Local stores Paul-Andr Mellis Oregon Summer School in Programming - - PowerPoint PPT Presentation

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: A

pimpurep

• B

= A

ppurep

• T (B)

Algebraic theories:

• perations

: 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 category

• f

algebraic theories

• the category
• f

finitary monads

4

Finitary monads

Alternative definition. A monad T : Set −→ Set is finitary when the diagram Set

id

• FinSet

i

• T◦ i
• Set

T

• 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
• [1]

e1

• e2
• 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: TTA

T ( alg )

• µ
• TA

alg

• TA

alg

A

TA

alg

• A

ηA

• id

A

10

Key theorem [ originating from Lawvere ]

The category of T-algebras is equivalent to the category of finite product preserving functors Θ op

T

−→ 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 A

impure

• B

is interpreted as a function V × A

pure

• V × B

thus as a function A

pure

• V ⇒ (V × B)

Hence, the state monad T : A → 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 updateval : 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 false v when the register is true

17

Annihilation lookup – update

A V

lookup

• A

updateV

• id

A

term = lookup       val → updateval term      

18

Annihilation lookup – update

Here, the map updateV : A −→ A V is the V-ary vector       update0 , · · · , updateV−1       also noted       val → updateval      

19

Annihilation lookup – update

x

=

x x true false

20

Interaction update – update

A

updateval1

• A

updateval2

• updateval2

A

updateval1 ◦ updateval2 = updateval2

21

Interaction update – update

val2 val1

=

val2

22

Interaction update – lookup

AV

lookup

• Aval
• A

updateval

• A

updateval

A

updateval ◦ lookup       x → term(x)       = updateval ◦ term(val)

23

Interaction update – lookup

x y true

=

true x

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 mnemoids by interpreting the update and lookup operations and by 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 → 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 lookuploc : AV −→ A updateloc,val : A −→ A

• ne for each location loc ∈ L.

A tensor product of algebraic theories

29

Annihilation lookup – update

AV

lookuploc

• A

updateloc,V

• id

A

lookuploc       val → updateloc,val term       = term

30

Annihilation lookup – update

x

=

t r u e f a l s e

x x

31

Interaction update – update

A

updateloc,val

• A

updateloc,val′

• updateloc,val′

A

updateloc,val1 ◦ updateloc,val2 = updateloc,val2

32

Interaction update – update

val1 val2

=

val2

33

Interaction update – lookup

AV

lookuploc

• Aval
• A

updateloc,val

• A

updateloc,val

A

updateloc,val ◦ lookuploc       x → term(x)       = updateloc,val ◦ term(val)

34

Interaction update – lookup

true

y x

=

true

x 35

Interaction update – lookup

false

y x

=

false

y 36

Commutation update – update

A

updateloc,val

• updateloc′,val′
• A

updateloc′,val′

• A

updateloc,val

A

updateloc,v updateloc′,v′ x = updateloc′,v′ updateloc,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

• n the category Inj of natural numbers and injections.

The action of the injections on A are induced by the operations collectloc : A[n] −→ A[n+1] defined for 0 ≤ loc ≤ n.

42

Local stores [Plotkin – Power]

The slightly intimidating monad TA : n → S[n] ⇒       p∈Inj S[p] × A[p] × Inj(n, p)      

• n 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 → 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]

id

• Σ Inj op

i

• T◦ i
• [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
• n

e1

• e2
• TA

T [q0, · · · , qj]

T f2

• 48

Motivating example

• Theorem. The local state monad

TA : n → 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 D D D

=

D D 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)

• 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

Letמ‌ be the graded theory associated to the monad of local stores. Fact.

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

D(A + B)

• DA + DB

D(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] updateval : A[1] −→ A[1] freshval : 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

=

true x x y false

=

false y

63

Model of a graded theory

A model of a graded theoryמ‌ 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 theoryמ‌ 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

=

t r u e f a l s e

x x

The equation D(A × B) DA × DB means that space commutes with time ramification !!!

Interaction update – update

val1 val2

=

val2

68

Interaction update – lookup

true

y x

=

true

x 69

Interaction update – lookup

false

y x

=

false

y 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 val1 val2

=

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

=

val D 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

s

• t
• 1

s

• t
• 2

s

• t
• · · ·

defined by the equations s ◦ s = s ◦ t t ◦ s = t ◦ t An ω-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 A0

f1

A1

f2

A2

f3

· · ·

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] =

1 · · · n

in the category Graph.

80

The nerve functor

This induces a fully faithful functor nerve :

A

−→

• Θ0

which transports every object A of the category A into the presheaf nerve : Θ0 −→ Set p →

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

id

• Θ0

i0

• T◦i0
• A

T

• 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:
• Θ0

A

nerve

• id
• Θ0

i0

• T◦i0
• A

T

• 84

A purely combinatorial formulation

For every object A, the canonical morphism p∈Θ0

A(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
• i0n

e1

• e2
• TA

Ti0q

T f2

• 87

Model of an algebraic theory

A model A of a Lawvere theory L is a presheaf A : Θop

L

−→ Set such that the induced presheaf Θop

i0

Θop

L 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

• Θ0

∃L

• ΘL

L∗

• Θ0

nerve

• nerve
• A

F

• A

commute up natural isomorphism.

92

Algebraic theories with arities

Set

• Θop

L

• nerve(A)
• Θop

L left Kan extension

• 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