View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Queen Mary Research Online Linear and Affine Typing of Continuation-Passing Style Berdine, Joshua James For additional information about this publication click this link. http://qmro.qmul.ac.uk/jspui/handle/123456789/5035 Information about this research object was correct at the time of download; we occasionally make corrections to records, please therefore check the published record when citing. For more information contact [email protected] ISSN 1470-5559 Linear and Affine Typing of Continuation-Pass- ing Style Joshua James Berdine RR-04-04 January 2004 Department of Computer Science 1 Linear and Affine Typing of Continuation-Passing Style Joshua James Berdine Submitted for the degree of Doctor of Philosophy Queen Mary, University of London 2004 2 3 Linear and Affine Typing of Continuation-Passing Style Joshua James Berdine Abstract In this dissertation we show that linear and affine type systems for continuation-passing style support correct and tight refinements of standard continuation semantics. In particular, a wide variety of control constructs admit typing disciplines which ensure linear or affine use of the control context in their continuation semantics. This refinement of standard continuation semantics using restricted types is an exploitation of the stylized use of continuations many control behaviors exhibit. Continuations are the raw material of control and can be used to explain a wide variety of control behaviors, including call- ing/returning (procedures), raising/handling (exceptions), jumping/labeling (goto and labels), process switching (coroutines), backtracking (amb and fail), and capturing/invoking first-class continuations (call/cc, or callcc and throw). However, in all but the last case, continuations are not themselves intrinsic to the control construct, instead they are “behind the scenes,”imple- menting the control construct. In other words, except for first-class continuations, each control behavior is simply an idiom of continuation usage, and hence the continuations are used in a stylized fashion. Linear or affine use of control contexts; by which we mean, roughly, that con- trol contexts cannot be duplicated or, in the linear case, discarded; captures this stylized usage in all the above-mentioned cases bar backtracking. We also investigate several cases where even storable labels (plus goto) admit restricted interpretations. Submitted for the degree of Doctor of Philosophy Queen Mary, University of London 2004 4 Contents 1 Introduction 10 1.1 Note to the Reader . 10 1.2 Using a Continuation Twice . 10 1.3 Constraining CPS: Linear Use of Control Contexts . 12 1.4 Vantage Point and Plan of Attack . 13 1.4.1 Continuation semantics . 14 1.4.2 Refining with linearity . 16 1.5 Motivations . 16 1.5.1 Conceptual subtleties . 17 1.5.2 Mechanized reasoning . 17 1.5.3 Loss of precision . 17 1.6 Contributions . 18 2 Procedures, Simply-Typed 22 2.1 Source Language . 22 2.1.1 Syntax . 22 2.1.2 Type system . 23 2.2 Target Language (= R + (·) → (·) + (·) ( (·)) . 23 2.2.1 Syntax . 23 2.2.2 Type system . 24 2.3 Refined CPS Transformation . 25 2.4 Soundness . 27 2.5 Conclusion . 29 3 Procedures, Untyped 31 3.1 Source Language . 31 3.1.1 Syntax . 31 3.1.2 Operational semantics . 32 3.2 Target Language (+= X + µX. (·)) ......................... 32 3.2.1 Equational theory . 32 3.2.2 Recursive types . 32 3.3 Refined CPS Transformation . 33 3.4 Soundness . 34 3.5 Adequacy . 37 3.6 Completeness . 37 3.6.1 Target sublanguage . 39 3.6.2 Extracting the range of CPS ......................... 40 3.6.3 DS transformation . 41 3.6.4 No junk . 42 3.7 Other CPS Transformations . 44 3.8 Conclusion . 45 5 4 Exceptions 47 4.1 Source Language . 47 4.1.1 Syntax . 47 4.1.2 Operational semantics . 48 4.2 Target Language (+= (·) & (·)) ........................... 49 4.2.1 Syntax . 49 4.2.2 Equational theory . 49 4.2.3 Type system . 50 4.3 Refined CPS Transformation . 50 4.4 Soundness . 53 4.5 Completeness . 55 4.5.1 Target sublanguage . 56 4.5.2 DS transformation . 57 4.5.3 No junk . 58 4.6 Conclusion . 64 5 Delimited Continuations, and Control Contexts versus Continuations 65 5.1 Delimited Continuations . 65 5.2 Refined Delimited Continuation Interpretation of Procedures . 68 5.3 Control Contexts versus Continuations . 70 5.4 Delimited Continuation Interpretation of First-Class Continuations . 71 5.5 Background and Related Work . 72 5.6 Conclusion . 73 6 A Simple Command Language 74 6.1 Source Syntax . 74 6.2 Target Language (+= N) .............................. 75 6.2.1 Syntax . 75 6.2.2 Equational theory . 75 6.2.3 Type system . 75 6.3 Direct Semantics . 75 6.3.1 States . 76 6.3.2 Expressions . 76 6.3.3 Commands . 77 6.4 Refined CPS Transformation . 77 6.5 Soundness . 78 6.6 Adequacy . 78 6.7 Conclusion . 79 7 Forward Jumps 80 7.1 Source Syntax . 80 7.2 Target Language (+= 1 + &n (·)) ......................... 81 7.2.1 Syntax . 81 7.2.2 Equational theory . 82 7.2.3 Type system . 82 7.3 Standard CPS Transformation . 83 7.4 Refined CPS Transformation . 84 7.5 Soundness . 85 7.6 Adequacy . 86 7.7 Conclusion . 86 6 Contents 8 Backward Jumps 87 8.1 Source Language . 87 8.1.1 Syntax . 87 8.1.2 Standard CPS transformation . 88 8.2 Refined CPS Transformation . 89 8.2.1 Commands . 89 8.2.2 Programs . 89 8.3 Soundness . 92 8.4 Adequacy . 92 8.5 Hacked Linear CPS Transformation . 96 8.6 Conclusion . 98 9 Coroutines 99 9.1 Source Language . 99 9.1.1 Syntax . 99 9.1.2 Direct semantics . 99 9.2 Refined CPS Transformation . ..