abortplaying the role of
shiftthe role of
Prompts as local exceptions
shiftas a green
The tutorial on delimited continuations was given together with Kenichi Asai (Ochanomizu University, Japan) in the evening before the Continuation Workshop 2011.
The concept of continuations arises naturally in programming: a
conditional branch selects a continuation from the two possible
futures; raising an exception discards a part of the continuation; a
goto continues with the continuation. Although
continuations are implicitly manipulated in every language,
manipulating them explicitly as first-class objects is rarely used
because of the perceived difficulty.
This tutorial aims to give a gentle introduction to
continuations and a taste of programming with first-class delimited
continuations using the control operators
reset. Assuming no prior knowledge on continuations,
the tutorial helps participants write simple co-routines and
non-deterministic searches. The tutorial should make it easier to
understand and appreciate the talks at CW 2011.
We assume basic familiarity with functional programming languages, such as OCaml, Standard ML, Scheme, and Haskell. No prior knowledge of continuations is needed. Participants are encouraged to bring their laptops and program along.
CW2011 Tutorial Session. September 23, 2011
Tutorial notes for OchaCaml and Haskell
A sample shift/reset code in Haskell, in the
Cont monad -- the monad for delimited control
The complete code for the Haskell portion of the tutorial
ACM SIGPLAN Continuation Workshop 2011 (co-located with ICFP 2011)
Tokyo, Japan. Saturday, September 24, 2011.
This tutorial-like Haskell code illustrates the application of delimited control for non-deterministic search. We apply different search strategies to the same non-deterministic program without re-writing it. A non-deterministic computation is reified into a lazy search tree, which can then be examined in different ways. We write non-deterministic search strategies as standard depth-first, breadth-first, etc., tree traversals.
The search tree is the ordinary tree data type, with branches constructed on demand. The tree is potentially infinite, as is the case in the example below.
data SearchTree a = Leaf a | Node [() -> SearchTree a]We implement three tree traversals, which collect the values from leaf nodes into a list:
dfs, bfs, iter_deepening :: SearchTree a -> [a]
Cont monad from the standard monad transformer library
and its operations
implement two primitives: non-deterministically choosing a value
from a finite list, and reifying a computation into a
choose :: [a] -> Cont (SearchTree w) a reify :: Cont (SearchTree a) a -> SearchTree aOther non-deterministic operations --
mplus(to join two computations),
choose'(to choose from a potentially infinite list) -- are all written in terms of
The running example non-deterministically computes all Pythagorean triples, naively:
ex = do x <- choose' [1..] y <- choose' [1..] z <- choose' [1..] if x*x + y*y == z*z then return (x,y,z) else failureWe show the first five found triples:
test3d = take 5 . dfs . reify $ ex -- diverges!! test3b = take 5 . bfs . reify $ ex -- [(3,4,5),(4,3,5),(6,8,10),(8,6,10),(5,12,13)] test3i = take 5 . iter_deepening . reify $ ex -- [(3,4,5),(4,3,5),(6,8,10),(8,6,10),(5,12,13)]Depth-first search expectedly diverges. Breadth-first and iterative deepening are both complete strategies and both find the answer if it exists. Expectedly iterative deepening takes much less memory than breadth-first search.
The complete Haskell code with comments and tests. The comments tell why the search tree is defined as it is, with a thunk.
Embedded probabilistic programming
The paper explains the reification of non-deterministic programs as lazy search trees. We use the same technique here, only in Haskell rather than OCaml, and without probabilities.
Preventing memoization in (AI) search problems
The explanation of the trick to prevent unwelcome implicit memoizations
This technical report shows that the delimited continuation
etc. are macro-expressible in terms of each other. The report thus
confirms the result first established by Chung-chieh Shan in ``Shift
to Control'' (Scheme Workshop, 2004). The report uses a more uniform
technique that lets us skip an arbitrary number of
prompts. The report formally proves that
shift indeed has its standard reduction
semantics. It is a common knowledge that first-class continuations are
quite tricky -- and delimited continuations are trickier
still. Therefore, a formal proof is a necessity.
The report shows the simplest known Scheme implementations of
control0 (similar to
cupto). The method in the
report lets us design 700 more delimited control operators, which
compose stack fragments in arbitrary ways.
Technical Report TR611, Department of Computer Science, Indiana
Scheme code with the simplest implementation of
control0 and other
delimited continuation operators in terms of
reset. The code includes a large set of test cases.
``Lambda the Ultimate'' discussion thread, esp. on the meaning of
The master SXML file of the report
Writing LaTeX/PDF mathematical papers with SXML
We present a simple CGI framework for web programming with nested transactions. The framework uses the unmodified OCaml system and an arbitrary, unmodified web server (e.g., Apache). The library makes writing web applications (CGI scripts) as straightforward as writing interactive console applications using read and printf. We write the scripts in the natural question-answer, storytelling style, with the full use of lexical scope, exceptions, mutable data and other imperative features (if necessary). The scripts can even be compiled and run as interactive console applications. With a different implementation of basic primitives for reading and writing, the console programs become CGI scripts.
Our library depends on the delimcc library of persistent delimited continuations. The captured delimited continuations can be stored on disk, to be later loaded and resumed in a different process. Alternatively, serialized captured continuations can be inserted as an encoded string into a hidden field of the response web form. The use of continuations lets us avoid iterations, relying instead on the `Back button.' Delimited continuations naturally support `thread-local' scope and are quite compact to serialize. The library works with the unmodified OCaml system as it is.
Delimited continuations help us implement nested transactions. The simple blog application demonstrates that a user may repeatedly go back-and-forth between editing and previewing their blog post, perhaps in several windows. The finished post can be submitted only once.
The demonstration of the library at the Continuation Fest 2008
The extended version of the talk, Clicking on Delimited Continuations, has been given at FLOLAC in July 2008. The extended version includes a detailed introduction to delimited continuations.
The source code for the library of delimited-continuation--based CGI programming with form validation and nested transactions. The library includes the complete code for the Continuation Fest demos.
We present Church-style call-by-name lambda-calculus with delimited control operators shift/reset and first-class contexts. In addition to the regular lambda-abstractions -- permitting substitutions of general, even effectful terms -- the calculus also supports strict lambda-abstractions. The latter can only be applied to values. The demand for values exerted by reset and strict functions determines the evaluation order. The calculus most closely corresponds to the familiar call-by-value shift/reset calculi and embeds the latter with the help of strict functions.
The calculus is typed, assigning types both to terms and to contexts. Types abstractly interpret operational semantics, and thus concisely describe all the effects that could occur in the evaluation of a term. Pure types are given to the terms whose evaluation incurs no effect, i.e., includes no shift-transitions, in any context and in any environment binding terms' free variables, if any. A term whose evaluation may include shift-transitions has an effectful type, which describes the eventual answer-type of the term along with the delimited context required for the evaluation of the term. Control operators may change the answer type of their context.
Twelf code that implements the dynamic semantics (the
eval* relation) and the type
teval relation). The
teval relation is
deterministic and terminating, thus constructively proving that the
type system for our Church-style calculus is decidable. The code
includes a large number of examples of evaluating terms and
determining their types.
Call-by-name linguistic side effects
ESSLLI 2008 Workshop on Symmetric calculi and Ludics for the semantic interpretation. 4-7 August, 2008. Hamburg, Germany.
Compilation by evaluation as syntax-semantics interface
Linguistics turns out to offer the first interesting application of the typed call-by-name shift/reset. The paper develops the calculus in several steps, presenting the syntax and the dynamic semantics of the final calculus in Figure 3 and the type system in Figures 4 and 5. The paper details several sample reductions and type reconstructions, and discusses the related work.
A Substructural Type System for Delimited Continuations
That TLCA 2007 paper introduced the abstract interpretation technique for reconstructing the effect type of a term in a calculus of delimited control. The technique progressively reduces a term to its abstract form, i.e., the type. The TCLA paper used a call-by-value calculus with a so-called dynamic control operator,
shift0. Here we apply the technique to the call-by-name
calculus with the static control operator
Simply typed lambda-calculus has strong normalization property:
the sequence of reductions of any term terminates. If we add delimited
control operators with typed prompts (e.g.,
strong normalization property no longer holds. A single typed prompt
already leads to non-termination. The following example has been
designed by Chung-chieh Shan, from the example of non-termination of
simply typed lambda-calculus with dynamic binding. It uses the OCaml
delimited control library. The function
loop is essentially
fun () -> Omega: its inferred type is
unit -> 'a, and consequently, the evaluation of
loop () loops forever.
let loop () = let p = new_prompt () in let delta () = shift p (fun f v -> f v v) () in push_prompt p (fun () -> let r = delta () in fun v -> r) delta ;;
Chung-chieh Shan also offered the explanation: the answer
type being an arrow type hides a recursive type. In other words,
the type of
unit -> 'a, hides the answer
type and the fact the function is impure.
Olivier Danvy has kindly pointed out the similar non-terminating example that he and Andrzej Filinski designed in 1998 using their version of shift implemented in SML/NJ. Their example too relied on the answer type being an arrow type.
Carl A. Gunter, Didier R'emy and Jon G. Riecke: A Generalization of Exceptions and Control in ML-Like Languages. Proc. Functional Programming Languages and Computer Architecture Conf., June 26-28, 1995, pp. 12-23.
The paper that introduced
cupto, the first
delimited control operator with an explicitly typed prompt.
Delimited Dynamic Binding
Reformulation of the above in terms of shift and simply typed lambda-calculus.
Simply typed lambda-calculus with dynamic binding is not strongly normalizing
A differently-formulated proof: representing general recursive types
[The Abstract of the paper]
We propose type systems that abstractly interpret small-step rather than big-step operational semantics. We treat an expression or evaluation context as a structure in a linear logic with hypothetical reasoning. Evaluation order is not only regulated by familiar focusing rules in the operational semantics, but also expressed by structural rules in the type system, so the types track control flow more closely. Binding and evaluation contexts are related, but the latter are linear.
We use these ideas to build a type system for delimited continuations. It lets control operators change the answer type or act beyond the nearest dynamically-enclosing delimiter, yet needs no extra fields in judgments and arrow types to record answer types. The typing derivation of a direct-style program desugars it into continuation-passing style.
Joint work with Chung-chieh Shan.
Type checking as small-step abstract evaluation
Detailed discussion of the two main slogans of the paper:
The extended (with Appendices) version of the paper published in Proc. of Int. Conf. on Typed Lambda Calculi and Applications, Paris, June 26-28, 2007 -- LNCS volume 4583.
Chung-chieh Shan. Slides of the TLCA 2007 Presentation, Jun 26, 2007.
Commented Twelf code accompanying the paper
The code implements type checking -- of simply-typed lambda-calculus for warm-up, and of the main lambda-xi0 calculus -- and contains numerous tests and sample derivations.
In the recent paper `Typed Dynamic Control Operators for Delimited Continuations' Kameyama and Yonezawa exhibited a divergent term in their polymorphically typed calculus for prompt/control. Hence the latter calculus, in contrast to Asai and Kameyama's polymorphically typed shift/reset calculus, is not strongly normalizing. Unlike the untyped case, typed control is not macro-expressible in terms of shift. Kameyama and Yonezawa conjectured that the (typed) fixpoint operator is expressible in their calculus too. The conjecture is correct: Here is the derivation of the fixpoint combinator, using the notation of their paper. The combinator is not fully polymorphically typed however: the result type must be populated.
f be a pure function of the type
a -> a and
d be a value of the type
a (in the paper,
d is written as a black
dot). As in the paper, we write
# for prompt. The
#( control k.(f #(k d; k d)) ; control k.(f #(k d; k d)) )appears to be well-typed. It reduces to
#(f #(k d; k d)) where k u = u; control k.(f #(k d; k d)) then #(f #(f #(k d; k d))) eventually to #(f #(f #(f ..... )))Since we are in a call-by-value language, the above result is not terribly useful, but it is a good start. We only need an eta-expansion: Suppose
fis of the type
(a->b) -> (a->b). Let
dbe any value of the type
a->b: this is the witness that the return type is populated. We build the term
FX = #( control k.(f (\x . #(k d; k d) x)) ; control k.(f (\x . #(k d; k d) x)) )that is well-typed and expands as
#(f (\x . #(k d; k d) x) ) where k u = u; control k.(f (\x . #(k d; k d) x)) and then f (\x . #(k d; k d) x) we notice that k d = control k.(f (\x . #(k d; k d) x)) so we get f (\x . FX x)Thus we obtain
FX x = f (\x . FX x) x, which means
FXis the call-by-value fixpoint of
Without access to the implementation of Kameyama and
Yonezawa's calculus, we can test this expression using the
cupto-derived control. The latter is implemented in OCaml. We cannot
test the typing of our fix, since the type system of cupto is too
coarse. We can test the dynamic behavior however. To avoid passing the
witness that the result type is populated, we set the result type to
a->a, which is obviously populated, by the identity
open Delimcc let control p f = take_subcont p (fun sk () -> push_prompt p (fun () -> (f (fun c -> push_subcont sk (fun () -> c))))) let fix f = let p = new_prompt () in let d = fun x -> x in let delta u = control p (fun k -> f (fun x -> push_prompt p (fun () -> (k d; k d)) x)) in push_prompt p (fun () -> (delta d; delta d));; (* val fix : (('a -> 'a) -> 'a -> 'a) -> 'a -> 'a = <fun> *) let fact self n = if n <= 1 then 1 else n * self (pred n);; fix fact 5;; (* 120 *)
Yukiyoshi Kameyama and Takuo Yonezawa:
Typed Dynamic Control Operators for Delimited Continuations (draft Oct. 21, 2007).
Kenichi Asai and Yukiyoshi Kameyama:
Polymorphic Delimited Continuations
Proc. Fifth Asian Symposium on Programming Languages and Systems (APLAS 2007), LNCS
A general recursive type is usually defined (see
Kameyama and Yonezawa) as
\mu X. F[X] where
X may appear negatively (i.e., contravariantly) in
X appears only positively (as in the type of
\mu X. (1 + Int * X))), the resulting type
is often called inductive.
A general recursive type, e.g.,
\mu X. X->Int->Int can be characterized by the following signature:
module type RecType = sig type t (* an abstract type *) val wrap : (t->int->int) -> t val unwrap : t -> (t->int->int) endprovided that
(unwrap . wrap)is the identity. If we have an implementation of this signature, we can transcribe a term such as
\x. x xfrom the untyped lambda-calculus to the typed one.
ML supports one implementation of
iso-recursive (data)types. However, there is another implementation,
using ML exceptions. Since exceptions are a particular case of
delimited control, we obtain another proof that simply
typed lambda-calculus with a cupto-like delimited
control is not strongly normalizing.
Call-by-need, or lazy, evaluation is call-by-name evaluation with the memoization of the result. A lazy expression is not evaluated until its result is needed. At that point, the expression is evaluated and the result is memoized. Thus a lazy expression is evaluated at most once. Lazy expressions may nest: a lazy expression may include other lazy expressions.
We implement lazy evaluation without any mutation or other destructive operations -- essentially in call-by-value lambda-calculus with shift and reset.
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