APLAS 2012 "A Dynamic Interpretation of the CPS Hierarchy"

A DynamicInterpretation

of the CPSHierarchy

Marek Materzok,Dariusz Biernacki

Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

A Dynamic Interpretationof the CPS Hierarchy

Marek Materzok, Dariusz BiernackiInstitute of Computer Science, University of Wroc law

APLAS 2012Dec 12, 2012


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Structure of the talk

I Short introduction to continuations

I Introducing shift0/$ operators

I The CPS Hierarchy

I Interpreting the CPS Hierarchy with shift0/$

I Conclusion

This is a continuation of our previous work (ICFP’11).


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

What are continuations?

I Continuations are representations of the rest of theprogram execution.

I Control operators are a means of altering a program’scontrol flow.

I They can be thought of as capturing and restoring theprogram’s control stack, making continuations firstclass.

I The operator call/cc, which captures ,,full”continuations, is well known and implemented in e.g.Scheme and SML/NJ.


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Evaluation contexts

Evaluation context is a ,,term with a hole”:

if sq(2) = 4 then 1 else 0

I cyan part – evaluation context

It is a formal representation of the continuation.


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Partial evaluation contexts

One can consider partial contexts:

if sq(2) = 4 then 1 else 0

Partial context is a prefix of the full context.

I cyan part – evaluation context

I yellow part – partial evaluation context


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Delimited control operators

I They allow to reify partial contexts as functions, just ascall/cc reifies entire contexts.

I Examples are Felleisen’s control/prompt and Danvy andFilinski’s shift/reset.

I Delimited control has lots of applications, includingasynchronous I/O, representing monads, Webprogramming, mobile code, linguistics, and so on.


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Delimited control operators

Delimited control operators usually come in pairs:

I the delimiter, which marks where a context begins,

I capture operator, which reifies the context up to thedynamically nearest delimiter.

Example:

1 + 〈2 + Sf.f(f 3) 〉


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Delimited control operators shift/reset

I The most known and well explored delimited controloperators.

I The shift operator captures the context up to (andincluding) the nearest delimiter and resumes executionin an empty context.

1 + 〈2 ∗ Sf. 3 + Sg.f(g 4) 〉

The term above evaluates to 15: f gets the yellow context,g gets the cyan one. Notice the “implicit” delimiter createdby a shift.


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

The abstract machine for shift/reset

T denotes a stack of evaluation contexts.

〈λx.e, E · T 〉e ⇒ 〈λx.e, E · T 〉a〈E′, E · T 〉e ⇒ 〈E′, E · T 〉a〈e1 e2, E · T 〉e ⇒ 〈e1, E e2 · T 〉e

(1) 〈Sf.e, E · T 〉e ⇒ 〈e{E/f}, • · T 〉e(2) 〈〈e〉, T 〉e ⇒ 〈e, • · T 〉e

〈v,E e2 · T 〉a ⇒ 〈e2, v E · T 〉e〈v, (λx.e) E · T 〉a ⇒ 〈e{v/x}, E · T 〉e

(3) 〈v,E′ E · T 〉a ⇒ 〈v,E′ · E · T 〉a(4) 〈v, • · T 〉a ⇒ 〈v, T 〉a


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈Sf.e, E · T 〉e ⇒ 〈e{E/f}, • · T 〉e

Sf.eE[�]. . .

⇒e{E/f}�. . .

〈〈e〉, T 〉e ⇒ 〈e, • · T 〉e

〈e〉. . .

⇒e

�. . .


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈v,E′ E · T 〉a ⇒ 〈v,E′ · E · T 〉a

v

E[E′ �]. . .

⇒

v

E′[�]E[�]. . .

〈v, • · T 〉a ⇒ 〈v, T 〉a

v

�. . .

⇒ v. . .


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Delimited control operators shift0/reset0

I A variant of shift/reset operators (also by Danvy andFilinski).

I When shift0 executes, the execution resumes in thesurrounding context.

I This allows the shift0 operator to “reach” beyond thenearest surrounding delimiter.

1 + 〈2 + 〈3 + S0f.S0g.f (g (g 4)) 〉〉

The term above evaluates to 12. (f gets theyellow context , g gets the cyan one .)


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

The abstract machine for shift0/reset0

Only single change from shift/reset:

〈Sf.e, E · T 〉e ⇒ 〈e{E/f}, • · T 〉e

Sf.eE[�]. . .

⇒e{E/f}�. . .

is changed to:

〈S0f.e, E · T 〉e ⇒ 〈e{E/f}, T 〉e

S0f.eE[�]. . .

⇒ e{E/f}. . .


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Delimited control operators shift0/$

I A variant of shift0/reset0 operators, inspired by Kiselyovand Shan’s work (TLCA’07).

I The $ operator is a delimiter with a “chain link” to afunction which is executed in place (not inside!) of thedelimiter when the delimited context evaluatescompletely.

(λx.x+ 2) $ (λy.S0f.f (f y)) $ 1 + S0g.2 ∗ g 1

The term above evaluates to 14:

I g gets the yellow context , which gets applied to 1,

I y gets the value 2,

I f gets the cyan context joined with 2 ∗ ,

I f (f y) gets evaluated.


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

The abstract machine for shift0/reset0

Has to take care of the chain links!

〈λx.e, w$E · T 〉e ⇒ 〈λx.e, w$E · T 〉a〈v$E′, w$E · T 〉e ⇒ 〈v$E′, w$E · T 〉a〈e1 e2, w$E · T 〉e ⇒ 〈e1, w$(E e2) · T 〉e〈Sf.e, w$E · T 〉e ⇒ 〈e{w$E/f}, • · T 〉e

(1) 〈e1 $ e2, w$E · T 〉e ⇒ 〈e1, w$(E $ e2) · T 〉e〈v, w$(E e2) · T 〉a ⇒ 〈e2, w$(v E) · T 〉e

〈v, w$((λx.e) E) · T 〉a ⇒ 〈e{v/x}, w$E · T 〉e〈v, w$((w′$E′) E) · T 〉a ⇒ 〈v, w′$E′ · w$E · T 〉a

(2) 〈v, w$(E $ e2) · T 〉a ⇒ 〈e2, v$ • ·w$E · T 〉e(3) 〈v, (λx.e)$ • ·T 〉a ⇒ 〈e{v/x}, T 〉a(4) 〈v, (w$E)$ • ·T 〉a ⇒ 〈v, w$E · T 〉a


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

The abstract machine for shift0/$

〈e1 $ e2, w$E · T 〉e ⇒ 〈e1, w$(E $ e2) · T 〉e

e1 $ e2w E[�]

. . .⇒

e1w E[� $ e2]

. . .

〈v, w$(E $ e2) · T 〉a ⇒ 〈e2, v$ • ·w$E · T 〉e

v

w E[� $ e2]. . .

⇒

e2v �w E[�]

. . .


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

The abstract machine for shift0/$

〈v, (λx.e)$ • ·T 〉a ⇒ 〈e{v/x}, T 〉av

λx.e �. . .

⇒ e{v/x}. . .

〈v, (w$E)$ • ·T 〉a ⇒ 〈v, w$E · T 〉a

v

w E[�] �. . .

⇒v

w E[�]. . .

Notice that λx.w $E[x] behaves like (w$E)!


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

More about shift0/$

The operators are equally expressible as shift0/reset0, even inthe typed setting!

〈e〉 ≈ (λx.x) $ ee1 $ e2 ≈ (λw.〈(λv.S0k.w v) e2〉) e1

(I will be using 〈e〉 as a shorthand for (λx.x) $ e)But the $ is very convenient...

S0k.k $ e ≈ e(λx.v $E[x]) $ e ≈ v $E[e]


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

The CPS Hierarchy

I Introduced by Danvy and Filinski, enables expressingnested control effects.

I The n-th level of the CPS Hierarchy has n pairs ofdelimited control operators: Sik. and 〈〉i.

I They give access to the layers of continuationsconstructed by iterated CPS translation.

I The Sik. operator captures the context up to thenearest 〈〉j (where j ≥ i), which is not removed (theexecution resumes inside it). The captured context isdelimited by 〈〉i.


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

The abstract machine for the CPS Hierarchy

I Has n+ 1 layers of continuations.

I The first level has one evaluation context.

I Every next level has stacks of continuations of theprevious level.

〈Sik.e, E, T1, . . . , Ti−1, Ti, . . . , Tn〉e⇒ 〈e{(E, T1, . . . , Ti−1)/k}, •, •1, . . . , •i−1, Ti, . . . , Tn〉e〈〈e〉i, E, T1, . . . , Ti−1, Ti, . . . , Tn〉e⇒ 〈e, •, •1, . . . , •i−1, ((E · T1) . . . · Ti−1) · Ti, Ti+1, . . . , Tn〉e〈v, •, •1, . . . , •i−1, ((E · T1) . . . · Ti−1) · Ti, Ti+1, . . . , Tn〉a⇒ 〈v,E, T1, . . . , Ti−1, Ti, . . . , Tn〉a〈v, (E′, T ′1, . . . , T ′i−1) E, T1, . . . , Ti−1, Ti, . . . , Tn〉a⇒ 〈v,E′, T ′1, . . . , T ′i−1, ((E · T1) . . . · Ti−1) · Ti, Ti+1, . . . , Tn〉a


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈S2k.e, E, T1, T2〉e ⇒ 〈e{(E, T1)/k}, •, •1, T2〉e

S2k.eE[�]T1T2

⇒

e{(E, T1)/k}�•1T2

〈〈e〉2, E, T1, T2〉e ⇒ 〈e, •, •1, (E · T1) · T2〉e

〈e〉2E[�]T1T2

⇒

e

�•1

T2 T1 – E


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈v, •, •1, (E · T1) · T2〉a ⇒ 〈v,E, T1, T2〉av

�•1

T2 T1 – E

⇒

v

E[�]T1T2

〈v, (E′, T ′1) E, T1, T2〉a ⇒ 〈v,E′, T ′1, (E · T1) · T2〉a

vE[(E′, T ′

1) �]

T1T2

⇒

v

E′[�]T ′1

T2 T1 – E


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Embedding the CPS Hierarchy in shift0/$

Structure seemingly not similar, but... Idea:

I Layers – stack depth

I Continuation stacks – $ chain links

Sik.e = S0k.S0k1. . . .S0ki−1.〈 i. . . 〈e{λx.S0k′.S0k′1. . . .S0k′i.(λy.k′i $ . . . $ k1 $ k x) $ ki−1 $ . . . $ k

′1 $ k

′ x/k}〉 . . .〉

〈e〉i = S0k.S0k1. . . .S0ki.(λx.ki $ . . . $ k1 $ k x) $ 〈 i. . . 〈e〉 . . .〉

Let’s see how this works!


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈〈e〉2, E, T1, T2〉e ⇒ 〈e, •, •1, (E · T1) · T2〉e

〈e〉2E[�]T1T2

⇒

e

�•1

T2 T1 – E

〈S0k.S0k1.S0k2.(λx.k2 $ k1 $ k x) $ 〈〈e〉〉,((λx.x)$E) · (w1$•) · (w2$•) · ((λx.x)$•)〉e

λx.x E[�]w1 �w2 �λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈〈e〉2, E, T1, T2〉e ⇒ 〈e, •, •1, (E · T1) · T2〉e

〈e〉2E[�]T1T2

⇒

e

�•1

T2 T1 – E

〈(λx.(w2$•) $ (w1$•) $ ((λx.x)$E) x) $ 〈〈e〉〉,((λx.x)$•)〉e

λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈〈e〉2, E, T1, T2〉e ⇒ 〈e, •, •1, (E · T1) · T2〉e

〈e〉2E[�]T1T2

⇒

e

�•1

T2 T1 – E

〈〈〈e〉〉,((λx.(w2$•) $ (w1$•) $ ((λx.x)$E) x)$•) · ((λx.x)$•)〉e

w2 � (w1$•) $ (id$E) � �λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈〈e〉2, E, T1, T2〉e ⇒ 〈e, •, •1, (E · T1) · T2〉e

〈e〉2E[�]T1T2

⇒

e

�•1

T2 T1 – E

〈e, ((λx.x)$•) · ((λx.x)$•)·((λx.(w2$•) $ (w1$•) $ ((λx.x)$E) x)$•) · ((λx.x)$•)〉e

λx.x �λx.x �

w2 � (w1$•) $ (id$E) � �λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈v, •, •1, (E · T1) · T2〉a ⇒ 〈v,E, T1, T2〉av

�•1

T2 T1 – E

⇒

v

E[�]T1T2

〈v, ((λx.x)$•) · ((λx.x)$•)·((λx.(w2$•) $ (w1$•) $ ((λx.x)$E) x)$•) · ((λx.x)$•)〉a

λx.x �λx.x �

w2 � (w1$•) $ (id$E) � �λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈v, •, •1, (E · T1) · T2〉a ⇒ 〈v,E, T1, T2〉av

�•1

T2 T1 – E

⇒

v

E[�]T1T2

〈v,((λx.(w2$•) $ (w1$•) $ ((λx.x)$E) x)$•) · ((λx.x)$•)〉a

w2 � (w1$•) $ (id$E) � �λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈v, •, •1, (E · T1) · T2〉a ⇒ 〈v,E, T1, T2〉av

�•1

T2 T1 – E

⇒

v

E[�]T1T2

〈(w2$•) $ (w1$•) $ ((λx.x)$E) v,((λx.x)$•)〉e

λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈v, •, •1, (E · T1) · T2〉a ⇒ 〈v,E, T1, T2〉av

�•1

T2 T1 – E

⇒

v

E[�]T1T2

〈(((λx.x)$E) v,((w1$•)$•) · ((w2$•)$•) · ((λx.x)$•)〉e

w1 � �w2 � �

λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions


〈v, •, •1, (E · T1) · T2〉a ⇒ 〈v,E, T1, T2〉av

�•1

T2 T1 – E

⇒

v

E[�]T1T2

〈v,((λx.x)$E) · ((w1$•)$•) · ((w2$•)$•) · ((λx.x)$•)〉a

λx.x E[�]w1 � �w2 � �

λx.x �


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Shift0 is more powerful

csort l =let insert a =

let cinsert k = S0f.case f [] of[] → 〈〈a :: k ()〉〉[b] →

if a < bthen cinsert (λx.f $ k x)else f $ 〈a :: k ()〉

in S0k. cinsert kin 〈〈 foldr (λx.λl′. insert x; l′ ) [] l 〉〉


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Summary of the results

We have shown the correspondence to hold for:

I Abstract machines (as shown)

I CPS translations (βη-equality)

I Reduction semantics

I Type systems (with the Biernacka et al. type system –PPDP’11)

We have also shown a shift0/$ program which cannotcorrespond to any CPS Hierarchy program.


of the CPSHierarchy


Introduction

Evaluation contexts

Delimited control

Shift0/$

CPS Hierarchy

Embedding

Conclusions

Conclusions

What does this result give us?

I The shift0/reset0 operators have more expressive powerthan the CPS Hierarchy.

I We can use the much simpler shift0/reset0 toimplement the CPS Hierarchy.

I The result can lead to a level-polymorphic type systemfor the Hierarchy.

Thank you for your attention!

APLAS 2012 "A Dynamic Interpretation of the CPS Hierarchy"

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Transcript of APLAS 2012 "A Dynamic Interpretation of the CPS Hierarchy"