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14
Minimal Classical Logic and Control Operators
 In ICALP: Annual International Colloquium on Automata, Languages and Programming, volume 2719 of LNCS
, 2003
"... We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a \natural" implementation of this logic is Parigot's classical natural deduction. ..."
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Cited by 30 (5 self)
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We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a \natural" implementation of this logic is Parigot's classical natural deduction.
A typetheoretic foundation of delimited continuations. Higher Order Symbol
 Comput
, 2009
"... Abstract. There is a correspondence between classical logic and programming language calculi with firstclass continuations. With the addition of control delimiters, the continuations become composable and the calculi become more expressive. We present a finegrained analysis of control delimiters a ..."
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Cited by 14 (6 self)
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Abstract. There is a correspondence between classical logic and programming language calculi with firstclass continuations. With the addition of control delimiters, the continuations become composable and the calculi become more expressive. We present a finegrained analysis of control delimiters and formalise that their addition corresponds to the addition of a single dynamicallyscoped variable modelling the special toplevel continuation. From a type perspective, the dynamicallyscoped variable requires effect annotations. In the presence of control, the dynamicallyscoped variable can be interpreted in a purely functional way by applying a storepassing style. At the type level, the effect annotations are mapped within standard classical logic extended with the dual of implication, namely subtraction. A continuationpassingstyle transformation of lambdacalculus with control and subtraction is defined. Combining the translations provides a decomposition of standard CPS transformations for delimited continuations. Incidentally, we also give a direct normalisation proof of the simplytyped lambdacalculus with control and subtraction.
2005, ‘A ProofTheoretic Foundation of Abortive Continuations (Extended version
"... Abstract. We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce’s law without enforcing Ex Falso Quodlibet. We show that a “natural ” implementation of this logic is Parigot’s classical natural deduction. We then move on to the comp ..."
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Cited by 9 (5 self)
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Abstract. We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce’s law without enforcing Ex Falso Quodlibet. We show that a “natural ” implementation of this logic is Parigot’s classical natural deduction. We then move on to the computational side and emphasize that Parigot’s λµ corresponds to minimal classical logic. A continuation constant must be added to λµ to get full classical logic. The extended calculus is isomorphic to a syntactical restriction of Felleisen’s theory of control that offers a more expressive reduction semantics. This isomorphic calculus is in correspondence with a refined version of Prawitz’s natural deduction.
Control Reduction Theories: the Benefit of Structural Substitution
 UNDER CONSIDERATION FOR PUBLICATION IN J. FUNCTIONAL PROGRAMMING
"... The historical design of the callbyvalue theory of control relies on the reification of evaluation contexts as regular functions and on the use of ordinary term application for jumping to a continuation. To the contrary, the λCtp control calculus, developed by the authors, distinguishes between ju ..."
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Cited by 7 (4 self)
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The historical design of the callbyvalue theory of control relies on the reification of evaluation contexts as regular functions and on the use of ordinary term application for jumping to a continuation. To the contrary, the λCtp control calculus, developed by the authors, distinguishes between jumps and terms. This alternative calculus, which derives from Parigot’s λµcalculus, works by direct structural substitution of evaluation contexts. We review and revisit the legacy theories of control and argue that λCtp provides an observationally equivalent but smoother theory. In an additional note contributed by Matthias Felleisen, we review the story of the birth of control calculi during the mid to late eighties at Indiana University.
Soundness and Principal Contexts for a Shallow Polymorphic Type System based on Classical Logic
"... In this paper we investigate how to adapt the wellknown notion of MLstyle polymorphism (shallow polymorphism) to a term calculus based on a CurryHoward correspondence with classical sequent calculus, namely, theX icalculus. We show that the intuitive approach is unsound, and pinpoint the precise ..."
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In this paper we investigate how to adapt the wellknown notion of MLstyle polymorphism (shallow polymorphism) to a term calculus based on a CurryHoward correspondence with classical sequent calculus, namely, theX icalculus. We show that the intuitive approach is unsound, and pinpoint the precise nature of the problem. We define a suitably refined type system, and prove its soundness. We then define a notion of principal contexts for the type system, and provide an algorithm to compute these, which is proved to be sound and complete with respect to the type system. In the process, we formalise and prove correctness of generic unification, which generalises Robinson’s unification to shallowpolymorphic types. Key words: CurryHoward, classical logic, generic unification, principal types, cut elimination 1.
λµPRL – A Proof Refinement Calculus for Classical Reasoning
 in Computational Type Theory Diploma thesis, Institut für Informatik, Universität Potsdam
, 2009
"... Abstract. We present a hybrid proof calculus λµPRL that combines the propositional fragment of computational type theory with classical reasoning rules from the λµcalculi. The calculus supports the topdown development of proofs as well as the extraction of proof terms in a functional programming l ..."
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Abstract. We present a hybrid proof calculus λµPRL that combines the propositional fragment of computational type theory with classical reasoning rules from the λµcalculi. The calculus supports the topdown development of proofs as well as the extraction of proof terms in a functional programming language extended by a nonconstructive binding operator. It enables a user to employ a mix of constructive and classical reasoning techniques and to extract algorithms from proofs of specification theorems that are fully executable if classical arguments occur only in proof parts related to the validation of the algorithm. We prove the calculus sound and complete for classical propositional logic, introduce the concept of µsafe terms to identify proof terms corresponding to constructive proofs and show that the restriction of λµPRL to µsafe proof terms is sound and complete for intuitionistic propositional logic. We also show that an extension of λµPRL to arithmetical and firstorder expressions is isomorphic to Murthy’s calculus P ROGK.
Intuitionistic Control Logic
, 2012
"... We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. The new constant requires a simpley ..."
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We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. The new constant requires a simpleyetsignificant modification of intuitionistic logic both semantically and prooftheoretically. We define a Kripkestyle semantics as well as a topological space interpretation in which the new constant is given a precise denotation. We define a sequent calculus and prove cutelimination. We then formulate a natural deduction proof system with a term calculus, one that gives a direct, computational interpretation of contraction. This calculus shows that ICL is fully capable of typing programming language control constructs such as call/cc while maintaining intuitionistic implication as a genuine connective.
An Intuitionistic Logic for Sequential Control
"... We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. We define Kripke models for ICL and ..."
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We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. We define Kripke models for ICL and show how they translate to several other forms of semantics. We define a sequent calculus and prove cutelimination. We then formulate a natural deduction proof system with a term calculus that gives a direct, computational interpretation of contraction. This calculus shows that ICL is fully capable of typing programming language control operators such as call/cc while maintaining intuitionistic implication as a genuine connective. 1
unknown title
, 905
"... An arithmetical proof of the strong normalization for the λcalculus with recursive equations on types ..."
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An arithmetical proof of the strong normalization for the λcalculus with recursive equations on types