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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: =

SLIDE 1

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

SLIDE 2

Why higher-order?

Teleportation should be the same:

=

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 2 / 20

SLIDE 3

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

SLIDE 4

Why higher-order?

Teleportation should be the same:

=

Regardless of whether you are teleporting A ψ a state A f A∗

r a function

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 2 / 20

SLIDE 5

Why higher-order?

+ My Files My Music My Files My Papers

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 3 / 20

SLIDE 6

Why higher-order?

+ My Files My Music My Files My Papers + f b f a∗

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 3 / 20

SLIDE 7

Terms

term ::= variable | constant | term ⊗ term | term∗

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 4 / 20

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Terms

term ::= variable | constant | term ⊗ term | term∗ i.e. x, y, z | c | t1 ⊗ t2 | f∗, (t1 ⊗ t2)∗

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 4 / 20

SLIDE 9

Types

type ::= atomic | type ⊗ type | type∗

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 5 / 20

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Types

type ::= atomic | type ⊗ type | type∗ i.e. A, B, C | A ⊗ B | (A ⊗ B)∗, A∗ ⊗ A

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 5 / 20

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Linear negation

Involutive: (a∗)∗ ≡ a and (A∗)∗ ≡ A

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 6 / 20

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Linear negation

Involutive: (a∗)∗ ≡ a and (A∗)∗ ≡ A Is ⊗ equal to & ?

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 6 / 20

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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

SLIDE 14

The soup

A set of connections between equityped terms. These connections correspond to connecting wires in a categorical diagram: S = {x1 : x2, f : x2∗ ⊗ x3, x3 : x4} f 3 4 2 2 1

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 7 / 20

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Switchboard

Used in compliance with the CC-Attribution license of the original from: https://www.flickr.com/photos/glenbledsoe/6245440290/

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SLIDE 16

Sequents

t1 : A1, t2 : A2, . . . , tn : An ⊢S t : B

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Reconstructing λ

λa.b := a∗ ⊗ b A ⊸ B := A∗ ⊗ B

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Representing connections between wires

f g A C B∗ B E D

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Representing connections between wires

f g A C B∗ B E D a : A, c : C ⊢S d ⊗ e : D ⊗ E where S = f : λ (a ⊗ b∗) .d, g : λ (b ⊗ c) .e

Philip Atzemoglou (University of Oxford)

The dagger lambda calculus QPL 2014 12 / 20

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Sequent rules

Id, x : A ⊢ x : A Γ, a : A, b : B ⊢S c : C ⊗L, Γ, a ⊗ b : A ⊗ B ⊢S c : C Γ ⊢S1 a : A ∆ ⊢S2 b : B ⊗R, Γ, ∆ ⊢S1∪S2 a ⊗ b : A ⊗ B Γ ⊢S1 a : A a′ : A, ∆ ⊢S2 b : B Cut, Γ, ∆ ⊢S1∪S2∪{a:a′} b : B a : A, Γ ⊢S b : B Curry, Γ ⊢S a∗ ⊗ b : A∗ ⊗ B a : A ⊢S b : B Negation, a∗ : A∗ ⊢S∗ b∗ : B∗ Γ, a : A, b : B, ∆ ⊢ c : C Exchange, Γ, b : B, a : A, ∆ ⊢ c : C Γ ⊢S∪{i∗:1} b : B λΓ, i : I, Γ ⊢S b : B Γ ⊢S∪{i∗:1} b : B ρΓ. Γ, i : I ⊢S b : B

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 13 / 20

SLIDE 21

†-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

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†-flip

a : A ⊢S b : B

†-flip

b : B ⊢S∗ a : A a : A ⊢S b : B

Negation

a∗ : A∗ ⊢S∗ b∗ : B∗

Uncurry

b : B, a∗ : A∗ ⊢S∗

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

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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 ∪ {DA : 1} S ∪ {1 : 1} − → S

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 15 / 20

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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 ∪ {DA : 1} S ∪ {1 : 1} − → S Our soup rules also contain a consumption rule: Γ ⊢S∪{t:u} b : B − →

Γ ⊢S b : B

[t/u], if u has no constants [u/t], if t has no constants

Philip Atzemoglou (University of Oxford) The dagger lambda calculus QPL 2014 15 / 20

SLIDE 25

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}∗

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Properties

Subject reduction Consistency Strong normalisation Confluence

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SLIDE 27

Curry-Howard-Lambek correspondence

Dagger Lambda Calculus Categorical Quantum Logic Dagger Compact Categories

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Curry-Howard-Lambek correspondence

Dagger Lambda Calculus Categorical Quantum Logic Dagger Compact Categories Internal language for dagger compact categories

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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