Fibrational Units of Measure Timothy Revell and N. Ghani, R. - - PowerPoint PPT Presentation

Fibrational Units of Measure

Timothy Revell

and N. Ghani, R. Atkey, S. Staton

University of Strathclyde

1

Motivation

2

Motivation

• Units of Measure m, s, kg...etc

2

Motivation

• Units of Measure m, s, kg...etc
• Used for measurement, e.g., X is twice as much as a metre

2

Motivation

• Units of Measure m, s, kg...etc
• Used for measurement, e.g., X is twice as much as a metre
• Single quantity can have many different units, e.g., metres,

inches, nautical mile, ...etc.

2

Motivation

• Units of Measure m, s, kg...etc
• Used for measurement, e.g., X is twice as much as a metre
• Single quantity can have many different units, e.g., metres,

inches, nautical mile, ...etc.

• Class of representations is a dimension, e.g., Length (L),

Mass (M), Time (T)

2

Motivation

• Units of Measure m, s, kg...etc
• Used for measurement, e.g., X is twice as much as a metre
• Single quantity can have many different units, e.g., metres,

inches, nautical mile, ...etc.

• Class of representations is a dimension, e.g., Length (L),

Mass (M), Time (T)

• Normally pick out base units for dimensions, e.g., SI Base

units include kg, m, s, K,...etc Derived Units kgm−2s−2

2

Dimensional Analysis

3

Dimensional Analysis

• +, −, ≤ of two quantities with different dimensions gives a

dimension error

3

Dimensional Analysis

• +, −, ≤ of two quantities with different dimensions gives a

dimension error

• ×, ÷ of two quantities with different dimensions is OK, but

gives new units

3

Dimensional Analysis

• +, −, ≤ of two quantities with different dimensions gives a

dimension error

• ×, ÷ of two quantities with different dimensions is OK, but

gives new units Quantification of units allows us to express this.

3

Dimensional Analysis

• +, −, ≤ of two quantities with different dimensions gives a

dimension error

• ×, ÷ of two quantities with different dimensions is OK, but

gives new units Quantification of units allows us to express this.

• × : ∀u1.∀u2.num(u1) → num(u2) → num(u1 · u2)

3

Background

4

Background

• Relational Parametricity and Units of Measure

by A. J. Kennedy

4

Background

• Relational Parametricity and Units of Measure

by A. J. Kennedy

• Introduces type system and relational semantics

4

Background

• Relational Parametricity and Units of Measure

by A. J. Kennedy

• Introduces type system and relational semantics
• ... But no category theory.

4

Outline

• Type System for Units of Measure
• Categorical Semantics
• Examples and Theorems
• Parametricity

5

A Type System for Units of Measure

Unit Context ∆ = u1, ..., un

6

A Type System for Units of Measure

Unit Context ∆ = u1, ..., un Types e ∈ Ab(∆)

∆ ⊢ e ∆ ⊢ e ∆ ⊢ num(e) Type

6

A Type System for Units of Measure

Unit Context ∆ = u1, ..., un Types e ∈ Ab(∆)

∆ ⊢ e ∆ ⊢ e ∆ ⊢ num(e) Type ∆ ⊢ T Type ∆ ⊢ U Type ∆ ⊢ T × U Type ∆ ⊢ T Type ∆ ⊢ U Type ∆ ⊢ T → U Type

6

A Type System for Units of Measure

Unit Context ∆ = u1, ..., un Types e ∈ Ab(∆)

∆ ⊢ e ∆ ⊢ e ∆ ⊢ num(e) Type ∆ ⊢ T Type ∆ ⊢ U Type ∆ ⊢ T × U Type ∆ ⊢ T Type ∆ ⊢ U Type ∆ ⊢ T → U Type ∆, u ⊢ T Type ∆ ⊢ ∀u.T Type

6

A Type System for Units of Measure

Typing Context Γ = x1 : T1, ..., xm : Tm

7

A Type System for Units of Measure

Typing Context Γ = x1 : T1, ..., xm : Tm Terms

Usual STλC

7

A Type System for Units of Measure

Typing Context Γ = x1 : T1, ..., xm : Tm Terms

Usual STλC

and

∆ ⊢ Γ ctxt ∆, u, Γ ⊢ t : T ∆; Γ ⊢ Λu.t : ∀u.T ∆ ⊢ e ∆, Γ ⊢ t : ∀u.T ∆; Γ ⊢ te : T[e/u]

7

Some Constants

Then add constants, popular choices include...

8

Some Constants

Then add constants, popular choices include...

• 0 : ∀u.num(u)

8

Some Constants

Then add constants, popular choices include...

• 0 : ∀u.num(u)
• 1 : num(1)

8

Some Constants

Then add constants, popular choices include...

• 0 : ∀u.num(u)
• 1 : num(1)
• + : ∀u.num(u) → num(u) → num(u)

8

Some Constants

Then add constants, popular choices include...

• 0 : ∀u.num(u)
• 1 : num(1)
• + : ∀u.num(u) → num(u) → num(u)
• × : ∀u1.∀u2.num(u1) → num(u2) → num(u1 · u2)

8

Where are the Fibrations?

9

Fibrational Semantics

10

Fibrational Semantics

A fibration p : E → B with enough structure such that

10

Fibrational Semantics

A fibration p : E → B with enough structure such that

• B - Unit contexts and expressions

10

Fibrational Semantics

A fibration p : E → B with enough structure such that

• B - Unit contexts and expressions
• E - Types and terms

10

A Bit More Detail ...

Start with a fibration p : E → B

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G
• i.e. u1, u2 ⊢ u1 · u−1

2 (g1, g2) = g1 · g−1 2

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G
• i.e. u1, u2 ⊢ u1 · u−1

2 (g1, g2) = g1 · g−1 2

• ∆; Γ ⊢ T Type ∈ E∆

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G
• i.e. u1, u2 ⊢ u1 · u−1

2 (g1, g2) = g1 · g−1 2

• ∆; Γ ⊢ T Type ∈ E∆
• Fibred CCC structure

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G
• i.e. u1, u2 ⊢ u1 · u−1

2 (g1, g2) = g1 · g−1 2

• ∆; Γ ⊢ T Type ∈ E∆
• Fibred CCC structure
• Let num ∈ EG, then ∆ ⊢ num(e) = ∆ ⊢ e∗num.

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G
• i.e. u1, u2 ⊢ u1 · u−1

2 (g1, g2) = g1 · g−1 2

• ∆; Γ ⊢ T Type ∈ E∆
• Fibred CCC structure
• Let num ∈ EG, then ∆ ⊢ num(e) = ∆ ⊢ e∗num.
• Quantification given by π∗ ⊣ ∀, where π : ∆, u → ∆.

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G
• i.e. u1, u2 ⊢ u1 · u−1

2 (g1, g2) = g1 · g−1 2

• ∆; Γ ⊢ T Type ∈ E∆
• Fibred CCC structure
• Let num ∈ EG, then ∆ ⊢ num(e) = ∆ ⊢ e∗num.
• Quantification given by π∗ ⊣ ∀, where π : ∆, u → ∆.
• Terms define morphisms ∆; Γ ⊢ t : T : Γ → T ∈ E∆.

11

A Bit More Detail ...

Start with a fibration p : E → B

• G ∈ Ab(B) using Abelian Group Object
• ∆ ⊢ e : G|∆| → G
• i.e. u1, u2 ⊢ u1 · u−1

2 (g1, g2) = g1 · g−1 2

• ∆; Γ ⊢ T Type ∈ E∆
• Fibred CCC structure
• Let num ∈ EG, then ∆ ⊢ num(e) = ∆ ⊢ e∗num.
• Quantification given by π∗ ⊣ ∀, where π : ∆, u → ∆.
• Terms define morphisms ∆; Γ ⊢ t : T : Γ → T ∈ E∆.

Definition

We call (p, G, num) a UoM-fibration.

11

...Or More Concisely

12

...Or More Concisely

Definition

A UoM-fibration (p : E → B, G, num) is given by

12

...Or More Concisely

Definition

A UoM-fibration (p : E → B, G, num) is given by

• A λ1-fibration p : E → B

12

...Or More Concisely

Definition

A UoM-fibration (p : E → B, G, num) is given by

• A λ1-fibration p : E → B
• An exponentiable Abelian group object G and

12

...Or More Concisely

Definition

A UoM-fibration (p : E → B, G, num) is given by

• A λ1-fibration p : E → B
• An exponentiable Abelian group object G and
• An object num ∈ EG

12

UoM-Fibration Examples

13

UoM-Fibration Examples

• Syntax of UoM

(p : E → LAb, 1, num)

where E = Types and Terms

13

UoM-Fibration Examples

• Syntax of UoM

(p : E → LAb, 1, num)

where E = Types and Terms

• UoM-Fibrations from the Usual Fibrations

Codomain, Suboject and Relations fibration over Set are

λ1-fibrations with simple products.

13

UoM-Fibration Examples

• Syntax of UoM

(p : E → LAb, 1, num)

where E = Types and Terms

• UoM-Fibrations from the Usual Fibrations

Codomain, Suboject and Relations fibration over Set are

λ1-fibrations with simple products.

• Choose Abelian group object, G

13

UoM-Fibration Examples

• Syntax of UoM

(p : E → LAb, 1, num)

where E = Types and Terms

• UoM-Fibrations from the Usual Fibrations

Codomain, Suboject and Relations fibration over Set are

λ1-fibrations with simple products.

• Choose Abelian group object, G
• Choose object in fibre above G

13

UoM-Fibration Examples

• Syntax of UoM

(p : E → LAb, 1, num)

where E = Types and Terms

• UoM-Fibrations from the Usual Fibrations

Codomain, Suboject and Relations fibration over Set are

λ1-fibrations with simple products.

• Choose Abelian group object, G
• Choose object in fibre above G
• Unit Erasure Semantics

(p : E → 1, ∗, num)

e.g. E = cpo and num = Q⊥

13

Theorems About UoM-Fibrations

14

Theorems About UoM-Fibrations

Theorem

• (p : E → B, G, X) UoM-fibration
• A finite products
• F : A → B product preserving functor
• G′ ∈ A an Abelian group object with

FG′ = G

14

Theorems About UoM-Fibrations

Theorem

• (p : E → B, G, X) UoM-fibration
• A finite products
• F : A → B product preserving functor
• G′ ∈ A an Abelian group object with

FG′ = G Then (F ∗p, G′, (G′, X)) is a UoM-fibration.

14

Theorems About UoM-Fibrations

Theorem

• (p : E → B, G, X) UoM-fibration
• A finite products
• F : A → B product preserving functor
• G′ ∈ A an Abelian group object with

FG′ = G Then (F ∗p, G′, (G′, X)) is a UoM-fibration. F ∗E ✲ E

A

F ∗p ❄ F ✲ B p ❄

14

Theorems About UoM-Fibrations ctd...

15

Theorems About UoM-Fibrations ctd...

Theorem

Any UoM-fibration can be converted into a UoM-fibration with LAb in the base.

15

Theorems About UoM-Fibrations ctd...

Theorem

Any UoM-fibration can be converted into a UoM-fibration with LAb in the base. F(1) = G F ∗E ✲ E LAb F ∗p ❄ F ✲ B p ❄

15

Another Example

Recall an Abelian group G can be thought of as a category G

16

Another Example

Recall an Abelian group G can be thought of as a category G

• Ob(G) = ∗

16

Another Example

Recall an Abelian group G can be thought of as a category G

• Ob(G) = ∗
• G(∗, ∗) = G

16

Another Example

Recall an Abelian group G can be thought of as a category G

• Ob(G) = ∗
• G(∗, ∗) = G

A G-Set is a functor φ : G → Set, i.e.,

16

Another Example

Recall an Abelian group G can be thought of as a category G

• Ob(G) = ∗
• G(∗, ∗) = G

A G-Set is a functor φ : G → Set, i.e.,

• φ∗ ∈ Set, which we denote |φ|

16

Another Example

Recall an Abelian group G can be thought of as a category G

• Ob(G) = ∗
• G(∗, ∗) = G

A G-Set is a functor φ : G → Set, i.e.,

• φ∗ ∈ Set, which we denote |φ|
• φg : |φ| → |φ|.

16

Another Example

Recall an Abelian group G can be thought of as a category G

• Ob(G) = ∗
• G(∗, ∗) = G

A G-Set is a functor φ : G → Set, i.e.,

• φ∗ ∈ Set, which we denote |φ|
• φg : |φ| → |φ|.

Definition

We call the functor p : Ab-Set → Ab the Ab-Set fibration, where Ab-SetG = [G, Set]

16

Another Example

Recall an Abelian group G can be thought of as a category G

• Ob(G) = ∗
• G(∗, ∗) = G

A G-Set is a functor φ : G → Set, i.e.,

• φ∗ ∈ Set, which we denote |φ|
• φg : |φ| → |φ|.

Definition

We call the functor p : Ab-Set → Ab the Ab-Set fibration, where Ab-SetG = [G, Set]

Theorem

The Ab-Set fibration is a λ1-fibration with simple products. Hence, for choices G ∈ Ab, num ∈ Ab-SetG

(p : Ab-Set → Ab, G, num)

is a UoM-fibration

16

Theorem About Fibrations

17

Theorem About Fibrations

Theorem

Let E and B be categories with finite products.

17

Theorem About Fibrations

Theorem

Let E and B be categories with finite products.

• Suppose that [

] : B → Cat is a product preserving functor.

17

Theorem About Fibrations

Theorem

Let E and B be categories with finite products.

• Suppose that [

] : B → Cat is a product preserving functor.

• p : E → B is a fibration with EX := [X] → D and hence

reindexing is given by precomposition

17

Theorem About Fibrations

Theorem

Let E and B be categories with finite products.

• Suppose that [

] : B → Cat is a product preserving functor.

• p : E → B is a fibration with EX := [X] → D and hence

reindexing is given by precomposition

• i.e., for any f : X → Y ∈ B,

f ∗(φ : [Y] → D) = φ ◦ [f].

Then, the reindexing of any projection map πX : X × Y → X has a right adjoint π∗

X ⊣ Ran[π] , which satisfies the Beck-Chevalley

condition.

17

Sketch Proof

18

Sketch Proof

Lemma

For π : X × Y → X in B and φ : [X] × [Y] → D in EX×Y then

18

Sketch Proof

Lemma

For π : X × Y → X in B and φ : [X] × [Y] → D in EX×Y then

(Ran[π]φ)x = lim

y∈[Y] φ(x, y)

18

Sketch Proof

Keep in mind: (Ran[π]φ)x = limy∈[Y] φ(x, y)

19

Sketch Proof

Keep in mind: (Ran[π]φ)x = limy∈[Y] φ(x, y) Want to show:

19

Sketch Proof

Keep in mind: (Ran[π]φ)x = limy∈[Y] φ(x, y) Want to show: For any f : X → X ′ in B and

ψ : [X ′] × [Y] → D

in EX′×Y

(RanπX (f × id)∗ψ)x ∼ = (f ∗RanπX′ψ)x

19

Sketch Proof

Keep in mind: (Ran[π]φ)x = limy∈[Y] φ(x, y) Want to show: For any f : X → X ′ in B and

ψ : [X ′] × [Y] → D

in EX′×Y

(RanπX (f × id)∗ψ)x ∼ = (f ∗RanπX′ψ)x

Use Lemma:

19

Sketch Proof

Keep in mind: (Ran[π]φ)x = limy∈[Y] φ(x, y) Want to show: For any f : X → X ′ in B and

ψ : [X ′] × [Y] → D

in EX′×Y

(RanπX (f × id)∗ψ)x ∼ = (f ∗RanπX′ψ)x

Use Lemma:

(RanπX (f × id)∗ψ)x ∼ = lim

y∈Y(f × id)∗φ(x, y) ∼

= lim

y∈Y φ(fx, y)

19

Sketch Proof

Keep in mind: (Ran[π]φ)x = limy∈[Y] φ(x, y) Want to show: For any f : X → X ′ in B and

ψ : [X ′] × [Y] → D

in EX′×Y

(RanπX (f × id)∗ψ)x ∼ = (f ∗RanπX′ψ)x

Use Lemma:

(RanπX (f × id)∗ψ)x ∼ = lim

y∈Y(f × id)∗φ(x, y) ∼

= lim

y∈Y φ(fx, y)

(f ∗RanπX′ψ)x ∼ = lim

y∈Y φ(fx, y)

19

Summary of Last Few Slides

20

Summary of Last Few Slides

• If fibration such that reindexing is given by precomposition

20

Summary of Last Few Slides

• If fibration such that reindexing is given by precomposition
• AND right adjoints are given by right Kan extensions

20

Summary of Last Few Slides

• If fibration such that reindexing is given by precomposition
• AND right adjoints are given by right Kan extensions
• Then quantification satisfies BC

20

Summary of Last Few Slides

• If fibration such that reindexing is given by precomposition
• AND right adjoints are given by right Kan extensions
• Then quantification satisfies BC
• We use this to show the Ab-Set fibration is a UoM-fibration

20

Results in the Ab-Set Fibration

21

Results in the Ab-Set Fibration

Lemma

Suppose u ⊢ S, T Type, then

|∀u.S → T| ∼ = Nat(S, T)

21

Results in the Ab-Set Fibration

Lemma

Suppose u ⊢ S, T Type, then

|∀u.S → T| ∼ = Nat(S, T) Proof.

By end formula for a Kan extension.

21

Results in the Ab-Set Fibration ctd...

Lemma

Let t : ∀u.num(u) → num(un) for some m, n ∈ N,

22

Results in the Ab-Set Fibration ctd...

Lemma

Let t : ∀u.num(u) → num(un) for some m, n ∈ N, then for x ∈ |num(u)|

t(g · x) = gn · (tx) ∀g ∈ G

22

Results in the Ab-Set Fibration ctd...

Lemma

Let t : ∀u.num(u) → num(un) for some m, n ∈ N, then for x ∈ |num(u)|

t(g · x) = gn · (tx) ∀g ∈ G Proof.

Use previous lemma to see t ∈ G-Set(num(u), num(un))

22

Results in the Ab-Set Fibration ctd...

Lemma

Let t : ∀u.num(u) → num(un) for some m, n ∈ N, then for x ∈ |num(u)|

t(g · x) = gn · (tx) ∀g ∈ G Proof.

Use previous lemma to see t ∈ G-Set(num(u), num(un)) Naturality gives result.

22

Results in the Ab-Set Fibration ctd...

Corollary

There is no non-trivial term of type ∀u.num(u2) → num(u).

23

Results in the Ab-Set Fibration ctd...

Corollary

There is no non-trivial term of type ∀u.num(u2) → num(u).

Proof.

Consider (p : Ab-Set → Ab, Z2, Z2),

23

Results in the Ab-Set Fibration ctd...

Corollary

There is no non-trivial term of type ∀u.num(u2) → num(u).

Proof.

Consider (p : Ab-Set → Ab, Z2, Z2), Then if there were a term t : ∀u.num(u2) → num(u), then

t(g2 · x) = g · (tx) ∀g ∈ Z2

23

Results in the Ab-Set Fibration ctd...

Corollary

There is no non-trivial term of type ∀u.num(u2) → num(u).

Proof.

Consider (p : Ab-Set → Ab, Z2, Z2), Then if there were a term t : ∀u.num(u2) → num(u), then

t(g2 · x) = g · (tx) ∀g ∈ Z2

Which does not hold, because If t0 = 1 then

t(1 + 1 + 0) = 1 + t(0)

23

Results in the Ab-Set Fibration ctd...

Corollary

There is no non-trivial term of type ∀u.num(u2) → num(u).

Proof.

Consider (p : Ab-Set → Ab, Z2, Z2), Then if there were a term t : ∀u.num(u2) → num(u), then

t(g2 · x) = g · (tx) ∀g ∈ Z2

Which does not hold, because If t0 = 1 then

t(1 + 1 + 0) = 1 + t(0)

If t1 = 1 then

t(1 + 1 + 1) = 1 + t(1)

23

Parametric UoM-Fibrations

24

Parametric UoM-Fibrations

• B - unit erasure semantics

24

Parametric UoM-Fibrations

• B - unit erasure semantics
• R : LAb → B product preserving functor

24

Parametric UoM-Fibrations

• B - unit erasure semantics
• R : LAb → B product preserving functor

Rel(E) ✲ E LAb × B p ❄ R × ∆ ✲ B × B × B _ × _ × _ ✲ B u ❄ LAb ❄ 1 ❄

24

Parametric UoM-Fibrations

• B - unit erasure semantics
• R : LAb → B product preserving functor

Rel(E) ✲ E LAb × B p ❄ R × ∆ ✲ B × B × B _ × _ × _ ✲ B u ❄ LAb ❄ 1 ❄ Rel(E)n = {(n, B, P) | B ∈ B, P ∈ ER(n)×B×B}

24

Parametric UoM-Fibrations ctd...

25

Parametric UoM-Fibrations ctd...

Theorem (r : Rel(E) → LAb, 1, num), for a choice of num, is a UoM-fibration.

25

Parametric UoM-Fibrations ctd...

Theorem (r : Rel(E) → LAb, 1, num), for a choice of num, is a UoM-fibration.

Rel(E) ✲ E LAb × B p ❄ R × ∆ ✲ B × B × B _ × _ × _ ✲ B u ❄ LAb ❄ 1 ❄

25

Why Parametric?

26

Why Parametric?

For ∆ ⊢ T Type, where |∆| = n

26

Why Parametric?

For ∆ ⊢ T Type, where |∆| = n

T = (n, T0, T1)

26

Why Parametric?

For ∆ ⊢ T Type, where |∆| = n

T = (n, T0, T1)

with T0 ∈ B and T1 ∈ EGn×T0×T0.

26

Why Parametric?

For ∆ ⊢ T Type, where |∆| = n

T = (n, T0, T1)

with T0 ∈ B and T1 ∈ EGn×T0×T0.

26

Why Parametric?

For ∆ ⊢ T Type, where |∆| = n

T = (n, T0, T1)

with T0 ∈ B and T1 ∈ EGn×T0×T0.

• T0 as the unit-erasure

semantics of T

26

Why Parametric?

For ∆ ⊢ T Type, where |∆| = n

T = (n, T0, T1)

with T0 ∈ B and T1 ∈ EGn×T0×T0.

• T0 as the unit-erasure

semantics of T

• T1 as the relational

semantics of T.

26

Why Parametric?

For ∆ ⊢ T Type, where |∆| = n

T = (n, T0, T1)

with T0 ∈ B and T1 ∈ EGn×T0×T0.

• T0 as the unit-erasure

semantics of T

• T1 as the relational

semantics of T. 1

Γ1

❥

⇓ t1

T1

✯Pn 1

=

❄

Γ0

❥

⇓ t0

T0

✯ B p′ ❄

26

Conclusions and Future Work

27

Conclusions and Future Work

• Units of Measure can be given a fibrational semantics

27

Conclusions and Future Work

• Units of Measure can be given a fibrational semantics
• Nice model using G-sets can exploit naturality properties of

UoM

27

Conclusions and Future Work

• Units of Measure can be given a fibrational semantics
• Nice model using G-sets can exploit naturality properties of

UoM

• There exist parametric UoM-fibrations

27

Conclusions and Future Work

28

Conclusions and Future Work

• Look more at role of G-sets c.f. nominal sets

28

Conclusions and Future Work

• Look more at role of G-sets c.f. nominal sets
• Invariance properties and symmetries

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Conclusions and Future Work

• Look more at role of G-sets c.f. nominal sets
• Invariance properties and symmetries
• ...write thesis...

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Thanks for listening.

timothy.revell@strath.ac.uk @timothyrevell

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