Local states in string diagrams Paul-Andr Mellis CNRS & - - PowerPoint PPT Presentation

Local states in string diagrams

Paul-André Melliès

CNRS & Université Paris Diderot RTA-TLCA 2014 Vienna Summer Logic 14 −→ 17 July 2014

Algebraic effects

A seminal idea by Power and Plotkin

Algebraic presentations of effects

We want to reason about programs with effects like states, exceptions... Computational monads: A pimpurep

• B

= A ppurep

• T (B)

Equational theories:

• perations

: A n −→ A and equations

The state monad

A program accessing one register with a set of values Val = {true, false} A impure

• B

is interpreted as a function Val × A pure

• Val × B

thus as a function A pure

• Val ⇒ (Val × B)

Hence, the state monad T : A → Val ⇒ (Val × A)

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

Algebraic effects

Key idea : Effects are performed by algebraic operations on types The algebraic theory is presented by a family of n-ary operations

• peration

: A × · · · × A

• n

−→ A together with a family of well-chosen equations.

States

Similarly, memory calls are performed by a binary operation lookup : A × A −→ A and two unary operations update true : A −→ A update false : A −→ A The three equations of the theory are: term = lookup

• update true (term) , update false (term)
• updateval1 ◦ updateval2

= updateval2 updateval ◦ lookup

• term(true), term( false)
• =

updateval (term(val))

Creation lookup – update

A × A

lookup

• A

updatetrue,false

• id

A

term = lookup

• update true (term) , update false (term)

Creation lookup – update

Here, the map updatetrue,false : A −→ A × A is the pair

• updatetrue

, update false

• also noted
• val → updateval

Creation lookup-update

root

x

=

x

false

root

true

x

Interaction update – update

A

updateval1

• A

updateval2

• updateval2

A

updateval1 ◦ updateval2 = updateval2

Interaction update-update

val

root

 val

x

=

root

val

x

Interaction update – lookup

A × A

lookup

• Aval
• A

updateval

• A

updateval

A

updateval ◦ lookup

• term(true) , term(false)
• =

updateval ◦ term(val)

Interaction update – lookup

x y

true

root

=

x

true

root x y

false

root

=

y

false

root

Key theorem [ Plotkin & Power 2002 ]

the category of mnemoids is equivalent to the category of algebras of the state monad This provides an algebraic presentation of the state monad

A proof based on rewriting

[LICS 2010] A clean bridge between rewriting and semantics

From algebraic theories to monads

Fact. Every algebraic theory T induces a monad T : Set −→ Set which transports every set A to its free model TA defined as T A =

• terms with variables in A modulo the equations of T

Finitary monads

• Definition. A monad

T : Set −→ Set is called finitary when it preserves filtered colimits. Important fact. There is an equivalence of categories the category

• f

algebraic theories

• the category
• f

finitary monads

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.

Finitary monads

Hence, a finitary monad T : Set −→ Set is entirely described by its restriction to finite sets [n] =

• x0 , . . . , xn−1
• .

A proof by rewriting in two steps

Step 1. establish that the state monad T : A → Val ⇒ (Val × A) : Set −→ Set is a finitary monad. Step 2. show that it computes the free mnemoid on finite sets [n] =

• x0 , . . . , xn−1

Canonical form theorem [LICS 2010]

Every term with n variables is equivalent to a term of the form lookup

• val → update f(val) xg(val)
• which interprets the function

Val −→

n

• Val + · · · + Val

val → injg(val) f(val)

Canonical form theorem [LICS 2010]

p

wal

root

val

q x x

Proof by rewriting

Step 1. use the creation rule to rewrite term

create

−→ lookup

• val → updateval term
• Step 2. use the interaction rules to rewrite

updateval term

∗

−→ updatewal xp for some value wal and some variable xp. Step 3. conclude...

Proof by rewriting

Corollary. The state monad T : Set −→ Set coincides with the free mnemoid monad when restricted to finite sets. Important consequence: One recovers in this way the original theorem by Plotkin & Power.

Remark: the equation below is redundant

A × A × A × A

lookup×lookup

• [π11,π22]
• A × A

lookup

• A × A

lookup

A

lookup

• val1 → lookup [ val2 → term (val1, val2) ]
• =

lookup

• val → term (val, val)

The local state monad

A more sophisticated example by Plotkin and Power

Presheaf models

Key idea: interpret a type A as a family of sets A0 A1 · · · An · · · indexed by natural numbers, where each set An contains the programs of type A which have access to n variables.

Presheaf models

This defines a covariant presheaf An : Inj −→ Set

• n the category Inj of natural numbers and injections.

Every injection f : p −→ q induces a function Af : Ap −→ Aq

• btained by renaming every register i ∈ [p] by the register f(i) ∈ [q].

Local stores [Plotkin & Power 2002]

The slightly intimidating monad L A : n → S n ⇒ p∈Inj S p × Ap × Inj (n, p)

• n the presheaf category [Inj, Set] where the contravariant presheaf

S p = Val p describes the states available at degree p.

Our methodology here

⊲ Show that the monad is finitary in some sense. ⊲ Establish a similar canonical form theorem for local states. Fine, but what is the appropriate notion of finitary monad here ?

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)

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.

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.

Main result of the paper

Theorem. The local state monad L A : n → S n ⇒ p∈Inj S p × Ap × Inj (n, p)

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

Main result of the paper

In particular, the local state monad L : [Inj, Set] −→ [Inj, Set] is entirely described by its restriction L ◦ i : Σ Inj op

i

−→ [Inj, Set]

L

−→ [Inj, Set] to the subcategory Σ Inj op of graded arities.

An algebraic presentation of local states

Update, lookup, fresh, permute, discharge

The Lawvere theory of a graded monad

The Lawvere theory Θ T associated to a graded monad T : [Inj, Set] −→ [Inj, Set] has graded arities as objects [ p0, · · · , pm] and morphisms defined as Θ T ( [ p0, · · · , pm] , [ q0, · · · , qn] ) = [Inj, Set] ( [ p0, · · · , pm] , T [ q0, · · · , qn] )

The Lawvere theory of the local state monad

The Lawvere theoryמ‌ associated to the local state monad L : [Inj, Set] −→ [Inj, Set] has graded arities as objects [ p0, · · · , pm] and morphisms defined as

מ‌([ p0 , · · · , pm ] , [ q0 , · · · , qn ] )

= [Inj, Set] ( [ p0, · · · , pm] , L [ q0, · · · , qn] )

Key observation

The local state monad L : [Inj, Set] −→ [Inj, Set] is the composite L = F ◦ B

• f two graded monads

F , B : [Inj, Set] −→ [Inj, Set] related by a distributivity law λ : B ◦ F ⇒ F ◦ B. A phenomenon already noticed by Staton [2010]

The two component monads

The first monad F A : n → Sn ⇒ ( Sn × An ) is the global state monad applied on each fiber n while B = ℓ∗ ◦ ∃ℓ : [Inj, Set] −→ [Inj, Set] is the change-of-basis monad associated to a functor ℓ : Inj −→ Res.

Local mnemoids

Definition. A local mnemoid is a family of sets A0 A1 · · · An · · · equipped with the following operations lookuploc : An × An −→ An updateloc,val : An −→ An freshloc,val : An+1 −→ An disposeloc : An −→ An+1 permuteloc : An −→ An satisfying a series of elementary equations.

An algebraic presentation of the fiber monad

Creation lookup – update

x

=

t r u e f a l s e

x x

Interaction update – update

val1 val2

=

val2

Interaction update – lookup

true

y x

=

true

x

Commutation update – update

val′ val

=

val val′

An algebraic presentation of the basis monad

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

Modules over the category Inj

A category A equipped with an endofunctor D :

A

−→

A

and two natural transformations permute : D ◦ D −→ D ◦ D dispose : Id −→ D depicted as follows in the language of string diagrams:

D D D D D

Yang-Baxter equation

D D D D D D

=

D D D D D D

Symmetry

D D D D

=

D D D D

Interaction dispose – permute

D D D

=

D D D D D D

=

D D D

Allocation

One adds a fresh operator freshval : D −→ Id for each value val ∈ {true, false} depicted in string diagrams as:

val D

Garbage collect: fresh – collect

val

=

Identity

Interaction fresh – permutation

D D D val

=

D D D val

Commutation fresh – fresh

D val D val

=

D val D val

An algebraic presentation of the distributivity law

Interaction fresh – update

D D D D D D val val D

=

D D D D D D val D

Commutation fresh – update

D D D D D D val D val

=

D D D D D D val D val

Commutation fresh – update

D D D x y val

=

D D D x y val val

Interaction collect – update

D D D D D D val D

=

D D D D D D val D

Interaction permute – lookup

D D D D D D D D val

=

D D D D D D D D val

Local mnemoids

Definition. A local mnemoid is a family of sets A0 A1 · · · An · · · equipped with the following operations lookuploc : An × An −→ An updateloc,val : An −→ An freshloc,val : An+1 −→ An disposeloc : An −→ An+1 permuteloc : An −→ An satisfying the equations above.

Theorem [ revisiting Plotkin & Power 2002 ]

the category of local mnemoids is equivalent to the category of algebras of the local state monad Lookup – Update – Discharge – Allocate – Permute

Thank you !

Effects on the wire

val

η

The update operation in string diagrams

Effects on the wire

η

left branch

L R η

right branch

L R

The lookup operation in string diagrams

Effects on the wire

κ+ κ+ ε B A R A B R R L L L