[PPT] - Dont try this at home: No-Go Theorems for Distributive Laws Maaike PowerPoint Presentation

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Intro Plotkin Algebra to the Rescue Results

Don’t try this at home: No-Go Theorems for Distributive Laws

Maaike Zwart & Dan Marsden

University of Oxford

27 March 2018

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Intro Plotkin Algebra to the Rescue Results

Overview

Introduction
Monads
Distributive laws
Previously broken distributive laws
Plotkin’s counterexample
General No-Go theorems: the algebraic approach
Generalized Plotkin
And more...
Results

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Intro Plotkin Algebra to the Rescue Results

Monads

A monad is a triple xT, η, µy, with T an endofunctor and η : 1 ñ T, µ : TT ñ T natural transformations, such that:

T TT TTT TT TT T TT T

ηT Tη Id µ Tµ µT µ µ µ

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Intro Plotkin Algebra to the Rescue Results

Monads

A monad is a triple xT, η, µy, with T an endofunctor and η : 1 ñ T, µ : TT ñ T natural transformations, such that:

T TT TTT TT TT T TT T

ηT Tη Id µ Tµ µT µ µ µ

List: L
LpXq set of all finite lists.
ηXpxq “ rxs
µX concatenation.

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Intro Plotkin Algebra to the Rescue Results

Monads

A monad is a triple xT, η, µy, with T an endofunctor and η : 1 ñ T, µ : TT ñ T natural transformations, such that:

T TT TTT TT TT T TT T

ηT Tη Id µ Tµ µT µ µ µ

List: L
LpXq set of all finite lists.
ηXpxq “ rxs
µX concatenation.
Powerset P
PpXq set of all subsets.
ηXpxq “ txu
µX union.

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Intro Plotkin Algebra to the Rescue Results

Monads

A monad is a triple xT, η, µy, with T an endofunctor and η : 1 ñ T, µ : TT ñ T natural transformations, such that:

T TT TTT TT TT T TT T

ηT Tη Id µ Tµ µT µ µ µ

List: L
LpXq set of all finite lists.
ηXpxq “ rxs
µX concatenation.
Powerset P
PpXq set of all subsets.
ηXpxq “ txu
µX union.
Distribution D
DpXq set of all probability

distributions.

ηXpxq point distribution.
µX weighted average.

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Intro Plotkin Algebra to the Rescue Results

Monads

A monad is a triple xT, η, µy, with T an endofunctor and η : 1 ñ T, µ : TT ñ T natural transformations, such that:

T TT TTT TT TT T TT T

ηT Tη Id µ Tµ µT µ µ µ

List: L
LpXq set of all finite lists.
ηXpxq “ rxs
µX concatenation.
Powerset P
PpXq set of all subsets.
ηXpxq “ txu
µX union.
Distribution D
DpXq set of all probability

distributions.

ηXpxq point distribution.
µX weighted average.
More examples: Multiset,

Exception, Reader, Writer, ...

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Intro Plotkin Algebra to the Rescue Results

Composing Monads with Distributive Laws

We can compose monads with the help of a distributive law - Beck 1969 xTS, ηTηS, µTµS ¨ TλSy Where λ : ST Ñ TS is a natural transformation satisfying the following axioms.

T SST STS TSS ST TS ST TS S STT TST TTS ST TS ST TS

ηS T TηS µS T Sλ λS TµS λ λ SηT ηT S SµT λT Tλ µT S λ λ

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Intro Plotkin Algebra to the Rescue Results

Examples

There is a distributive law LP ñ PL. It works like the famous ‘times over plus’ distributivity: pa ` bq ˚ c “ a ˚ b ` a ˚ c rta, bu, tcus ÞÑ tra, cs, rb, csu Many more work like this: MM ñ MM, LM ñ ML, MP ñ PM, ...

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Intro Plotkin Algebra to the Rescue Results

Examples

There is a distributive law LP ñ PL. It works like the famous ‘times over plus’ distributivity: pa ` bq ˚ c “ a ˚ b ` a ˚ c rta, bu, tcus ÞÑ tra, cs, rb, csu Many more work like this: MM ñ MM, LM ñ ML, MP ñ PM, ... Some general results:

If T is a commutative monad, and S a monad defined by

linear equations, then there is a distributive law ST ñ TS -

Manes and Mulry 2007.

There are variations on the above theorem for affine and

relevant monads - Dahlqvist, Parlant and Silva 2018.

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Intro Plotkin Algebra to the Rescue Results

All that glisters is not gold

Distributive laws are often quite intuitive, like times over plus.

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Intro Plotkin Algebra to the Rescue Results

All that glisters is not gold

Distributive laws are often quite intuitive, like times over plus. However...

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Intro Plotkin Algebra to the Rescue Results

All that glisters is not gold

Distributive laws are often quite intuitive, like times over plus. However...

We thought we had a distributive law DD ñ DD.

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Intro Plotkin Algebra to the Rescue Results

All that glisters is not gold

Distributive laws are often quite intuitive, like times over plus. However...

We thought we had a distributive law DD ñ DD.

But we made a mistake, which was hard to spot.

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Intro Plotkin Algebra to the Rescue Results

All that glisters is not gold

Distributive laws are often quite intuitive, like times over plus. However...

We thought we had a distributive law DD ñ DD.

But we made a mistake, which was hard to spot.

We are not alone.
Several examples of mistakes in the literature.
According to Bonsangue, Hansen, Kurz, and Rot:

“It can be rather difficult to prove the defining axioms of a distributive law.”

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Intro Plotkin Algebra to the Rescue Results

All that glisters is not gold

Distributive laws are often quite intuitive, like times over plus. However...

We thought we had a distributive law DD ñ DD.

But we made a mistake, which was hard to spot.

We are not alone.
Several examples of mistakes in the literature.
According to Bonsangue, Hansen, Kurz, and Rot:

“It can be rather difficult to prove the defining axioms of a distributive law.”

What to do now? Is there a distributive law at all?

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Intro Plotkin Algebra to the Rescue Results

All that glisters is not gold

Distributive laws are often quite intuitive, like times over plus. However...

We thought we had a distributive law DD ñ DD.

But we made a mistake, which was hard to spot.

We are not alone.
Several examples of mistakes in the literature.
According to Bonsangue, Hansen, Kurz, and Rot:

“It can be rather difficult to prove the defining axioms of a distributive law.”

What to do now? Is there a distributive law at all?
Our goal: to find general principles that tell us when no

distributive law exists.

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Intro Plotkin Algebra to the Rescue Results

Previous Results

No distributive law DP ñ PD - Plotkin / Varacca and Winskel 2005
No distributive law PD ñ DP - Varacca 2003, without proof
No monad structure on PD - Dahlqvist and Neves 2018
No monad structure on PP - Klin and Salamanca 2018
No distributive law TP ñ PT, with T satisfying some

technical conditions.

Klin and Salamanca 2018

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Intro Plotkin Algebra to the Rescue Results

Previous Results

No distributive law DP ñ PD - Plotkin / Varacca and Winskel 2005
No distributive law PD ñ DP - Varacca 2003, without proof
No monad structure on PD - Dahlqvist and Neves 2018
No monad structure on PP - Klin and Salamanca 2018
No distributive law TP ñ PT, with T satisfying some

technical conditions.

Klin and Salamanca 2018

What is so special about powerset?

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Intro Plotkin Algebra to the Rescue Results

Plotkin’s Proof

Main idea is to chase a specially chosen element: Ξ “ ta, bu `

1 2 tc, du P DPpXq

round the naturality diagram: Ξ ? ? ? λX λX DPpf q PDpf q

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Intro Plotkin Algebra to the Rescue Results

Plotkin’s Proof

Main idea is to chase a specially chosen element: Ξ “ ta, bu `

1 2 tc, du P DPpXq

round the naturality diagram: Ξ ? ? ? λX λX DPpf q PDpf q

Cleverly choose functions so

that on the bottom row, the unit laws can be applied: f paq “ a f pcq “ a f pbq “ b f pdq “ b

SLIDE 22

Intro Plotkin Algebra to the Rescue Results

Plotkin’s Proof

Main idea is to chase a specially chosen element: Ξ “ ta, bu `

1 2 tc, du P DPpXq

round the naturality diagram: Ξ ta, bu ? ? λX λX DPpf q PDpf q

Cleverly choose functions so

that on the bottom row, the unit laws can be applied: f paq “ a f pcq “ a f pbq “ b f pdq “ b

SLIDE 23

Intro Plotkin Algebra to the Rescue Results

Plotkin’s Proof

Main idea is to chase a specially chosen element: Ξ “ ta, bu `

1 2 tc, du P DPpXq

round the naturality diagram: Ξ ta, bu ta, bu λX λX DPpf q PDpf q

Cleverly choose functions so

that on the bottom row, the unit laws can be applied: f paq “ a f pcq “ a f pbq “ b f pdq “ b

SLIDE 24

Intro Plotkin Algebra to the Rescue Results

Plotkin’s Proof

Main idea is to chase a specially chosen element: Ξ “ ta, bu `

1 2 tc, du P DPpXq

round the naturality diagram: Ξ ta, bu ta, bu λX λX DPpf q PDpf q

Cleverly choose functions so

that on the bottom row, the unit laws can be applied: f paq “ a f pcq “ a f pbq “ b f pdq “ b

Take an inverse image to

learn fact about λpΞq:

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Intro Plotkin Algebra to the Rescue Results

Plotkin’s Proof

Main idea is to chase a specially chosen element: Ξ “ ta, bu `

1 2 tc, du P DPpXq

round the naturality diagram: Ξ ta, bu ta, bu λX λX DPpf q PDpf q´1

Cleverly choose functions so

that on the bottom row, the unit laws can be applied: f paq “ a f pcq “ a f pbq “ b f pdq “ b

Take an inverse image to

learn fact about λpΞq:

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Intro Plotkin Algebra to the Rescue Results

Plotkin’s Proof

Main idea is to chase a specially chosen element: Ξ “ ta, bu `

1 2 tc, du P DPpXq

round the naturality diagram: Ξ ta, bu ta, bu λX λX DPpf q PDpf q´1

Cleverly choose functions so

that on the bottom row, the unit laws can be applied: f paq “ a f pcq “ a f pbq “ b f pdq “ b

Take an inverse image to

learn fact about λpΞq:

3 facts together get

contradiction.

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Intro Plotkin Algebra to the Rescue Results

Analysing the Proof

What is so special about Powerset?

“Going down”: idempotence, commutativity.
“Going up”: easy to take inverse image.

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Intro Plotkin Algebra to the Rescue Results

Analysing the Proof

What is so special about Powerset?

“Going down”: idempotence, commutativity.
“Going up”: easy to take inverse image.

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Intro Plotkin Algebra to the Rescue Results

Analysing the Proof

What is so special about Powerset?

“Going down”: idempotence, commutativity.
“Going up”: easy to take inverse image.

Idea: try an algebraic perspective!

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Intro Plotkin Algebra to the Rescue Results

A Quick Reminder

Algebraic theory:
Signature Σ and a set of variables give terms.
Axioms E and equational logic give equivalence of terms.

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Intro Plotkin Algebra to the Rescue Results

A Quick Reminder

Algebraic theory:
Signature Σ and a set of variables give terms.
Axioms E and equational logic give equivalence of terms.

Theory of complete semilattices: Σ “ tKp0q, _p2qu

K _ x “ x (left unit) x _ K “ x (right unit) x _ y “ y _ x (commutativity) x _ x “ x (idempotence)

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Intro Plotkin Algebra to the Rescue Results

A Quick Reminder

Algebraic theory:
Signature Σ and a set of variables give terms.
Axioms E and equational logic give equivalence of terms.

Theory of complete semilattices: Σ “ tKp0q, _p2qu

K _ x “ x (left unit) x _ K “ x (right unit) x _ y “ y _ x (commutativity) x _ x “ x (idempotence)

Monads arise from free/forgetful adjunction between Set and

category of pΣ, Eq-algebras.

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Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.

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Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.

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Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

SLIDE 36

Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

Theorem (Pir´
g and Staton): There is a distributive law

VP ñ PV iff there exits a composite theory.

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Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

Theorem (Pir´
g and Staton): There is a distributive law

VP ñ PV iff there exits a composite theory.

How to get general no-go theorem:

SLIDE 38

Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

Theorem (Pir´
g and Staton): There is a distributive law

VP ñ PV iff there exits a composite theory.

How to get general no-go theorem:
Proof by contradiction, so assume a composite theory exists.

SLIDE 39

Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

Theorem (Pir´
g and Staton): There is a distributive law

VP ñ PV iff there exits a composite theory.

How to get general no-go theorem:
Proof by contradiction, so assume a composite theory exists.
Start with a clever term ‘out of order’, a V-term of P-terms:

vpppa, bq, ppc, dqq “ p1rv 1

x1{x1s.

SLIDE 40

Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

Theorem (Pir´
g and Staton): There is a distributive law

VP ñ PV iff there exits a composite theory.

How to get general no-go theorem:
Proof by contradiction, so assume a composite theory exists.
Start with a clever term ‘out of order’, a V-term of P-terms:

vpppa, bq, ppc, dqq “ p1rv 1

x1{x1s.

Bring the V-part or the P-part of chosen term to a variable.

SLIDE 41

Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

Theorem (Pir´
g and Staton): There is a distributive law

VP ñ PV iff there exits a composite theory.

How to get general no-go theorem:
Proof by contradiction, so assume a composite theory exists.
Start with a clever term ‘out of order’, a V-term of P-terms:

vpppa, bq, ppc, dqq “ p1rv 1

x1{x1s.

Bring the V-part or the P-part of chosen term to a variable.
Using substitutions + idempotence.

SLIDE 42

Intro Plotkin Algebra to the Rescue Results

No-Go theorems algebraically, how does it work?

Need equivalent of distributive law: composite theory.
Idea: Use terms of V as variables for the terms of P.
That is, all terms are equal to a P-term of V-terms: prvx{xs,

without introducing any ‘extra’ equalities.

Theorem (Pir´
g and Staton): There is a distributive law

VP ñ PV iff there exits a composite theory.

How to get general no-go theorem:
Proof by contradiction, so assume a composite theory exists.
Start with a clever term ‘out of order’, a V-term of P-terms:

vpppa, bq, ppc, dqq “ p1rv 1

x1{x1s.

Bring the V-part or the P-part of chosen term to a variable.
Using substitutions + idempotence.
Need additional assumptions about variables to get useful

conclusions.

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Intro Plotkin Algebra to the Rescue Results

Generalised Plotkin Theorem

Theorem (Generalised Plotkin)

Let P and V be algebraic theories such that: Then, there is no composite theory of P and V. And so: no distributive law VP ñ PV for corresponding monads.

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Intro Plotkin Algebra to the Rescue Results

Generalised Plotkin Theorem

Theorem (Generalised Plotkin)

Let P and V be algebraic theories such that:

1. P has a binary term p which is commutative and idempotent.

Then, there is no composite theory of P and V. And so: no distributive law VP ñ PV for corresponding monads.

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Intro Plotkin Algebra to the Rescue Results

Generalised Plotkin Theorem

Theorem (Generalised Plotkin)

Let P and V be algebraic theories such that:

1. P has a binary term p which is commutative and idempotent.
2. In addition, p is such that if ppx1, x2q “ p1 then # varpp1q ď 2

Then, there is no composite theory of P and V. And so: no distributive law VP ñ PV for corresponding monads.

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Intro Plotkin Algebra to the Rescue Results

Generalised Plotkin Theorem

Theorem (Generalised Plotkin)

Let P and V be algebraic theories such that:

1. P has a binary term p which is commutative and idempotent.
2. In addition, p is such that if ppx1, x2q “ p1 then # varpp1q ď 2
3. V has a binary term v which is idempotent.

Then, there is no composite theory of P and V. And so: no distributive law VP ñ PV for corresponding monads.

SLIDE 47

Intro Plotkin Algebra to the Rescue Results

Generalised Plotkin Theorem

Theorem (Generalised Plotkin)

Let P and V be algebraic theories such that:

1. P has a binary term p which is commutative and idempotent.
2. In addition, p is such that if ppx1, x2q “ p1 then # varpp1q ď 2
3. V has a binary term v which is idempotent.
4. In addition, v is such that if vpy1, y2q “ v1 then # varpv1q ě 2

Then, there is no composite theory of P and V. And so: no distributive law VP ñ PV for corresponding monads.

SLIDE 48

Intro Plotkin Algebra to the Rescue Results

Generalised Plotkin Theorem

Theorem (Generalised Plotkin)

Let P and V be algebraic theories such that:

1. P has a binary term p which is commutative and idempotent.
2. In addition, p is such that if ppx1, x2q “ p1 then # varpp1q ď 2
3. V has a binary term v which is idempotent.
4. In addition, v is such that if vpy1, y2q “ v1 then # varpv1q ě 2
5. For all v1 in V, if v1 “ y then varpv1q “ tyu

Then, there is no composite theory of P and V. And so: no distributive law VP ñ PV for corresponding monads.

SLIDE 49

Intro Plotkin Algebra to the Rescue Results

Can we have more of these please?

What is so special about...

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Intro Plotkin Algebra to the Rescue Results

Can we have more of these please?

What is so special about... idempotence?

SLIDE 51

Intro Plotkin Algebra to the Rescue Results

Can we have more of these please?

What is so special about... idempotence? It reduces terms to a variable in a controlled way.

SLIDE 52

Intro Plotkin Algebra to the Rescue Results

Can we have more of these please?

What is so special about... idempotence? It reduces terms to a variable in a controlled way. Unitality axioms do this too!

SLIDE 53

Intro Plotkin Algebra to the Rescue Results

Can we have more of these please?

What is so special about... idempotence? It reduces terms to a variable in a controlled way. Unitality axioms do this too! ù ñ Many No-Go theorems:

P has idempotent term, V has idempotent term. (Plotkin)
P has idempotent term, V has term with units.
P has term with units, V has idempotent term.
P has term with units, V has term with units. (List monad!!)

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Intro Plotkin Algebra to the Rescue Results

Monads combinations we’ve seen before

Is there a distributive law of type Row˝Column ñ Column˝Row? Previous results: L M P D L Y Y M Y Y P N N D N

SLIDE 55

Intro Plotkin Algebra to the Rescue Results

Monads combinations we’ve seen before

Is there a distributive law of type Row˝Column ñ Column˝Row? New results: L M P D L N Y Y N M N Y Y N P N N N N D N N N

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Intro Plotkin Algebra to the Rescue Results

Extended Boom Hierarchy

associative commutative idempotent Tree Tr N N N Idempotent tree I N N Y Commutative tree C N Y N Commutative Idempotent tree CI N Y Y List L Y N N Associative Idempotent tree AI Y N Y Multiset M Y Y N Powerset P Y Y Y

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Intro Plotkin Algebra to the Rescue Results

Extended Boom Hierarchy

associative commutative idempotent Tree Tr N N N Idempotent tree I N N Y Commutative tree C N Y N Commutative Idempotent tree CI N Y Y List L Y N N Associative Idempotent tree AI Y N Y Multiset M Y Y N Powerset P Y Y Y

+ all non-empty-versions: without constants. 256 combinations of monads.

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Intro Plotkin Algebra to the Rescue Results

Results

Is there a distributive law of type Row˝Column ñ Column˝Row? Row and Column both without units: Tr` I` C` CI` L` AI` M` P` Tr` Y Y Y I` N N N N C` Y Y? Y Y CI` N N N N L` Y Y Y Y AI` N N N N M` Y Y Y P` N N N N

SLIDE 59

Intro Plotkin Algebra to the Rescue Results

More Results

Is there a distributive law of type Row˝Column ñ Column˝Row? Row with unit, Column without unit: Tr` I` C` CI` L` AI` M` P` Tr Y N N N Y Y I N N N N C Y N N N Y Y CI N N N N L Y N N Y N Y Y AI N N N N M Y N N N Y Y P N N N N

SLIDE 60

Intro Plotkin Algebra to the Rescue Results

Even More Results

Is there a distributive law of type Row˝Column ñ Column˝Row? Row without unit, Column with unit: Tr I C CI L AI M P Tr` Y Y I` N N N N N N C` Y Y CI` N N N N N N L` Y Y AI` N N N N N N M` Y Y P` N N N N N N

SLIDE 61

Intro Plotkin Algebra to the Rescue Results

Final slide with results

Is there a distributive law of type Row˝Column ñ Column˝Row? Row and Column both have units: Tr I C CI L AI M P Tr N N N N N N Y Y I N N N N N N N N C N N N N N N Y Y CI N N N N N N N N L N N N N N N Y Y AI N N N N N N N N M N N N N N N Y Y P N N N N N N N N