optimality for dynamic patternsblsk/docs/balabonski-talk-ppdp10.pdf · t. balabonski — optimality...

I

Optimal optimality?

Families and history

(Some syntax)

Recording the history

History and memory

T. Balabonski — Optimality for Dynamic Patterns PPDP 2010: July 28 — 1/20

Optimalityfor Dynamic Patterns

Thibaut Balabonski

Laboratoire PPSCNRS and Universite Paris Diderot

PPDP 2010: July 28

Computing with data structures

I

Optimal optimality?


(Some syntax)


History and memory


type ’a tree =

| Data of ’a

| Node of ’a tree * ’a tree

let rec upd f t = match t with

| Data a -> Data (f a)

| Node l r -> Node (upd f l) (upd f r)

Pattern matching:

• tests the shape.

• extracts substructures.

New polymorphisms

I

Optimal optimality?


(Some syntax)


History and memory


• Pattern polymorphism: parameters in patterns.

• Path polymorphism: shape irrelevance.

let rec upd c f t = match t with

| c d -> c (f d)

| x y -> (upd c f x) (upd c f y)

| a -> a

Instantiation of the parameter c

• Constructors like Data.

• Functions building patterns.

Dynam

icpa

ttern

s

New polymorphisms

I

Optimal optimality?


(Some syntax)


History and memory





| Data d -> Data (f d)


| a -> a




Dynam

icpa

ttern

s

New polymorphisms

I

Optimal optimality?


(Some syntax)


History and memory





| (fun x -> Node (Data x) (Data )) d ->


| a -> a




Dynam

icpa

ttern

s

New polymorphisms

I

Optimal optimality?


(Some syntax)


History and memory





| Node (Data d) (Data ) -> Node ...


| a -> a




Dynam

icpa

ttern

s

New polymorphisms

I

Optimal optimality?


(Some syntax)


History and memory





| c d -> c (f d)


| a -> a




Dynam

icpa

ttern

s

Collateral damage

I

Optimal optimality?


(Some syntax)


History and memory


Usual compile-time analyzes

| Node l (Data d) ->

| Node (Data d) (Node ) ->

| Node (Node l r) (Node ) ->

This requires static patterns!

Goal

Run-time sharing mechanismsfor dynamic patterns.

Roadmap


1 Is optimal sharing a reasonable solution?

2 Families of terms: learning from history

3 (Some syntax)

4 Recording the history

5 From history to memory

Is optimal sharing a reasonable solution?

There is some mismatch betweentheoretical optimality and practical efficiency.

Families of terms: learning from history

(Some syntax)


From history to memory


The trap of optimality


Sharing

Extra cost

Usual implementations

β-Optimality

Weak reduction

IOptimal optimality?


(Some syntax)


History and memory


Usual implementations

• Passing arguments to functions.

• No partial evaluation.

⇒ No evaluation under a binder (λ).

Confluent weak reduction

No reduction betweena bounded variable and its binder.


Terms with same history should not be considereddifferent.


(Some syntax)




Optimality and history

IFamilies and history

Optimal optimality?

(Some syntax)


History and memory


Optimality commandment

At each step you shall reduce a whole family.

• Family: all redexes sharing a common history.

• History: all events that contributed to thecreation of a redex.

• Redex: REDucible EXpression.

Screening the history

IFamilies and history

Optimal optimality?

(Some syntax)


History and memory


Which reduction steps are relevant?

Back to basics

• analysis of direct contribution.

• how can redexes be created?

And later

• record contributions in labels.

(Some syntax)

Digression: the Weak Pure Pattern Calculus.Simple syntax, complex reduction.






Syntactic trees

I(Some syntax)

Optimal optimality?



History and memory


@

A

λx

Bx x

Syntactic trees

I(Some syntax)

Optimal optimality?



History and memory


@

A

λθ

Bx x

Pxc

Pattern matching: a complex operation

I(Some syntax)

Optimal optimality?



History and memory


@

λθ

Bxi xj

PP

xi xj Ai Aj

A

B

Ai Aj

Failure

If A and P incompatible, reduction to ⊥.

Partial definition

Matching undefined for improper terms.

Pattern matching: a meta-operation


a/θ x = x ← a x ∈ θ

x/θ x = x 6∈ θ

a1a2/θ p1p2 = a1/θ p1 ] a2/θ p2 a1a2, p1p2 matchable

a/θ p = ⊥ a, p matchable

Undefined otherwise!

Matchable forms

• x t1 ... tn

• [θ]p b


Labels to record contributionsand define redex families.



(Some syntax)



History and contributions materialized

IRecording the history

Optimal optimality?


(Some syntax)

History and memory


Contributions can be recorded by labels.

@α

@β

cγ

N

Nδ

ε

• Family: all redexes sharing a common label.

Forward: The consequences of my deeds


@

λθ

xi xj

Ω

?

initial labels ¡, ¢, £...atomic labels α ::= ¡ | pΩq | |Ω, α|

labels Γ, ∆, Ω ::= α1α2 . . . αn



@

λθ

xi xj

Ω

λ

λ

?


labels Γ, ∆, Ω ::= α1α2 . . . αn



@

λθ

xi xj

Ω

W

λ

λ

?


labels Γ, ∆, Ω ::= α1α2 . . . αn



@

λθ

xi xj

Ω

?

|Ω, α|

pΩq


labels Γ, ∆, Ω ::= α1α2 . . . αn

Backward: Where do I come from?

IRecording the history

Optimal optimality?


(Some syntax)

History and memory


@

λθ

α

Γ

∆pat

∆arg

Name of the redex

Ω = α Γ ∆pat ∆arg

Pattern matching and history extraction


Y /θ α : Zm = α : Y /θ Zm m ∈ l ,wY /θ xl = (ε, x ← Y ) x ∈ θY /θ xw = (|Y |, x ← Y ) x ∈ θ

Y /θ ([τ ]p b)l = (ε,⊥)Y /θ ([τ ]p b)w = (|Y |,⊥)

α : Z/θ Xm = α : Z/θ Xm matchabilityx/θ xm = (ε, ) x 6∈ θ

A1A2/θ P1P2m = A1/θ P1w A2/θ P2l matchabilityY /θ Xm = (|Y ||X |,⊥) matchabilityY /θ Xm = undefined otherwise

|[θ]P B| = ε|x | = ε

|T1T2| = |T1||α : Z | = α|Z |

(∆2, σ2) (∆2,⊥) undef.(∆1, σ1) (∆1∆2, σ1 ] σ2) (∆1∆2,⊥) undef.(∆1,⊥) (∆1,⊥) (∆1,⊥) (∆1,⊥)undef. undef. undef. undef.


From labelled terms to directed acyclic graphs.



(Some syntax)



Labels as memory locations


@α

@γ

y ι

N

Nδ

δ

?@α

@γ

y ι

N ′

N ′δ′

δ′

@

@y

N

@

@y

N ′

Sharing property: equal labels ⇒ equal terms.



@α

@γ

y ι

N

Nδ

δ

?@α

@γ

y ι

N ′

N ′δ′

δ′

@

@y

N

@

@y

N ′

Graph reduction




@α

@γ

y ι

N

Nδ

δ ?@α

@γ

y ι

N ′

N ′δ′

δ′

@

@y

N

@

@y

N ′

Graph reduction

Labelled development?


From labels to graph reduction

IHistory and memory

Optimal optimality?


(Some syntax)



Complete labelled reduction:development of all redexes with a given label.

Sharing theorem

Complete labelled reductionpreserves

the sharing property.

Hypothesis: start with initial labels.Key: contribution is characterized by labels.

Conclusion

I

Optimal optimality?


(Some syntax)


History and memory


Present resultsA causality analysis can yield

• a concrete DAG implementation.

• a proof of optimality of call-by-need.

Through PPC, this work reaches as well λ-calculusand pattern matching a la ML.

Future work

• Sharing in functions with multiple cases.

• Effective reduction strategy.

A broader picture

I

Optimal optimality?


(Some syntax)


History and memory


Pure PatternCalculus

Sharing inWeak PPCPPDP’10

ExplicitMatchingHOR’10

Call-by-needAlternatives

EfficientEvaluation

...and questions...

optimality for dynamic patternsblsk/docs/balabonski-talk-ppdp10.pdf · t. balabonski — optimality...

Documents