DOCSLIB.ORG

Continuations in Natural Language

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Continuations in Natural Language Continuations in Natural Language [Extended Abstract] Chris Barker University of California, San Diego 9500 Gillman Drive La Jolla, CA, USA [email protected] ABSTRACT As diverse as these applications are, they all involve the Computer scientists, logicians and functional programmers design and analysis of artificial languages (i.e., programming have studied continuations in laboratory settings for years. languages and logical languages), as opposed to natural lan- As a result of that work, continuations are now accepted guages (the sort of languages that humans characteristically as an indispensable tool for reasoning about control, order use when communicating with each other). We might won- of evaluation, classical versus intuitionistic proof, and more. der, then, whether continuations are relevant for the study But all of the applications just mentioned concern artifi- of natural languages. This is the master question addressed cial languages; what about natural languages, the languages by this paper: spoken by humans in their daily life? Do natural languages • [Relevance] Are there phenomena in natural language get by without any of the marvelous control operators pro- that can profitably be analyzed using continuations? vided by continuations, or can we find continuations in the wild? This paper argues yes: that an adequate and com- Obviously, I believe that the answer is \yes"! I will try plete analysis of natural language must recognize and rely to persuade you to believe likewise by presenting four case on continuations. In support of this claim, I identify four studies, each of which has an analysis in terms of continua- independent linguistic phenomena for which a simple CPS- tions. based description provides an insightful analysis. An affirmative answer to the master question of relevance leads to a number of subquestions, including the following. 1. INTRODUCTION • [Universality] Even if some natural languages make The applications of continuations to date are remarkably use of continuation, do all languages make use of con- diverse. As representative examples, I will mention three tinuations? strands of research here: first, continuations provide an order- independent way of describing evaluation order in formal Languages differ in many ways. For instance, all languages languages. For instance, Plotkin [10] shows how various make a distinction between nouns and verbs, but not all Continuation-Passing Style (CPS) transforms can model eval- languages make a distinction between singular nouns and uation disciplines such as call-by-name or call-by-value. Sec- plural nouns. I have drawn the case studies below exclu- ond, Griffin [5], Murthy [9] and others show that continu- sively from English, since that simplifies the exposition, but ations are intimately involved in characterizing the compu- we must remain alert to the possibility that English may tational content of classical (as opposed to intuitionistic) be unusual or even unique in the relevant respects. In fact, proofs. Third, continuations provide a useful tool for pro- however, all of the phenomena discussed below have close grammers who want to use powerful control operators such analogs in many other languages. Coordination in particu- as call/cc in Scheme (and its analogs in other functional lar is widespread: as far as I know, every language provides programming languages); conversely, Queinnec [11] shows some way of saying something equivalent to John saw Mary how to use continuations in order to write in a direct style and Tom, where the coordinating conjunction and combines and still be assured that programs will behave in a reason- two noun phrases (Mary and Tom) into a single complex able way when it is the program's users who have access to noun phrase Mary and Tom. If an adequate analysis of co- powerful control operators (such as the `back' button on a ordination depends on continuations, as I argue below, that web browser). would strongly suggest that every natural language makes essential use of continuations. • [Distribution] Which specific control operators do Permission to make digital or hard copies of all or part of this work for natural languages make use of? Which control op- personal or classroom use is granted without fee provided that copies are erators are more `natural'? not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to A variety of control operators that make use of continuations republish, to post on servers or to redistribute to lists, requires prior specific have been proposed over the years, some of which differ in permission and/or a fee. Fourth ACM-SIGPLAN Continuation Workshop ’04 Venice, Italy fairly subtle ways: call/cc versus control (C), shift and Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. reset versus fcontrol and run, and so on. Sometimes one operator can be expressed in terms of another; neverthe- jection applies for formal languages: `x == x' means some- less, if one or the other version is more prevalent in natural thing very different from `x == 3', yet it still makes sense to language, that might suggest that some operators are more treat them both as boolean expressions. `natural' than others. In the discussion below, I will ar- In some more elaborate linguistic treatments, sentences gue that C and fcontrol/run are remarkably well-suited to denote functions from times and worlds to truth values, with handling quantification and focus. an analogous shift for expressions of other types. In the par- lance of linguistics, a treatment in terms of truth values is • [Delimitation] Does natural language prefer delim- `extensional', and a system with times and worlds is `in- ited or undelimited continuations? tensional'. Shan [13] shows that intensionalization can be rendered as a monad. Intensionality is not crucial in any of Historically, the first continuation operators were undelim- the discussions below, and the types will be complex enough ited (e.g., call/cc or J). Felleisen [4, 3] proposed delim- anyway, so I will use an extensional semantics on which sen- ited continuations (sometimes called `composable' continua- tences denote truth values. tions) such as control (`C') and prompt (`%'). Interestingly, the natural-language phenomena discussed in this paper all 2.2 Methodology: formal grammar fragments seem to require some form of delimitation. It is standard in mainstream linguistics when analyzing a particular type of expression to provide a formal grammar 2. PRELIMINARIES approximating the syntax and the semantics of the expres- The intrepid reader is welcome to skip directly to the case sions under study, and that is the approach that I will take in studies, backtracking to this section only if some assumption this paper. Although I intend the grammar fragment devel- or technique seems puzzling. oped below to capture something genuine about the nature Theoretical computer scientists think deeply about the of the natural language expressions, it is important to bear nature of formal languages. Many of the tools and tech- in mind that any formal grammar is at best an approxima- niques that are relevant for the study and the design of for- tion of natural language, i.e., a hypothesis, not a definition, mal languages apply also to the study of natural language, and the degree to which the formal treatment accurately re- including lexical analysis, parsing, and denotational seman- flects the behavior of the natural language expression will tics. As a result, computer scientists have highly developed always be an empirical issue. intuitions about languages and about good ways to analyze In the fragment, each expression will be assigned mem- them. These intuitions will go a long way towards under- bership in some syntactic category, and the name of each standing the issues and phenomena discussed below; after syntactic category will also serve to indicate the semantic all, one of my main goals in this paper is to emphasize sim- type of the meanings of expressions in that category. ilarities between formal languages and natural languages. A concrete example will help make this clear. Here is a Nevertheless, there are significant differences between com- direct-style starting point that the fragment below will build puter science approaches and linguistic approaches. I have on: made some effort to present a discussion of natural language (1) Some direct-style lexical entries: in a way that makes sense to a computer scientist|but only up to a point. To the extent that the assumptions and tech- Syntactic category niques agree with standard practice in computer science, Expression (= type) Semantic value well and good; but when there are differences, I will depend John e j on the reader to trust that the linguistic assumptions are co- Mary e m herent and well motivated, and to keep an open mind about left e!t left what is the \right" way to model language. saw e!(e!t) saw 2.1 Declarative sentences denote truth values Intuitively, e is the category of expressions that denote in- dividuals. Since the semantic job of the proper name John As a starting point, it is necessary to think of natural lan- is to refer to a particular individual (in this case, the in- guage utterances as expressing computations. Natural lan- dividual j), the word John is a member of the category e. guage is clearly capable of specifying algorithms: just think Expressions in a category with a complex label of the form of a recipe in a cookbook. Natural language can also express A!B will denote a function from meanings of type A to imperatives (Shut the door!), queries, and can even impinge meanings of type B. Expressions in category t denote truth on the real world in a direct way that formal languages typ- values (booleans), so that an intransitive verb such as left ically do not (I hereby declare you husband and wife).
Load more

Recommended publications
  • Continuation-Passing Style
    Continuation-Passing Style CS 6520, Spring 2002 1 Continuations and Accumulators In our exploration of machines for ISWIM, we saw how to explicitly track the current continuation for a computation (Chapter 8 of the notes). Roughly, the continuation reflects the control stack in a real machine, including virtual machines such as the JVM. However, accurately reflecting the \memory" use of textual ISWIM requires an implementation different that from that of languages like Java. The ISWIM machine must extend the continuation when evaluating an argument sub-expression, instead of when calling a function. In particular, function calls in tail position require no extension of the continuation. One result is that we can write \loops" using λ and application, instead of a special for form. In considering the implications of tail calls, we saw that some non-loop expressions can be converted to loops. For example, the Scheme program (define (sum n) (if (zero? n) 0 (+ n (sum (sub1 n))))) (sum 1000) requires a control stack of depth 1000 to handle the build-up of additions surrounding the recursive call to sum, but (define (sum2 n r) (if (zero? n) r (sum2 (sub1 n) (+ r n)))) (sum2 1000 0) computes the same result with a control stack of depth 1, because the result of a recursive call to sum2 produces the final result for the enclosing call to sum2 . Is this magic, or is sum somehow special? The sum function is slightly special: sum2 can keep an accumulated total, instead of waiting for all numbers before starting to add them, because addition is commutative.
    [Show full text]
  • Denotational Semantics
    Denotational Semantics CS 6520, Spring 2006 1 Denotations So far in class, we have studied operational semantics in depth. In operational semantics, we define a language by describing the way that it behaves. In a sense, no attempt is made to attach a “meaning” to terms, outside the way that they are evaluated. For example, the symbol ’elephant doesn’t mean anything in particular within the language; it’s up to a programmer to mentally associate meaning to the symbol while defining a program useful for zookeeppers. Similarly, the definition of a sort function has no inherent meaning in the operational view outside of a particular program. Nevertheless, the programmer knows that sort manipulates lists in a useful way: it makes animals in a list easier for a zookeeper to find. In denotational semantics, we define a language by assigning a mathematical meaning to functions; i.e., we say that each expression denotes a particular mathematical object. We might say, for example, that a sort implementation denotes the mathematical sort function, which has certain properties independent of the programming language used to implement it. In other words, operational semantics defines evaluation by sourceExpression1 −→ sourceExpression2 whereas denotational semantics defines evaluation by means means sourceExpression1 → mathematicalEntity1 = mathematicalEntity2 ← sourceExpression2 One advantage of the denotational approach is that we can exploit existing theories by mapping source expressions to mathematical objects in the theory. The denotation of expressions in a language is typically defined using a structurally-recursive definition over expressions. By convention, if e is a source expression, then [[e]] means “the denotation of e”, or “the mathematical object represented by e”.
    [Show full text]
  • Bisimulations for Delimited-Control Operators
    Logical Methods in Computer Science Volume 15, Issue 2, 2019, pp. 18:1–18:57 Submitted Apr. 24, 2018 https://lmcs.episciences.org/ Published May 24, 2019 BISIMULATIONS FOR DELIMITED-CONTROL OPERATORS DARIUSZ BIERNACKI a, SERGUE¨I LENGLET b, AND PIOTR POLESIUK a a University of Wroclaw e-mail address: fdabi,[email protected] b Universit´ede Lorraine e-mail address: [email protected] Abstract. We present a comprehensive study of the behavioral theory of an untyped λ-calculus extended with the delimited-control operators shift and reset. To that end, we define a contextual equivalence for this calculus, that we then aim to characterize with coinductively defined relations, called bisimilarities. We consider different styles of bisimilarities (namely applicative, normal-form, and environmental) within a unifying framework, and we give several examples to illustrate their respective strengths and weaknesses. We also discuss how to extend this work to other delimited-control operators. 1. Introduction Delimited-control operators. Control operators for delimited continuations enrich a pro- gramming language with the ability to delimit the current continuation, to capture such a delimited continuation, and to compose delimited continuations. Such operators have been originally proposed independently by Felleisen [26] and by Danvy and Filinski [21], with nu- merous variants designed subsequently [34, 66, 32, 25]. The applications of delimited-control operators range from non-deterministic programming [21, 45], partial evaluation [58, 19], and normalization by evaluation [24] to concurrency [34], mobile code [89], linguistics [84], oper- ating systems [43], and probabilistic programming [44]. Several variants of delimited-control operators are nowadays available in mainstream functional languages such as Haskell [25], OCaml [42], Racket [29], and Scala [76].
    [Show full text]
  • An Operational Foundation for Delimited Continuations in the Cps Hierarchy
    Logical Methods in Computer Science Vol. 1 (2:5) 2005, pp. 1–39 Submitted Apr. 1, 2005 www.lmcs-online.org Published Nov. 8, 2005 AN OPERATIONAL FOUNDATION FOR DELIMITED CONTINUATIONS IN THE CPS HIERARCHY MALGORZATA BIERNACKA, DARIUSZ BIERNACKI, AND OLIVIER DANVY BRICS, Department of Computer Science, University of Aarhus IT-parken, Aabogade 34, DK-8200 Aarhus N, Denmark e-mail address: {mbiernac,dabi,danvy}@brics.dk Abstract. We present an abstract machine and a reduction semantics for the lambda- calculus extended with control operators that give access to delimited continuations in the CPS hierarchy. The abstract machine is derived from an evaluator in continuation- passing style (CPS); the reduction semantics (i.e., a small-step operational semantics with an explicit representation of evaluation contexts) is constructed from the abstract machine; and the control operators are the shift and reset family. We also present new applications of delimited continuations in the CPS hierarchy: finding list prefixes and normalization by evaluation for a hierarchical language of units and products. 1. Introduction The studies of delimited continuations can be classified in two groups: those that use continuation-passing style (CPS) and those that rely on operational intuitions about con- trol instead. Of the latter, there is a large number proposing a variety of control opera- tors [5,37,40,41,49,52,53,65,70,74,80] which have found applications in models of control, concurrency, and type-directed partial evaluation [8,52,75]. Of the former, there is the work revolving around the family of control operators shift and reset [27–29,32,42,43,55,56,66,80] which have found applications in non-deterministic programming, code generation, par- tial evaluation, normalization by evaluation, computational monads, and mobile comput- ing [6,7,9,17,22,23,33,34,44,46,48,51,57,59,61,72,77–79].
    [Show full text]
  • Continuation Passing Style for Effect Handlers
    Continuation Passing Style for Effect Handlers Daniel Hillerström1, Sam Lindley2, Robert Atkey3, and KC Sivaramakrishnan4 1 The University of Edinburgh, United Kingdom [email protected] 2 The University of Edinburgh, United Kingdom [email protected] 3 University of Strathclyde, United Kingdom [email protected] 4 University of Cambridge, United Kingdom [email protected] Abstract We present Continuation Passing Style (CPS) translations for Plotkin and Pretnar’s effect han- dlers with Hillerström and Lindley’s row-typed fine-grain call-by-value calculus of effect handlers as the source language. CPS translations of handlers are interesting theoretically, to explain the semantics of handlers, and also offer a practical implementation technique that does not require special support in the target language’s runtime. We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation first by uncurrying it to yield a properly tail-recursive translation and second by making it higher-order in order to contract administrative redexes at translation time. We prove that the higher-order CPS translation simulates effect handler reduction. We have implemented the higher-order CPS translation as a JavaScript backend for the Links programming language. 1998 ACM Subject Classification D.3.3 Language Constructs and Features, F.3.2 Semantics of Programming Languages Keywords and phrases effect handlers, delimited control, continuation passing style Digital Object Identifier 10.4230/LIPIcs.FSCD.2017.18 1 Introduction Algebraic effects, introduced by Plotkin and Power [25], and their handlers, introduced by Plotkin and Pretnar [26], provide a modular foundation for effectful programming.
    [Show full text]
  • A Descriptive Study of California Continuation High Schools
    Issue Brief April 2008 Alternative Education Options: A Descriptive Study of California Continuation High Schools Jorge Ruiz de Velasco, Greg Austin, Don Dixon, Joseph Johnson, Milbrey McLaughlin, & Lynne Perez Continuation high schools and the students they serve are largely invisible to most Californians. Yet, state school authorities estimate This issue brief summarizes that over 115,000 California high school students will pass through one initial findings from a year- of the state’s 519 continuation high schools each year, either on their long descriptive study of way to a diploma, or to dropping out of school altogether (Austin & continuation high schools in 1 California. It is the first in a Dixon, 2008). Since 1965, state law has mandated that most school series of reports from the on- districts enrolling over 100 12th grade students make available a going California Alternative continuation program or school that provides an alternative route to Education Research Project the high school diploma for youth vulnerable to academic or conducted jointly by the John behavioral failure. The law provides for the creation of continuation W. Gardner Center at Stanford University, the schools ‚designed to meet the educational needs of each pupil, National Center for Urban including, but not limited to, independent study, regional occupation School Transformation at San programs, work study, career counseling, and job placement services.‛ Diego State University, and It contemplates more intensive services and accelerated credit accrual WestEd. strategies so that students whose achievement in comprehensive schools has lagged might have a renewed opportunity to ‚complete the This project was made required academic courses of instruction to graduate from high school.‛ 2 possible by a generous grant from the James Irvine Taken together, the size, scope and legislative design of the Foundation.
    [Show full text]
  • RELATIONAL PARAMETRICITY and CONTROL 1. Introduction the Λ
    Logical Methods in Computer Science Vol. 2 (3:3) 2006, pp. 1–22 Submitted Dec. 16, 2005 www.lmcs-online.org Published Jul. 27, 2006 RELATIONAL PARAMETRICITY AND CONTROL MASAHITO HASEGAWA Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 Japan, and PRESTO, Japan Science and Technology Agency e-mail address: [email protected] Abstract. We study the equational theory of Parigot’s second-order λµ-calculus in con- nection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the tar- get calculus induces a natural notion of equivalence on the λµ-terms. On the other hand, the unconstrained relational parametricity on the λµ-calculus turns out to be inconsis- tent. Following these facts, we propose to formulate the relational parametricity on the λµ-calculus in a constrained way, which might be called “focal parametricity”. Dedicated to Prof. Gordon Plotkin on the occasion of his sixtieth birthday 1. Introduction The λµ-calculus, introduced by Parigot [26], has been one of the representative term calculi for classical natural deduction, and widely studied from various aspects. Although it still is an active research subject, it can be said that we have some reasonable un- derstanding of the first-order propositional λµ-calculus: we have good reduction theories, well-established CPS semantics and the corresponding operational semantics, and also some canonical equational theories enjoying semantic completeness [16, 24, 25, 36, 39]. The last point cannot be overlooked, as such complete axiomatizations provide deep understand- ing of equivalences between proofs and also of the semantic structure behind the syntactic presentation.
    [Show full text]
  • Continuation Calculus
    Eindhoven University of Technology MASTER Continuation calculus Geron, B. Award date: 2013 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Continuation calculus Master’s thesis Bram Geron Supervised by Herman Geuvers Assessment committee: Herman Geuvers, Hans Zantema, Alexander Serebrenik Final version Contents 1 Introduction 3 1.1 Foreword . 3 1.2 The virtues of continuation calculus . 4 1.2.1 Modeling programs . 4 1.2.2 Ease of implementation . 5 1.2.3 Simplicity . 5 1.3 Acknowledgements . 6 2 The calculus 7 2.1 Introduction . 7 2.2 Definition of continuation calculus . 9 2.3 Categorization of terms . 10 2.4 Reasoning with CC terms .
    [Show full text]
  • 6.5 a Continuation Machine
    6.5. A CONTINUATION MACHINE 185 6.5 A Continuation Machine The natural semantics for Mini-ML presented in Chapter 2 is called a big-step semantics, since its only judgment relates an expression to its final value—a “big step”. There are a variety of properties of a programming language which are difficult or impossible to express as properties of a big-step semantics. One of the central ones is that “well-typed programs do not go wrong”. Type preservation, as proved in Section 2.6, does not capture this property, since it presumes that we are already given a complete evaluation of an expression e to a final value v and then relates the types of e and v. This means that despite the type preservation theorem, it is possible that an attempt to find a value of an expression e leads to an intermediate expression such as fst z which is ill-typed and to which no evaluation rule applies. Furthermore, a big-step semantics does not easily permit an analysis of non-terminating computations. An alternative style of language description is a small-step semantics.Themain judgment in a small-step operational semantics relates the state of an abstract machine (which includes the expression to be evaluated) to an immediate successor state. These small steps are chained together until a value is reached. This level of description is usually more complicated than a natural semantics, since the current state must embody enough information to determine the next and all remaining computation steps up to the final answer.
    [Show full text]
  • Delimited Continuations in Operating Systems
    Delimited continuations in operating systems Oleg Kiselyov1 and Chung-chieh Shan2 1 FNMOC [email protected] 2 Rutgers University [email protected] Abstract. Delimited continuations are the meanings of delimited evalu- ation contexts in programming languages. We show they offer a uniform view of many scenarios that arise in systems programming, such as a re- quest for a system service, an event handler for input/output, a snapshot of a process, a file system being read and updated, and a Web page. Ex- plicitly recognizing these uses of delimited continuations helps us design a system of concurrent, isolated transactions where desirable features such as snapshots, undo, copy-on-write, reconciliation, and interposition fall out by default. It also lets us take advantage of efficient implemen- tation techniques from programming-language research. The Zipper File System prototypes these ideas. 1 Introduction One notion of context that pervades programming-language research is that of evaluation contexts. If one part of a program is currently running (that is, being evaluated), then the rest of the program is expecting the result from that part, typically waiting for it. This rest of the program is the evaluation context of the running part. For example, in the program “1 + 2 × 3”, the evaluation context of the multiplication “2 × 3” is the rest of the program “1 + ”. The meaning of an evaluation context is a function that maps a result value to an answer. For example, the meaning of the evaluation context “1 + ” is the increment function, so it maps the result value 6 to the answer 7.
    [Show full text]
  • Compositional Semantics for Composable Continuations from Abortive to Delimited Control
    Compositional Semantics for Composable Continuations From Abortive to Delimited Control Paul Downen Zena M. Ariola University of Oregon fpdownen,[email protected] Abstract must fulfill. Of note, Sabry’s [28] theory of callcc makes use of Parigot’s λµ-calculus, a system for computational reasoning about continuation variables that have special properties and can only be classical proofs, serves as a foundation for control operations em- instantiated by callcc. As we will see, this not only greatly aids in bodied by operators like Scheme’s callcc. We demonstrate that the reasoning about terms with free variables, but also helps in relating call-by-value theory of the λµ-calculus contains a latent theory of the equational and operational interpretations of callcc. delimited control, and that a known variant of λµ which unshack- In another line of work, Parigot’s [25] λµ-calculus gives a sys- les the syntax yields a calculus of composable continuations from tem for computing with classical proofs. As opposed to the pre- the existing constructs and rules for classical control. To relate to vious theories based on the λ-calculus with additional primitive the various formulations of control effects, and to continuation- constants, the λµ-calculus begins with continuation variables in- passing style, we use a form of compositional program transforma- tegrated into its core, so that control abstractions and applications tions which preserves the underlying structure of equational the- have the same status as ordinary function abstraction. In this sense, ories, contexts, and substitution. Finally, we generalize the call- the λµ-calculus presents a native language of control.
    [Show full text]
  • Delimited Control with Multiple Prompts in Theory and Practice
    Delimited control with multiple prompts in theory and practice Paul Downen Zena M. Ariola University of Oregon fpdownen,[email protected] 1. Proposal hE[S V ]i 7! hV (λx.hE[x]i)i The versatile and expressive capabilities of delimited control and composable continuations have gained it attention in both the the- #(E[F V ]) 7! #(V (λx.E[x])) ory and practice of functional programming with effects. On the hE[S0 V ]i0 7! V (λx.hE[x]i0) more theoretical side, delimited control may be used as a basis for # (E[F V ]) 7! V (λx.E[x]) explaining and understanding a wide variety of other computational 0 0 effects, like mutable state and exceptions, based on Filinski’s [8, 9] observation that composable continuations can be used to repre- Figure 1. Four different pairs of delimiters and control operators. sent any monadic effect. On the more practical side, forms of de- limited control have been implemented in real-world programming languages [6, 10, 15], and used in the design of libraries like for represents a closed off sub-computation, where the F operator is creating web servers [10]. only capable of capturing the evaluation context 2 × ≤ 10. That However, the design space for adding composable continua- way, the delimiter protects the surrounding context from the control tions to a programming language is vast, and a number of dif- operator, so that even in the larger program ferent definitions of delimited control operators have been pro- if #(2 × (F (λk:M)) ≤ 10) then red else blue posed [3, 4, 7, 12, 21].
    [Show full text]