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This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin Watkins which led me to simplify some of this work. Finally, I would like to thank my other advisor, John Reynolds, for all his kindness and support over the last five years. Abstract This thesis introduces a new logical system, ordered linear logic, which combines reasoning with unrestricted, linear, and ordered hypotheses. The logic conservatively extends (intuitionistic) linear logic, which contains both unrestricted and linear hypotheses, with a notion of ordered hypotheses. Ordered hypotheses must be used exactly once, subject to the order in which they were assumed (i.e., their order cannot be changed during the course of a derivation). This ordering constraint allows for logical representations of simple data structures such as stacks and queues. We construct ordered linear logic in the style of Martin-L"of from the basic notion of a hypothetical judgement. We then show normalization for the system by constructing a sequent calculus presentation and proving cut-elimination of the sequent system.

Security-typed languages enforce secrecy or integrity policies by type-checking. This paper investigates continuation-passing style (CPS) as a means of proving that such languages enforce noninterference and as a rst step towards understanding their compilation. We present a low-level, secure calculus with higher-order, imperative features and linear continuations.

The shift and reset operators, proposed by Danvy and Filinski, are powerful control primitives for capturing delimited continuations. Delimited continuation is a similar concept as the standard (unlimited) continuation, but it represents part of the rest of the computation, rather than the whole rest of computation. In the literature, the semantics of shift and reset has been given by a CPS-translation only. This paper gives a direct axiomatization of calculus with shift and reset, namely, we introduce a set of equations, and prove that it is sound and complete with respect to the CPS-translation. We also introduce a calculus with control operators which is as expressive as the calculus with shift and reset, has a sound and complete axiomatization, and is conservative over Sabry and Felleisen's theory for first-class continuations.

First-class continuations are a powerful computational effect, allowing the programmer to express any form of jumping. Types and effect systems can be used to reason about continuations, both in the source language and in the target language of the continuation-passing transform. In this paper, we establish the connection between an effect system for first-class continuations and typed versions of continuationpassing style. A region in the effect system determines a local answer type for continuations, such that the continuation transforms of pure expressions are parametrically polymorphic in their answer types. We use this polymorphism to derive transforms that make use of effect information, in particular, a mixed linear/non-linear continuation-passing transform, in which expressions without control effects are passed their continuations linearly.

We present a direct implementation of the shift and reset control operators in the Scheme 48 system. The new implementation improves upon the traditional technique of simulating shift and reset via call/cc. Typical applications of these operators exhibit space savings and a significant overall performance gain. Our technique is based upon the popular incremental stack/heap strategy for representing continuations. We present implementation details as well as some benchmark measurements for typical applications.

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Modularity is an important pragmatic aspect of semantic descriptions. In denotational semantics, the issue of modularity has received much attention, and appropriate abstractions have been introduced, so that definitions of semantic functions may be independent of the details of how computations are modelled. In structural operational semantics (SOS), however, this issue has largely been neglected, and SOS descriptions of programming languages typically exhibit rather poor modularity -- for instance, extending the described language may entail a complete reformulation of the description of the original constructs. A propos

. We formally prove the stackability and linearity of exception handlers with ML-style semantics using a novel proof technique via an ordered logical framework (OLF). We first transform exceptions into continuation-passing-style (CPS) terms and formalize the exception properties as a judgement on the CPS terms. Then, rather than directly proving that the properties hold for terms, we prove our theorem for the representations of the CPS terms and transform in OLF. We rely upon the correctness of our representations to transfer the results back to the actual CPS terms and transform. Our work can be seen as two-fold: we present a theoretical justification of using the stack mechanism to implement exceptions of ML-like semantics; and we demonstrate the value of an ordered logical framework as a conceptual tool in the theoretical study of programming languages. 1 Introduction Exception handling facilities in modern languages like ML [MTHM97,LDG + 00] or Java allow the programmer to defin...

Continuations can be used to explain a wide variety of control behaviours, including calling/returning (procedures), raising/handling (exceptions), labelled jumping (goto statements), process switching (coroutines), and backtracking. However, continuations are often manipulated in a highly stylised way, and we show that all of these, bar backtracking, in fact use their continuations linearly ; this is formalised by taking a target language for cps transforms that has both intuitionistic and linear function types.