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Automating the Meta Theory of Deductive Systems
, 2000
"... not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, a ..."
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not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, and experimental results related to the areas of programming languages, type theory, and logics. Design: The metalogical framework extends the logical framework LF [HHP93] by a metalogic M + 2. This design is novel and unique since it allows higherorder encodings of deductive systems and induction principles to coexist. On the one hand, higherorder representation techniques lead to concise and direct encodings of programming languages and logic calculi. Inductive de nitions on the other hand allow the formalization of properties about deductive systems, such as the proof that an operational semantics preserves types or the proof that a logic is is a proof calculus whose proof terms are recursive functions that may be consistent.M +
Ordered Linear Logic and Applications
, 2001
"... 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 ..."
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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 MartinL&quot;of from the basic notion of a hypothetical judgement. We then show normalization for the system by constructing a sequent calculus presentation and proving cutelimination of the sequent system.
Natural Deduction for Intuitionistic NonCommutative Linear Logic
 Proceedings of the 4th International Conference on Typed Lambda Calculi and Applications (TLCA'99
, 1999
"... We present a system of natural deduction and associated term calculus for intuitionistic noncommutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment. ..."
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We present a system of natural deduction and associated term calculus for intuitionistic noncommutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment.
Relating Natural Deduction and Sequent Calculus for Intuitionistic NonCommutative Linear Logic
, 1999
"... We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natura ..."
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Cited by 29 (16 self)
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We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of noncommutative natural deduction and its associated calculus. INCLL conservatively extends linear logic with means to express sequencing, which has applications in functional programming, logical frameworks, logic programming, and natural language parsing. 1 Introduction Linear logic [11] has been described as a logic of state because it views linear hypotheses as resources which may be consumed in the course of a deduction. It thereby significantly extends the expressive power of both classical and intuitionistic logics, yet it does not offer means to express sequencing. Th...
On proving syntactic properties of CPS programs
, 1999
"... Higherorder program transformations raise new challenges for proving properties of their output, since they resist traditional, rstorder proof techniques. In this work, we consider (1) the \onepass" continuationpassing style (CPS) transformation, which is secondorder, and (2) the occur ..."
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Cited by 23 (8 self)
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Higherorder program transformations raise new challenges for proving properties of their output, since they resist traditional, rstorder proof techniques. In this work, we consider (1) the \onepass" continuationpassing style (CPS) transformation, which is secondorder, and (2) the occurrences of parameters of continuations in its output. To this end, we specify the onepass CPS transformation relationally and we use the proof technique of logical relations.
Formalizing Implementation Strategies for FirstClass Continuations
 in [31
, 2000
"... We present the first formalization of implementation strategies for firstclass continuations. The formalization hinges on abstract machines for continuationpassing style (CPS) programs with a special treatment for the current continuation, accounting for the essence of firstclass continuation ..."
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We present the first formalization of implementation strategies for firstclass continuations. The formalization hinges on abstract machines for continuationpassing style (CPS) programs with a special treatment for the current continuation, accounting for the essence of firstclass continuations. These abstract machines are proven equivalent to a standard, substitutionbased abstract machine. The proof techniques work uniformly for various representations of continuations. As a byproduct, we also present a formal proof of the two folklore theorems that one continuation identifier is enough for secondclass continuations and that secondclass continuations are stackable.
There and back again
 In ICFP ’02: Proceedings of the seventh ACM SIGPLAN international conference on Functional programming
, 2002
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Proving Syntactic Properties of Exceptions in an Ordered Logical Framework
"... . We formally prove the stackability and linearity of exception handlers with MLstyle semantics using a novel proof technique via an ordered logical framework (OLF). We first transform exceptions into continuationpassingstyle (CPS) terms and formalize the exception properties as a judgement on th ..."
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Cited by 9 (2 self)
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. We formally prove the stackability and linearity of exception handlers with MLstyle semantics using a novel proof technique via an ordered logical framework (OLF). We first transform exceptions into continuationpassingstyle (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 twofold: we present a theoretical justification of using the stack mechanism to implement exceptions of MLlike 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...
HigherOrder Rewriting and Partial Evaluation
 REWRITING TECHNIQUES AND APPLICATIONS, LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... We demonstrate the usefulness of higherorder rewriting techniques for specializing programs, i.e., for partial evaluation. More precisely, we demonstrate how casting program specializers as combinatory reduction systems (CRSs) makes it possible to formalize the corresponding program transformat ..."
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Cited by 7 (4 self)
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We demonstrate the usefulness of higherorder rewriting techniques for specializing programs, i.e., for partial evaluation. More precisely, we demonstrate how casting program specializers as combinatory reduction systems (CRSs) makes it possible to formalize the corresponding program transformations as metareductions, i.e., reductions in the internal "substitution calculus." For partialevaluation problems, this means that instead of having to prove on a casebycase basis that one's "twolevel functions" operate properly, one can concisely formalize them as a combinatory reduction system and obtain as a corollary that static reduction does not go wrong and yields a wellformed residual program.