The dagger lambda calculus Philip Atzemoglou University of Oxford Quantum Physics and Logic 2014 Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 1 / 20
Why higher-order? Teleportation should be the same: = Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 2 / 20
Why higher-order? Teleportation should be the same: = Regardless of whether you are teleporting A ψ a state Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 2 / 20
Why higher-order? Teleportation should be the same: = Regardless of whether you are teleporting A ∗ A A ψ f a state or a function Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 2 / 20
Why higher-order? + My Files My Files My Papers My Music Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 3 / 20
Why higher-order? + My Files + f My Files f My Papers a ∗ My Music b Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 3 / 20
Terms � term � ::= variable | constant | � term � ⊗ � term � | � term � ∗ Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 4 / 20
Terms � term � ::= variable | constant | � term � ⊗ � term � | � term � ∗ i . e . x , y , z | | t 1 ⊗ t 2 | f ∗ , ( t 1 ⊗ t 2 ) ∗ c Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 4 / 20
Types � type � ∗ � type � ::= atomic | � type � ⊗ � type � | Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 5 / 20
Types � type � ∗ � type � ::= atomic | � type � ⊗ � type � | ( A ⊗ B ) ∗ , A ∗ ⊗ A i . e . A , B , C | A ⊗ B | Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 5 / 20
Linear negation Involutive: ( a ∗ ) ∗ ≡ a and ( A ∗ ) ∗ ≡ A Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 6 / 20
Linear negation Involutive: ( a ∗ ) ∗ ≡ a and ( A ∗ ) ∗ ≡ A & Is ⊗ equal to ? Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 6 / 20
Linear negation Involutive: ( a ∗ ) ∗ ≡ a and ( A ∗ ) ∗ ≡ A & Is ⊗ equal to ? Almost: Planar negation: ( a ⊗ b ) ∗ := b ∗ ⊗ a ∗ and ( A ⊗ B ) ∗ := B ∗ ⊗ A ∗ Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 6 / 20
The soup A set of connections between equityped terms. These connections correspond to connecting wires in a categorical diagram: S = { x 1 : x 2 , f : x 2 ∗ ⊗ x 3 , x 3 : x 4 } 4 2 2 3 f 1 Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 7 / 20
Switchboard Used in compliance with the CC-Attribution license of the original from: https://www.flickr.com/photos/glenbledsoe/6245440290/ Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 8 / 20
Sequents t 1 : A 1 , t 2 : A 2 , . . . , t n : A n ⊢ S t : B Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 9 / 20
Reconstructing λ λ a . b := a ∗ ⊗ b A ⊸ B := A ∗ ⊗ B Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 10 / 20
Representing connections between wires D E g f B ∗ B A C Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 11 / 20
Representing connections between wires D E a : A , c : C ⊢ S d ⊗ e : D ⊗ E g f B ∗ B where � f � : λ ( a ⊗ b ∗ ) . d , S = : λ ( b ⊗ c ) . e g A C Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 12 / 20
Sequent rules Id, x : A ⊢ x : A a : A ⊢ S b : B Negation, a ∗ : A ∗ ⊢ S ∗ b ∗ : B ∗ Γ , a : A , b : B ⊢ S c : C ⊗ L , Γ , a ⊗ b : A ⊗ B ⊢ S c : C Γ , a : A , b : B , ∆ ⊢ c : C Exchange, Γ ⊢ S 1 a : A ∆ ⊢ S 2 b : B Γ , b : B , a : A , ∆ ⊢ c : C Γ , � ∆ ⊢ S 1 ∪ S 2 a ⊗ b : A ⊗ B ⊗ R , Γ ⊢ S ∪{ i ∗ :1 } b : B λ Γ , a ′ : A , ∆ ⊢ S 2 b : B Cut, Γ ⊢ S 1 a : A i : I , Γ ⊢ S b : B Γ , ∆ ⊢ S 1 ∪ S 2 ∪{ a : a ′ } b : B Γ ⊢ S ∪{ i ∗ :1 } b : B ρ Γ . a : A , Γ ⊢ S b : B Γ , i : I ⊢ S b : B Curry, Γ ⊢ S a ∗ ⊗ b : A ∗ ⊗ B Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 13 / 20
† -flip a : A ⊢ S b : B † -flip b : B ⊢ S ∗ a : A Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 14 / 20
† -flip a : A ⊢ S b : B Negation a ∗ : A ∗ ⊢ S ∗ b ∗ : B ∗ a : A ⊢ S b : B Uncurry b : B , a ∗ : A ∗ ⊢ S ∗ † -flip b : B ⊢ S ∗ a : A Exchange a ∗ : A ∗ , b : B ⊢ S ∗ Curry b : B ⊢ S ∗ a : A Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 14 / 20
Soup reduction The soup propagation rules are bifunctoriality , trace and cancellation : S ∪ { a ⊗ b : c ⊗ d } − → S ∪ { a : c , b : d } S ∪ { x : A x } − → S ∪ { D A : 1 } S ∪ { 1 : 1 } − → S Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 15 / 20
Soup reduction The soup propagation rules are bifunctoriality , trace and cancellation : S ∪ { a ⊗ b : c ⊗ d } − → S ∪ { a : c , b : d } S ∪ { x : A x } − → S ∪ { D A : 1 } S ∪ { 1 : 1 } − → S Our soup rules also contain a consumption rule : � � � [ t / u ] , if u has no constants Γ ⊢ S ∪{ t : u } b : B − → Γ ⊢ S b : B [ u / t ] , if t has no constants Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 15 / 20
Application Application is defined as a notational shorthand, representing a variable and a connection in the soup. The origins of the application affect the structure of its corresponding soup connection: ft : B , Γ ⊢ c : C := x : B , Γ ⊢ { f : t ∗ ⊗ x } ∗ c : C and Γ ⊢ ft : B := Γ ⊢ { f : t ∗ ⊗ x } x : B For an application originating inside our soup, we have: { ft : c } := { x : c } ∪ { f : t ∗ ⊗ x } and { c : ft } := { c : x } ∪ { f : t ∗ ⊗ x } ∗ Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 16 / 20
Properties Subject reduction Consistency Strong normalisation Confluence Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 17 / 20
Curry-Howard-Lambek correspondence Dagger Categorical Quantum Lambda Logic Calculus Dagger Compact Categories Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 18 / 20
Curry-Howard-Lambek correspondence Categorical Dagger Quantum Lambda Logic Calculus Dagger Compact Categories Internal language for dagger compact categories Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 19 / 20
Conclusion Future work: Extend to cover complementary classical structures and dualisers Support for the non-determinacy of measurements Higher-order representation for MBQC Thanks are due to: Samson Abramsky, Bob Coecke, Prakash Panangaden, Jonathan Barrett, ... and many others ... Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 20 / 20
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