Continuations

1
Continuations

• COS 441
• Princeton University
• Fall 2004

2
Webster Says

• Main Entry reify Pronunciation 'rA--"fI,
  'rE-Function transitive verbInflected Form(s)
  reified reifyingEtymology Latin res thing
  -- to regard (something abstract) as a material
  or concrete thing

3
First-Class Continuations

• Form of structured GOTO
• Can be used to implement exception
• Can be used to build co-routines or threads
• Available in Scheme, Ruby, and SML/NJ but not
  Standard ML
• Also useful as a programming abstraction for
  web-services!

4
Some Informal Examples

• (throw e1 to e2)?
• Invoke continuation (e2? cont) with the value
  of e1?
• (letcc x in e)?
• Evaluate e with x bound to a value of type ?
  cont. Expression evaluates to the value of e or
  result thrown to x. Can throw to x multiple times
  and rewind to this control point

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Example

• Compute product of list elements with short
  circuit when 0 is encountered

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Example

• Compute product of list elements with short
  circuit when 0 is encountered

7
Trickier Example

• Compose a function with a continuation

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Continuation Passing Style

• Can apply a global translation of a program and
  remove all uses of throw and letcc
• Wont study in detail in class
• CPS transform used in compilation of functional
  languages to machine code
• Understanding how continuations can be used is
  harder than understanding their semantics!

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First-Class Continuations
10
Frames and Stacks
11
Dynamic Semantics
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Dynamic Semantics

• Capturing a continuation can be a linear time
  operation if not implemented cleverly
• Clever implementations make continuation capture
  constant time
• Compare this to two stack exception handler
  approach

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Type Safety

• Must define when a machine state is well-formed
• Introduced the notion of types for stacks,
  frames, and also choose one arbitrary but fixed
  type for the empty stack ?ans

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Programmer Visible Rules
15
Auxiliary Types
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Internal Type-Checking Rules
17
Rules For Frames

• (?,?) frame Frame which expects a value of type
  ? that can be used to fill the hole to produce an
  expression of type ?

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Frame Rules (cont.)

• Harpers text has a small bug rules above are
  correct

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Example Continuation Capture

• (²,((letcc x in 1),2))
• ? ((,2)B²,letcc x in 1)
• ? ((,2)B²,1)
• ? ((1,)B²,2)
• ? (²,3)

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Example Thrown Continuation

• (²,((letcc x in (1,throw 2 to x)),3))
• ? ((,3)B²,letcc x in (1,throw 2 to x))
• ? ((,3)B²,(1,throw 2 to (,3)B²))
• ? ((,throw 2 to (,3)B²)B(,3)B²,1)
• ? ((1,)B(,3)B²,throw 2 to (,3)B²)
• ? ((throw to (,3)B²)B(1,)B(,3)B²,2)
• ? ((throw 2 to )B(1,)B(,3)B²,(,3)B²)
• ? ((,3)B²,2)
• ? ((2,)B²,3)
• ? (²,5)

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Stacks and Frames as HOF

• One big insight of the CPS transform is that you
  can represent stacks as HOF

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Producer Consumer

• fun p(n)
• (put(n)p(n1))

• fun c() let
• val n get()
• in printInt(n)
• c()
• end

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Push Model

• fun p(n)
• (c(n)p(n1))

• fun c(n)
• printInt(n)

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Pull Model

• fun p(n,s)
• (n,n1)

• fun c(n,s) let
• val (n,s) p(s)
• in printInt(n)
• c(s)
• end

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Coroutine Model

• fun p(n,s) let
• val s put(n,s)
• in p(n1,s)
• end

• fun c(s) let
• val (n,s) get(s)
• in printInt(n)
• c(s)
• end

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Implementation

• datatype state S of state cont
• val buf ref 0
• fun resume(S k)
• callcc(fn k' gt throw k (S k'))
• fun put(n,s) (buf nresume s)
• fun get(s) (!buf,resume s)

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Threads

• signature THREADS sig
• exception NoMoreThreads
• val fork (unit -gt unit) -gt unit
• val yield unit -gt unit
• val exit unit -gt a
• end
• User-level threads implemented with callcc

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Summary

• First-Class continuations very powerful construct
• Simple semantics but allows powerful control
  constructs
• Non-local exits
• Coroutines
• User-level threads