬ B ⊢ Right ¬ ¬¬ A , ¬¬ B ⊢ ¬¬ ( A ⊗ B ) Left ⊗ ¬¬ A ⊗ ¬¬ B ⊢ ¬¬ ( A ⊗ B ) λϕ. λψ. λ k . ψ ( λ y . ϕ ( λ x . k ( x , y )) ) rlsched = 15

The right-to-left scheduler ¬ ¬ A × ¬ ¬ B ⇒ ¬ ¬ A ⊗ B question question answer question answer answer λϕ. λψ. λ k . ψ ( λ y . ϕ ( λ x . k ( x , y )) ) rlsched = 16

Dialogue categories A functorial bridge between proofs and knots 17

Dialogue categories A monoidal category with a left duality A natural bijection between the set of maps A ⊗ B −→ ⊥ and the set of maps B −→ A ⊸ ⊥ A familiar situation in tensorial algebra 18

Dialogue categories A monoidal category with a right duality A natural bijection between the set of maps A ⊗ B −→ ⊥ and the set of maps A −→ ⊥ � B A familiar situation in tensorial algebra 19

Dialogue categories Definition. A dialogue category is a monoidal category C equipped with ⊲ an object ⊥ ⊲ two natural bijections : C ( A ⊗ B , ⊥ ) C ( B , A ⊸ ⊥ ) ϕ A , B −→ : C ( A ⊗ B , ⊥ ) C ( A , ⊥ � B ) ψ A , B −→ 20

� � � � Helical dialogue categories A dialogue category equipped with a family of bijections wheel A , B : C ( A ⊗ B , ⊥ ) −→ C ( B ⊗ A , ⊥ ) natural in A and B making the diagram associativity C (( B ⊗ C ) ⊗ A , ⊥ ) � C ( A ⊗ ( C ⊗ B ) , ⊥ ) wheel A , B ⊗ C wheel B , C ⊗ A C ( A ⊗ ( B ⊗ C )) C (( C ⊗ A ) ⊗ B , ⊥ ) associativity associativity wheel A ⊗ B , C C (( A ⊗ B ) ⊗ C , ⊥ ) � C ( C ⊗ ( A ⊗ B ) , ⊥ ) commutes. 21

Helical dialogue categories The wheel should be understood diagrammatically as: wheel x , y : f �→ f x y y x 22

The coherence diagram f wheel x wheel , y z y , z x y z x wheel x y ,z f f x y z z x y 23

� � � � � � An equivalent formulation A dialogue category equipped with a natural isomorphism turn A : A ⊸ ⊥ −→ ⊥ � A making the diagram below commute: ⊥ eval eval ( ⊥ � A ) ⊗ A B ⊗ ( B ⊸ ⊥ ) turn − 1 turn A B ( A ⊸ ⊥ ) ⊗ A B ⊗ ( ⊥ � B ) eval eval turn A ⊗ B � B ⊗ ( ⊥ � ( A ⊗ B )) ⊗ A B ⊗ (( A ⊗ B ) ⊸ ⊥ ) ⊗ A 24

The free dialogue category The objects of the category free-dialogue ( C ) are the formulas of tensorial logic: A , B :: = X | A ⊗ B | A ⊸ ⊥ | ⊥ � A | 1 where X is an object of the category C . The morphisms are the proofs of the logic modulo equality. 25

� � A proof-as-tangle theorem Every category C of atomic formulas induces a functor [ − ] such that [ − ] � free-ribbon ( C ⊥ ) free-dialogue ( C ) C where C ⊥ is the category C extended with an object ⊥ . Theorem. The functor [ − ] is faithful. −→ a topological foundation for game semantics 26

� � � � An illustration Imagine that we want to check that the diagram � turn x ⊥ � ⊥ � ( x ⊸ ⊥ ) ⊥ � ( ⊥ � x ) turn ⊥ twist � ( x ⊸ ⊥ ) � x ( ⊥ � x ) ⊸ ⊥ ⊥ � ( x ⊸ ⊥ ) η ′ η x commutes in every balanced dialogue category. 27

An illustration Equivalently, we want to check that the two derivation trees are equal: A ⊢ A left ⊸ A , A ⊸ ⊥ ⊢ ⊥ left ⊸ A , A ⊸ ⊥ ⊢ ⊥ twist A , A ⊸ ⊥ ⊢ ⊥ right � A ⊢ ⊥ � ( A ⊸ ⊥ ) A ⊢ A left ⊸ A , A ⊸ ⊥ ⊢ ⊥ braiding A ⊸ ⊥ , A ⊢ ⊥ A ⊢ A right � left � A ⊸ ⊥ ⊢ ⊥ � A ⊥ � A , A ⊢ ⊥ cut A ⊸ ⊥ , A ⊢ ⊥ braiding A , A ⊸ ⊥ ⊢ ⊥ right � A ⊢ ⊥ � ( A ⊸ ⊥ ) 28

An illustration equality of proofs ⇐⇒ equality of tangles 29

Dialogue chiralities A symmetric account of dialogue categories 30

� � The self-adjunction of negations Negation defines a pair of adjoint functors L C op C ⊥ R witnessed by the series of bijection: C op ( ¬ A , B ) C ( A , ¬ B ) C ( B , ¬ A ) � � 31

The symmetry of logic Eloise speaks to Abelard who speaks to Eloise who speaks to... 32

From categories to chiralities This leads to a slightly bizarre idea: decorrelate the category C from its opposite category C op So, let us define a chirality as a pair of categories ( A , B ) such that C op A C B � � for some category C . Here means equivalence of category � 33

� � � � Dialogue chiralities A dialogue chirality is a pair of monoidal categories ( A , � , true) ( B , � , false) with a monoidal equivalence ( − ) ∗ monoidal B op (0 , 1) A equivalence ∗ ( − ) together with an adjunction L A B ⊥ R 34

Dialogue chiralities and two natural bijections χ L � a | m ∗ � b � : � m � a | b � −→ m , a , b � a | b � m ∗ � χ R : � a � m | b � −→ m , a , b where the evaluation bracket A op × B : Set � − | − � −→ is defined as � a | b � : = A ( a , Rb ) 35

� � � Dialogue chiralities These are required to make the diagrams commute: χ L m � n � a | ( m � n ) ∗ � b � � ( m � n ) � a | b � [1] χ L χ L � � n � a | m ∗ � b � � � a | n ∗ � ( m ∗ � b ) � m n � m � ( n � a ) | b � 36

� � � Dialogue chiralities These are required to make the diagrams commute: χ R � a | b � ( m � n ) ∗ � m � n � a � ( m � n ) | b � [2] χ R χ R � � a � m | b � n ∗ � � � a | ( b � n ∗ ) � m ∗ � n m � ( a � m ) � n | b � 37

Dialogue chiralities These are required to make the diagrams commute: χ R χ L � � m � a | b � n ∗ � � � a | m ∗ � ( b � n ∗ ) � n m � ( m � a ) � n | b � [3] χ L χ R � � a | ( m ∗ � b ) � n ∗ � � � a � n | m ∗ � b � m n � m � ( a � n ) | b � 38

Chiralities as Frobenius monoids A bialgebraic account of dialogue categories 39

Frobenius monoids A Frobenius monoid F is a monoid and a comonoid satisfying m m d = = d m d A deep relationship with ∗ -autonomous categories discovered by Brian Day and Ross Street. 40