Continuation

The earliest description of continuations was made by Adriaan van Wijngaarden in September 1964. Wijngaarden spoke at the IFIP Working Conference on Formal Language Description Languages held in Baden bei Wien, Austria. As part of a formulation for an Algol 60 preprocessor, he called for a transformation of proper procedures into continuation-passing style,[1] though he did not use this name, and his intention was to simplify a program and thus make its result more clear.

Christopher Strachey, Christopher P. Wadsworth and John C. Reynolds brought the term continuation into prominence in their work in the field of denotational semantics that makes extensive use of continuations to allow sequential programs to be analysed in terms of functional programming semantics.[1]

Steve Russell[2] invented the continuation in his second Lisp implementation for the IBM 704, though he did not name it.[3]

Reynolds (1993) gives a complete history of the discovery of continuations.

First-class continuations are a language's ability to completely control the execution order of instructions. They can be used to jump to a function that produced the call to the current function, or to a function that has previously exited. One can think of a first-class continuation as saving the execution state of the program. It is important to note that true first-class continuations do not save program data – unlike a process image – only the execution context. This is illustrated by the "continuation sandwich" description:

Say you're in the kitchen in front of the refrigerator, thinking about a sandwich. You take a continuation right there and stick it in your pocket. Then you get some turkey and bread out of the refrigerator and make yourself a sandwich, which is now sitting on the counter. You invoke the continuation in your pocket, and you find yourself standing in front of the refrigerator again, thinking about a sandwich. But fortunately, there's a sandwich on the counter, and all the materials used to make it are gone. So you eat it. :-)[4]

In this description, the sandwich is part of the program data (e.g., an object on the heap), and rather than calling a "make sandwich" routine and then returning, the person called a "make sandwich with current continuation" routine, which creates the sandwich and then continues where execution left off.

Scheme was the first full production system providing first "catch"[1] and then call/cc. Bruce Duba introduced call/cc into SML.

Continuations are also used in models of computation including denotational semantics, the actor model, process calculi, and lambda calculus. These models rely on programmers or semantics engineers to write mathematical functions in the so-called continuation-passing style. This means that each function consumes a function that represents the rest of the computation relative to this function call. To return a value, the function calls this "continuation function" with a return value; to abort the computation it returns a value.

Functional programmers who write their programs in continuation-passing style gain the expressive power to manipulate the flow of control in arbitrary ways. The cost is that they must maintain the invariants of control and continuations by hand, which can be a highly complex undertaking (but see 'continuation-passing style' below).

 (define the-continuation #f)

 (define (test)
   (let ((i 0))
     ; call/cc calls its first function argument, passing
     ; a continuation variable representing this point in
     ; the program as the argument to that function.
     ;
     ; In this case, the function argument assigns that
     ; continuation to the variable the-continuation.
     ;
     (call/cc (lambda (k) (set! the-continuation k)))
     ;
     ; The next time the-continuation is called, we start here.
     (set! i (+ i 1))
     i))

 > (test)
 1
 > (the-continuation)
 2
 > (the-continuation)
 3
 > ; stores the current continuation (which will print 4 next) away
 > (define another-continuation the-continuation)
 > (test) ; resets the-continuation
 1
 > (the-continuation)
 2
 > (another-continuation) ; uses the previously stored continuation
 4

 ;;; A naive queue for thread scheduling.
 ;;; It holds a list of continuations "waiting to run".

   (define *queue* '())

   (define (empty-queue?)
     (null? *queue*))

   (define (enqueue x)
     (set! *queue* (append *queue* (list x))))

   (define (dequeue)
     (let ((x (car *queue*)))
       (set! *queue* (cdr *queue*))
       x))

   ;;; This starts a new thread running (proc).

   (define (fork proc)
     (call/cc
      (lambda (k)
        (enqueue k)
        (proc))))

   ;;; This yields the processor to another thread, if there is one.

   (define (yield)
     (call/cc
      (lambda (k)
        (enqueue k)
        ((dequeue)))))

   ;;; This terminates the current thread, or the entire program
   ;;; if there are no other threads left.

   (define (thread-exit)
     (if (empty-queue?)
         (exit)
         ((dequeue))))

   ;;; The body of some typical Scheme thread that does stuff:

   (define (do-stuff-n-print str)
     (lambda ()
       (let loop ((n 0))
         (format #t "~A ~A\n" str n)
         (yield)
         (loop (+ n 1)))))

   ;;; Create two threads, and start them running.
   (fork (do-stuff-n-print "This is AAA"))
   (fork (do-stuff-n-print "Hello from BBB"))
   (thread-exit)

 This is AAA 0
 Hello from BBB 0
 This is AAA 1
 Hello from BBB 1
 This is AAA 2
 Hello from BBB 2
 ...

One area that has seen practical use of continuations is in Web programming.[7][8] The use of continuations shields the programmer from the stateless nature of the HTTP protocol. In the traditional model of web programming, the lack of state is reflected in the program's structure, leading to code constructed around a model that lends itself very poorly to expressing computational problems. Thus continuations enable code that has the useful properties associated with inversion of control, while avoiding its problems. "Inverting back the inversion of control or, Continuations versus page-centric programming"[9] is a paper that provides a good introduction to continuations applied to web programming.

Support for continuations varies widely. A programming language supports re-invocable continuations if a continuation may be invoked repeatedly (even after it has already returned). Re-invocable continuations were introduced by Peter J. Landin using his J (for Jump) operator that could transfer the flow of control back into the middle of a procedure invocation. Re-invocable continuations have also been called "re-entrant" in the Racket language. However this use of the term "re-entrant" can be easily confused with its use in discussions of multithreading.

A more limited kind is the escape continuation that may be used to escape the current context to a surrounding one. Many languages which do not explicitly support continuations support exception handling, which is equivalent to escape continuations and can be used for the same purposes. C's setjmp/longjmp are also equivalent: they can only be used to unwind the stack. Escape continuations can also be used to implement tail call elimination.

One generalization of continuations are delimited continuations. Continuation operators like call/cc capture the entire remaining computation at a given point in the program and provide no way of delimiting this capture. Delimited continuation operators address this by providing two separate control mechanisms: a prompt that delimits a continuation operation and a reification operator such as shift or control. Continuations captured using delimited operators thus only represent a slice of the program context.

Continuations are the functional expression of the GOTO statement, and the same caveats apply.[10] While they are a sensible option in some special cases such as web programming, use of continuations can result in code that is difficult to follow. In fact, the esoteric programming language Unlambda includes call-with-current-continuation as one of its features solely because expressions involving it "tend to be hopelessly difficult to track down."[11] The external links below illustrate the concept in more detail.

In "Continuations and the nature of quantification", Chris Barker introduced the "continuation hypothesis", that

some linguistic expressions (in particular, QNPs [quantificational noun phrases]) have denotations that manipulate their own continuations.[12]

Barker argued that this hypothesis could be used to explain phenomena such as duality of NP meaning (e.g., the fact that the QNP "everyone" behaves very differently from the non-quantificational noun phrase "Bob" in contributing towards the meaning of a sentence like "Alice sees [Bob/everyone]"), scope displacement (e.g., that "a raindrop fell on every car" is interpreted typically as ${\displaystyle \forall c\exists r,{\mbox{fell}}(r,c)}$ rather than as ${\displaystyle \exists r\forall c,{\mbox{fell}}(r,c)}$), and scope ambiguity (that a sentence like "someone saw everyone" may be ambiguous between ${\displaystyle \exists x\forall y,{\mbox{saw}}(x,y)}$ and ${\displaystyle \forall y\exists x,{\mbox{saw}}(x,y)}$). He also observed that this idea is in a way just a natural extension of Richard Montague's approach in "The Proper Treatment of Quantification in Ordinary English" (PTQ), writing that "with the benefit of hindsight, a limited form of continuation-passing is clearly discernible at the core of Montague’s (1973) PTQ treatment of NPs as generalized quantifiers".

The extent to which continuations can be used to explain other general phenomena in natural language is a topic of current research.[13]

• Call-with-current-continuation
• Closure
• COMEFROM
• Continuation-passing style
• Control flow
• Coroutine
• Delimited continuation
• Denotational semantics
• GOTO
• Spaghetti stack
• Quajects, a type of object which allows selectable continuations (called 'callouts') to be set for methods on a per-object basis, through Dependency Injection.

• Peter Landin. A Generalization of Jumps and Labels Report. UNIVAC Systems Programming Research. August 1965. Reprinted in Higher Order and Symbolic Computation, 11(2):125-143, 1998, with a foreword by Hayo Thielecke.
• Drew McDermott and Gerry Sussman. The Conniver Reference Manual MIT AI Memo 259. May 1972.
• Daniel Bobrow: A Model for Control Structures for Artificial Intelligence Programming Languages IJCAI 1973.
• Carl Hewitt, Peter Bishop and Richard Steiger. A Universal Modular Actor Formalism for Artificial Intelligence IJCAI 1973.
• Christopher Strachey and Christopher P. Wadsworth. Continuations: a Mathematical semantics for handling full jumps Technical Monograph PRG-11. Oxford University Computing Laboratory. January 1974. Reprinted in Higher Order and Symbolic Computation, 13(1/2):135—152, 2000, with a foreword by Christopher P. Wadsworth.
• John C. Reynolds. Definitional Interpreters for Higher-Order Programming Languages Proceedings of 25th ACM National Conference, pp. 717–740, 1972. Reprinted in Higher-Order and Symbolic Computation 11(4):363-397, 1998, with a foreword.
• John C. Reynolds. On the Relation between Direct and Continuation Semantics Proceedings of Second Colloquium on Automata, Languages, and Programming. LNCS Vol. 14, pp. 141–156, 1974.
• Reynolds, John C. (1993). "The discoveries of continuations" (PDF). LISP and Symbolic Computation. 6 (3/4): 233–248.
• Gerald Sussman and Guy Steele. SCHEME: An Interpreter for Extended Lambda Calculus AI Memo 349, MIT Artificial Intelligence Laboratory, Cambridge, Massachusetts, December 1975. Reprinted in Higher-Order and Symbolic Computation 11(4):405-439, 1998, with a foreword.
• Robert Hieb, R. Kent Dybvig, Carl Bruggeman. Representing Control in the Presence of First-Class Continuations Proceedings of the ACM SIGPLAN '90 Conference on Programming Language Design and Implementation, pp. 66–77.
• Will Clinger, Anne Hartheimer, Eric Ost. Implementation Strategies for Continuations Proceedings of the 1988 ACM conference on LISP and Functional Programming, pp. 124–131, 1988. Journal version: Higher-Order and Symbolic Computation, 12(1):7-45, 1999.
• Christian Queinnec. Inverting back the inversion of control or, Continuations versus page-centric programming SIGPLAN Notices 38(2), pp. 57–64, 2003.

• ACM SIGPLAN Workshop on Continuations 2011 at the ICFP.
• Continuations for Curmudgeons by Sam Ruby
• Teach Yourself Scheme in Fixnum Days by Dorai Sitaram features a nice chapter on continuations.
• Continuations and Stackless Python by Christian Tismer
• On-line proceedings of the Fourth ACM SIGPLAN Workshop on Continuations
• On-line proceedings of the Second ACM SIGPLAN Workshop on Continuations
• Continuation, functions and jumps
• http://okmij.org/ftp/continuations/ by Oleg Kiselyov
• https://wiki.haskell.org/Continuations
• Rhino With Continuations
• Continuations in pure Java from the RIFE web application framework
• Debugging continuations in pure Java from the RIFE web application framework
• Comparison of generators, coroutines, and continuations, source of the above example