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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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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.
Callbyvalue is dual to callbyname, reloaded
 In Rewriting Technics and Application, RTA’05, volume 3467 of LNCS
, 2005
"... Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµcalculus of Parigot (1992). We give translations from the λµcalculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, ..."
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Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµcalculus of Parigot (1992). We give translations from the λµcalculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, translating from λµ to dual and then ‘reloading ’ from dual back into λµ yields a term equal to the original term. Composing the translations with duality on the dual calculus yields an involutive notion of duality on the λµcalculus. A previous notion of duality on the λµcalculus has been suggested by Selinger (2001), but it is not involutive. Note This paper uses color to clarify the relation of types and terms, and of source and target calculi. If the URL below is not in blue please download the color version from
Combining Derivations and Refutations for Cutfree Completeness in BiIntuitionistic Logic
, 2008
"... Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree se ..."
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Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree sequent calculus for biintuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. 1
A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC.
"... Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatm ..."
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Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatments of multipleconclusion systems (e.g., Parigot’s λ−µ calculus, or Urban and Bierman’s termcalculus) here the termassignment is distributed to all conclusions, and exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The present construction can be regarded as the construction of a free coCartesian
Intuitionistic Letcc via Labelled Deduction
, 2006
"... Intuitionistic logic can be presented as a calculus of labelled deduction on multipleconclusion sequents. The corresponding natural deduction system constitutes a type system for programs using letcc, which capture the current program continuation, restricted to enforce constructive validity. This ..."
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Intuitionistic logic can be presented as a calculus of labelled deduction on multipleconclusion sequents. The corresponding natural deduction system constitutes a type system for programs using letcc, which capture the current program continuation, restricted to enforce constructive validity. This allows us to develop a rich dependent type theory incorporating letcc, which is known to be highly problematic for computational interpretations of classical logic. Moreover
A Parigotstyle linear λcalculus for Full intuitionistic Linear Logic
, 2005
"... This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical l ..."
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This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical logic in that its sequentcalculus formulation has intrinsic multiple conclusions. From the intrinsic multiple conclusions comes the inspiration to modify Parigot’s natural deduction systems for classical logic, to produce a natural deduction formulation and a term assignment system for FILL. keywords: linear logic, λµcalculus, CurryHoward isomorphism 1
A Parigotstyle Linear lambdaCalculus for Full Intuitionistic Linear Logic
, 2003
"... This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classic ..."
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This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical logic in that its sequentcalculus formulation has intrinsic multiple conclusions. From the intrinsic multiple conclusions comes the inspiration to modify Parigot's natural deduction systems for classical logic, to produce a natural deduction formulation and a term assignment system for FILL.
Categorical Proof Theory of CoIntuitionistic Linear Logic
"... Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of ..."
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Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponent!, we build models of cointuitionistic logic in symmetric monoidal closed categories with additional structure, using a variant of Crolard’s term assignment to cointuitionistic logic in the construction of a free category. 1
Relating Sequent Calculi for Biintuitionistic Propositional Logic
"... Abstract. Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic ..."
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Abstract. Biintuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for biintuitionistic propositional logic: (1) a basic standardstyle sequent calculus that restricts the premises of implicationright and exclusionleft inferences to be singleconclusion resp. singleassumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Goré et al., where a complete class of cuts is encapsulated into special “unnest ” rules and (3) a cutfree labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standardstyle sequent calculus. 1
NATURAL DEDUCTION AND TERM ASSIGNMENT FOR COHEYTING ALGEBRAS IN POLARIZED BIINTUITIONISTIC LOGIC.
"... Abstract. We reconsider Rauszer’s biintuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of ..."
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Abstract. We reconsider Rauszer’s biintuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of polarized biintuitionistic logic (PBL) consists of two fragments, positive intuitionistic logic LJ⊃ ∩ and its dual LJ� � , extended with two negations partially internalizing the duality between LJ⊃ ∩ and LJ� �. Modal interpretations and Kripke’s semantics over bimodal preordered frames are considered and a Natural Deduction system PBN is sketched for the whole system. A stricter interpretation of the duality and a simpler natural deduction system is obtained when polarized biintuitionistic logic is interpreted over S4 rather than bimodal S4 (a logic called intuitionistic logic for pragmatics of assertions and conjectures ILPAC). The term assignment for the conjectural fragment LJ� � exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The duality is extended from formulas to proofs and it is shown that every computation in our calculus is isomorphic to a computation in the simply typed λcalculus. §1. Preface. We present a natural deduction system for propositional polarized biintuitionistic logic PBL, (a variant of) intuitionistic logic extended with a connective of subtraction A � B, read as “A but not B”, which is dual to implication. 1 The logic PBL is polarized in the sense that its expressions are regarded as expressing acts of assertion or of conjecture; implications and conjunctions are assertive, subtractions and disjunctions are conjectural. Assertions and conjectures are regarded as dual; moreover there are two negations, transforming assertions into conjectures and viceversa, in some sense internalizing the duality. Our notion of polarity isn’t just a technical device: it is rooted in an analysis of the structure of speechacts, following the viewpoint of the