Programming Languages in String Diagrams

[ four ]

Local stores

Paul-André Melliès

Oregon Summer School in Programming Languages

June 2011

Finitary monads

A monadic account of algebraic theories

Algebraic presentations of effects

We want to reason about programs with effects like states, excep-tions...

Computational monads:

Apimpurep

// B = Appurep

// T (B)

Algebraic theories:

operations : A n−→ A and equations

A useful correspondence

Recall that a monad

T : Set −→ Set

is finitary when it preserves the filtered colimit.

Important fact. There is an equivalence of categories

the categoryof

algebraic theories�

the categoryof

finitary monads

4

Finitary monads

Alternative definition. A monad

T : Set −→ Set

is finitary when the diagram

Set

id08

FinSet i//

T◦ i

BB���������������������

Set

T

ZZ55555555555555555555

exhibits the functor T as a left Kan extension of T ◦ i along i.

5

Finitary monad

A monad T on the category Set is finitary when every map

[1] −→ TA

factors as

[1] e−→ T [p]

T f−→ TA

for a pair of maps

[1] −→ T [p] [p] −→ A.

6

Finitary monad

Moreover, the factorization should be unique up to zig-zag:

T [p]

Tu

��

T f1

!!CCCCCCCCCCCCCCCCCCCCCCC

[1]

e1

=={{{{{{{{{{{{{{{{{{{{{{

e2

!!CCCCCCCCCCCCCCCCCCCCCC TA

T [q]

T f2

=={{{{{{{{{{{{{{{{{{{{{{{

7

The Kleisli category of a monad

The Kleisli category Set T associated to a monad

T : Set −→ Set

has sets as objects, and functions

f : A −→ T(B)

as morphisms from A to B.

An alternative formulation of the category of free algebras.

8

The Lawvere theory of a monad

The Lawvere theory Θ T associated to a monad

T : Set −→ Set

is the full subcategory of Set T with finite ordinals

[p] = { 0 , 1 , . . . , p − 1 }

as objects.

Remark. The category Θ T has finite sums.

9

Algebras of a monad

A set A equipped with a function

alg : TA −→ A

making the two diagrams below commute:

TTAT ( alg )

//

µ

��

TA

alg

��

TAalg

//A

TA

alg

��::::::::::::::::::::::::

A

ηA

BB������������������������ id //A

10

Key theorem [ originating from Lawvere ]

The category of T-algebras

is equivalent to

the category of finite product preserving functors

ΘopT −→ Set

This property holds for every finitary monad T.

11

Finitary monads as saturated theories

The finitary monad describes the equational classes of the theory :

Tn =

terms with n variables modulo equations

where

n =

x0 , x1 , · · · , xn−1

The monad often reveals a canonical form for the equational

theory.

Mnemoids

An algebraic presentation of the state monad

The state monad

A program accessing one register with a finite set V of values

Aimpure

// B

is interpreted as a function

V × Apure

// V × B

thus as a function

Apure

// V ⇒ (V × B)

Hence, the state monad

T : A 7→ V ⇒ (V × A)

14

What is an algebra of the state monad ?

An elegant answer by Plotkin and Power [2002]

Mnemoids

A mnemoid is a set A equipped with a V-ary operation

lookup : A V−→ A

and a unary operation

update〈val〉 : A −→ A

for each value val ∈ V .

16

Mnemoids

Typically, the lookup operation on the boolean space

V = { false , true }

is binary :

lookup : A × A −→ A

The intuition is that

lookup (u, v) =

{u when the register is falsev when the register is true

17

Annihilation lookup – update

A V

lookup

AAAAAAAAAAAAAAAAAAAAA

A

update〈V〉

>>}}}}}}}}}}}}}}}}}}}}} id //A

term = lookup

val 7→ update〈val〉 term

18

Annihilation lookup – update

Here, the map

update〈V〉 : A −→ A V

is the V-ary vector update〈0〉 , · · · , update〈V−1〉

also noted val 7→ update〈val〉

19

Annihilation lookup – update

x =x

x

true

false

20

Interaction update – update

A

update〈val1〉

��?????????????????????????

A

update〈val2〉

??������������������������� update〈val2〉 //A

update〈val1〉 ◦ update〈val2〉 = update〈val2〉

21

Interaction update – update

val2 val1 = val2

22

Interaction update – lookup

AV lookup//

Aval

��

A

update〈val〉

��

Aupdate〈val〉 //A

update〈val〉 ◦ lookup

x 7→ term(x)

= update〈val〉 ◦ term(val)

23

Interaction update – lookup

x

y

true = truex

24

Key theorem [ Plotkin & Power ]

the category of mnemoids

is equivalent to

the category of algebras of the state monad

Provides an algebraic presentation of the state monad

25

A proof in three steps

1. establish that the state monad T is finitary [easy]

2. construct a functor [easy]

ι : Θ M −→ Θ T

starting from the Lawvere theory Θ M of mnemoidsby interpreting the update and lookup operations andby checking that the equations are valid in Θ T

3. show that the functor ι is full and faithful [rewriting]

26

Store with several locations

Algebraic presentation of the state monad

The global state monad

The state monad is generally defined as

T : A 7→ S⇒ (S × A)

for a set of states

S = V L

induced by a set L of locations and a finite set V of values.

28

Global store [Plotkin & Power ]

A global store is a family of compatible mnemoidal structures

lookup〈loc〉 : AV−→ A

update〈loc,val〉 : A −→ A

one for each location loc ∈ L.

A tensor product of algebraic theories

29

Annihilation lookup – update

AV

lookup〈loc〉

@@@@@@@@@@@@@@@@@@@@@@@

A

update〈loc,V〉

>>~~~~~~~~~~~~~~~~~~~~~~~ id //A

lookup〈loc〉

val 7→ update〈loc,val〉 term

= term

30

Annihilation lookup – update

x =

true

falsex

x

31

Interaction update – update

A

update〈loc,val〉

��?????????????????????????

A

update〈loc,val′〉

??�������������������������

update〈loc,val′〉//A

update〈loc,val1〉 ◦ update〈loc,val2〉 = update〈loc,val2〉

32

Interaction update – update

val1val2 = val2

33

Interaction update – lookup

AVlookup〈loc〉 //

Aval

��

A

update〈loc,val〉

��

A update〈loc,val〉//A

update〈loc,val〉 ◦ lookup〈loc〉

x 7→ term(x)

= update〈loc,val〉 ◦ term(val)

34

Interaction update – lookup

true

y

x

= truex

35

Interaction update – lookup

false

y

x

= falsey

36

Commutation update – update

Aupdate〈loc,val〉 //

update〈loc′,val′〉

��

A

update〈loc′,val′〉

��

A update〈loc,val〉//A

update〈loc,v〉 update〈loc′,v′〉 x=

update〈loc′,v′〉 update〈loc,v〉 x

when loc , loc′

37

Commutation update – update

val′

val = val

val′

38

Corollary [ Plotkin & Power ]

the category of objects with a global store

is equivalent to

the category of algebras of the state monad

39

Graded monads

A stratified version of finitary monads

Presheaf models

Key idea: interpret a type A as a family of sets

A[0] A[1] · · · A[n] · · ·

indexed by natural numbers, where each set

A[n]

contains the programs of type A which have access to n variables.

41

Presheaf models

This defines a covariant presheaf

A[n] : Inj −→ Set

on the category Inj of natural numbers and injections.

The action of the injections on A are induced by the operations

collect〈loc〉 : A[n] −→ A[n+1]

defined for 0 ≤ loc ≤ n.

42

Local stores [Plotkin – Power]

The slightly intimidating monad

TA : n 7→ S[n]⇒

∫ p∈Inj S[p]× A[p] × Inj(n, p)

on the presheaf category [Inj,Set] where the contravariant presheaf

S[p] = V p

describes the states available at degree p.

43

Graded arities

A graded arity is defined as a finite sum of representables

[ p0 , · · · , pn ] = 〈0〉 + · · · + 〈0〉︸ ︷︷ ︸p0 times

+ · · · + 〈n〉 + · · · + 〈n〉︸ ︷︷ ︸pn times

where each representable presheaf

〈n〉 : Inj −→ Set

is defined as

〈n〉 = p 7→ Inj(n, p)

44

Graded arities

This defines a full and faithful functor

Σ Inj op−→ Inj op

which generalizes the full and faithful functor

FinSet −→ Set

encountered in the case of finitary monads.

45

Graded monads

Definition. A monad

T : [Inj,Set] −→ [Inj,Set]

is graded when the diagram

[Inj,Set]

id19

Σ Inj opi

//

T◦ i

AA������������������������

[Inj,Set]

T

^^=========================

exhibits the functor T as a left Kan extension of T ◦ i along i.

46

Graded monad

A monad T on the category [Inj,Set] is graded when every map

〈n〉 −→ TA

factors as

〈n〉 e−→ T [p0, . . . , pk]

T f−→ TA

for a pair of maps

〈n〉 −→ T [p0, . . . , pk] [p0, . . . , pk] −→ A.

47

Graded monad

Moreover, the factorization should be unique up to zig-zag:

T [p0, · · · , pi]

Tu

��

T f1

%%KKKKKKKKKKKKKKKKKKKKKKKK

〈n〉

e1

99ssssssssssssssssssssssss

e2%%JJJJJJJJJJJJJJJJJJJJJJJ TA

T [q0, · · · , q j]

T f2

99tttttttttttttttttttttttt

48

Motivating example

Theorem. The local state monad

TA : n 7→ S[n]⇒

∫ p∈Inj S[p]× A[p] × Inj(n, p)

is a graded monad on the presheaf category [Inj,Set].

49

Graded theories

A stratified version of algebraic theories

The Lawvere theory of a graded monad

The Lawvere theory Θ T associated to a graded monad

T : [Inj,Set] −→ [Inj,Set]

is the full subcategory of [Inj,Set] T with the graded arities

[ p0 , · · · , pn ]

as objects.

Remark. The category Θ T has finite sums.

51

Modules over the category Inj

An Inj-module is a category A equipped with an action

� : Inj ×A −→ A

(m,A) m � A

satisfying the expected properties:

(p + q) � A = p � (q � A) 0 � A = A

52

Modules over the category Inj

A category A equipped with an endofunctor

D : A −→ A

and two natural transformations

permute : D ◦D −→ D ◦D collect : Id −→ D

depicted as follows in the language of string diagrams:

D

D

D

D

D

53

Yang-Baxter equation

D

D

D

D

D

D

=D

D

D

D

D

D

54

Symmetry

D

D

D

D

=D

D

D

D

55

Interaction dispose – permute

D

D D

=D

D D

DD

D

=DD

D

56

Graded theories

A graded theory is a category with finite products with objects

[ p0 , · · · , pn ] = 〈0〉 × · · · × 〈0〉︸ ︷︷ ︸p0 times

× · · · × 〈n〉 × · · · × 〈n〉︸ ︷︷ ︸pn times

equipped with an action of the category Inj satisfying

D(A × B) � DA × DB D(1) � 1

In particular

D [ p0 , · · · , pn ] = [ 0 , p0 , · · · , pn ]

57

The trivial graded theory

Σ Inj op is the free Inj-module with finite sums such that

D(A + B) � DA + DB D(0) � 0

Π Inj is the free Inj-module with finite products such that

D(A × B) � DA × DB D(1) � 1

58

The graded theory of local stores

Let |מ be the graded theory associated to the monad of localstores.

Fact.

|מ is the free Inj-module with finite sums such that

D(A + B) � DA + DB D(0) � 0

such that D is a dynamic mnemoid.

59

Dynamic mnemoids

A dynamic mnemoid is a pair of sets

A[0] A[1]

equipped with the following operations

lookup : AV[1] −→ A[1]

update〈val〉 : A[1] −→ A[1]

fresh〈val〉 : A[1] −→ A[0]

collect : A[0] −→ A[1]

satisfying a series of basic equations.

60

Garbage collect : fresh – dispose

val = id

61

Interaction fresh – update

val2 val1 = val2

62

Interaction fresh – lookup

x

y

true = truex

x

y

false = falsey

63

Model of a graded theory

A model of a graded theory |מ is a functor

M : |מ −→ [Inj,Set]

which preserves the finite products and the action of D.

A stratified version of the usual algebraic theories

64

The graded theory of local mnemoids

Main theorem [after Plotkin and Power]

The category of models of the theory |מ of local mnemoids

is equivalent to

the category of algebras of the local state monad

65

Local stores in string diagrams

A graphical account of memory calls

Annihilation lookup – update

x =

true

falsex

x

The equation D(A × B) � DA ×DB means

that space commutes with time ramification !!!

Interaction update – update

val1val2 = val2

68

Interaction update – lookup

true

y

x

= truex

69

Interaction update – lookup

false

y

x

= falsey

70

Interaction fresh – dispose

val

D

D

D

D

D

D

=D

D

D

D

D

D

71

Interaction fresh – update

D

D

D

D

D

D

D

val1val2 = D

D

D

D

D

D

D

val2

72

Interaction fresh – lookup

true

D

D

D

y

x

= true

D

D

D

y

73

Interaction fresh – lookup

false

D

D

D

y

x

= false

D

D

D

y

74

Interaction fresh – permutation

val

D

D

D

D

D

D

D

= valD

D

D

D

D

D

D

75

The Segal condition

A basic tool in higher dimensional algebra

76

In higher dimensional algebra

Exactly the same categorical combinatorics !!!

77

In higher dimensional algebra

A globular set is defined as a contravariant presheaf over the cate-gory

0s //

t// 1

s //

t// 2

s //

t// · · ·

defined by the equations

s ◦ s = s ◦ t t ◦ s = t ◦ t

Anω-category is then defined as an algebra of a particular monad T.

Is the notion of ω-category equational ?

78

Level 1 : the free category monad

A category C is a graph where every path

A0f1 // A1

f2 // A2f3 // · · ·

fn // An

defines an edge

A0◦n ( f1, ··· , fn)

// An

where the operations ◦n are required to be associative.

the arities of the theory are finite paths rather than finite sets

Category with arities

A fully faithful and dense functor

i0 : Θ0 −→ A

where Θ0 is a small category.

Typical example: the full subcategory Θ0 of linear graphs

[n] = 0 // 1 // · · · // n

in the category Graph.

80

The nerve functor

This induces a fully faithful functor

nerve : A −→ Θ̂0

which transports every object A of the category A into the presheaf

nerve : Θ0 −→ Set

p 7→ A( i0p , A )

81

The nerve functor

Definition of a monad T with arity

◦ the functor T is the left kan extension of T ◦ i0 along the functor i0

A

id08

Θ0 i0//

T◦i0

EE������������������������

A

T

XX111111111111111111111111

This enables to reconstruct T from its restriction to the arities

83

Monads with arities

◦ this left kan extension is preserved by the nerve functor:

Θ̂0

A

nerve

OO

id08

Θ0 i0//

T◦i0

EE�������������������������

A

T

YY2222222222222222222222222

84

A purely combinatorial formulation

For every object A, the canonical morphism

∫ p∈Θ0A(i0n,Ti0p) ×A(i0p,A) −→ A(i0n,TA)

is an isomorphism.

85

Unique factorization up to zig-zag

Every map

n −→ TA

in the category A decomposes as

n e−→ Tp

T f−→ TA

A finite number of words encounters a finite number of letters

A finite path in the free category TA encounters a finite path in A

86

Unique factorization up to zig-zag

The factorization should be unique up to zig-zag of

Ti0p

Ti0u

��

T f1

!!DDDDDDDDDDDDDDDDD

i0n

e1

=={{{{{{{{{{{{{{{{

e2!!CCCCCCCCCCCCCCCC

TA

Ti0qT f2

==zzzzzzzzzzzzzzzzz

87

Model of an algebraic theory

A model A of a Lawvere theory L is a presheaf

A : ΘopL

−→ Set

such that the induced presheaf

Θop0

i0 //ΘopL

A //Set

is representable along i0.

88

Key theorem

The category of T-modules

is equivalent to

the category of models of the theory ΘT

This property holds for every monad T with arities Θ0.

89

Algebraic theories with arities

A 2-dimensional approach to Lawvere theories

90

Algebraic theory with arities

An algebraic theory L with arities

i0 : Θ0 −→ A

is an identity-on-object functor

L : Θ0 −→ ΘL

such that...

91

Algebraic theories with arities

... there exists a functor

F : A −→ A

making the diagram

Θ̂0∃L // Θ̂L

L∗ // Θ̂0

nerve

OO

nerve

OO

AF // A

commute up natural isomorphism.

92

Algebraic theories with arities

Set

/7

Θop0 L

//

nerve(A)

EE

ΘopL

left Kan extension

YY4444444444444444444444444

This simply means that the free construction works...

93

Theorem

the category of algebraic theories with arity (A, i0)

is equivalent to

the category of monads with arity (A, i0)

This generalizes the traditional correspondence

algebraic theories ∼ finitary monads

Also extends the result by Power on enriched Lawvere theories

94