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A New Deconstructive Logic: Linear Logic
, 1995
"... The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different a ..."
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Cited by 102 (11 self)
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The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...
A Mixed Linear and NonLinear Logic: Proofs, Terms and Models (Preliminary Report)
, 1994
"... Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (`of course') modality. Benton, Bierman, Hyland and de Paiva have given a term assignment system for ILL and an associated notion of categorical model in which the ! modality is modelled by a comonad satis ..."
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Cited by 96 (3 self)
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Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (`of course') modality. Benton, Bierman, Hyland and de Paiva have given a term assignment system for ILL and an associated notion of categorical model in which the ! modality is modelled by a comonad satisfying certain extra conditions. Ordinary intuitionistic logic is then modelled in a cartesian closed category which arises as a full subcategory of the category of coalgebras for the comonad. This paper attempts to explain the connection between ILL and IL more directly and symmetrically by giving a logic, term calculus and categorical model for a system in which the linear and nonlinear worlds exist on an equal footing, with operations allowing one to pass in both directions. We start from the categorical model of ILL given by Benton, Bierman, Hyland and de Paiva and show that this is equivalent to having a symmetric monoidal adjunction between a symmetric monoidal closed category and a cartesian closed category. We then derive both a sequent calculus and a natural deduction presentation of the logic corresponding to the new notion of model.
A Judgmental Analysis of Linear Logic
, 2003
"... We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, ext ..."
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Cited by 49 (27 self)
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We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of doublenegation translation.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 38 (11 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Strong Normalisation of CutElimination in Classical Logic
, 2000
"... In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos ..."
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Cited by 35 (4 self)
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In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos et al., our procedure requires no special annotations on formulae and allows cutrules to pass over other cutrules. In order to adapt the notion of symmetric reducibility candidates for proving the strong normalisation property, we introduce a novel term assignment for sequent proofs of classical logic and formalise cutreductions as term rewriting rules.
CallbyName, CallbyValue, CallbyNeed, and the Linear Lambda Calculus
, 1994
"... Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second corresponds t ..."
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Cited by 28 (5 self)
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Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second corresponds to callbyvalue. We further show that if the target of the translation is taken to be an affine calculus, where ! controls contraction but weakening is allowed everywhere, then the second translation corresponds to a callbyneed calculus, as recently defined by Ariola, Felleisen, Maraist, Odersky, and Wadler. Thus the different calling mechanisms can be explained in terms of logical translations, bringing them into the scope of the CurryHoward isomorphism.
Permutability of Proofs in Intuitionistic Sequent Calculi
, 1996
"... We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deductio ..."
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Cited by 23 (4 self)
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We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising. Key words: intuitionistic logic, proof theory, natural deduction, sequent calculus. 1 Introduction There is a folklore theorem that two intuitionistic sequent calculus derivations are "really the same" iff they are interpermutable, using permutations as described by Kleene in [13]. Our purpose here is to make precise and prove such a "permutability theorem". Prawitz [18] showed how intuitionistic sequent calculus derivations determine natural deductions, via a mapping ' from LJ to NJ (here we consider only ...
On the Linear Decoration of Intuitionistic Derivations
, 1993
"... We define an optimal proofbyproof embedding of intuitionistic sequent calculus into linear logic and analyse the (purely logical) linearity information thus obtained. 1 Introduction Uniform translations of intuitionistic into linear logic, with their plethoric use of exponentials, are bound to gi ..."
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Cited by 13 (1 self)
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We define an optimal proofbyproof embedding of intuitionistic sequent calculus into linear logic and analyse the (purely logical) linearity information thus obtained. 1 Introduction Uniform translations of intuitionistic into linear logic, with their plethoric use of exponentials, are bound to give only `universal linearity information' about proofs. This paper aims at displaying the structure of `specific linearity information ' hidden in a given derivation. How can we apply this to intuitionistic proofs? We have to build a translation into linear logic such that reductions of the intuitionistic proof can be simulated by reductions of its linear image. A necessary condition for this to hold, is that the `skeleton' of the original proof is preserved by the translation. We call translations with this property `decorations '. Specifically, we construct a proofbyproof embedding of IL into LL (formulated as sequent calculi) such that: 1/ the skeleton of the original proof is preserve...
On the Fine Structure of the Exponential Rule
 Advances in Linear Logic
, 1993
"... We present natural deduction systems for fragments of intuitionistic linear logic obtained by dropping weakening and contractions also on !prefixed formulas. The systems are based on a twodimensional generalization of the notion of sequent, which accounts for a clean formulation of the introduction ..."
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Cited by 11 (4 self)
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We present natural deduction systems for fragments of intuitionistic linear logic obtained by dropping weakening and contractions also on !prefixed formulas. The systems are based on a twodimensional generalization of the notion of sequent, which accounts for a clean formulation of the introduction/elimination rules of the modality. Moreover, the different subsystems are obtained in a modular way, by simple conditions on the elimination rule for !. For the proposed systems we introduce a notion of reduction and we prove a normalization theorem. 1. Introduction Proof theory of modalities is a delicate subject. The shape of the rules governing the different modalities in the overpopulated world of modal logics is often an example of what a good rule should not be. In the context of sequent calculus, if we want cut elimination, we are often forced to accept rules which are neither left nor right rules, and which completely destroy the deep symmetries the calculus is based upon. In the c...
A Pragmatic Interpretation Of Substructural Logics
"... Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from ..."
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Cited by 11 (8 self)
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Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from mixed contexts of assertions and obligations and we prove the cutelimination theorem for it. For the formalization of reallife normative systems it is essential to consider inferences from mixed contexts of assertions and obligations, and also of assertions justi able relatively to a given state of information and obligations valid in a given normative system. In order to provide a formalization of the notion of causal implication and its interaction with obligations, the sequents of ILP have two areas in the antecedent, expressing the relevant and the ordinary intuitionistic consequence relation, respectively. To provide a pragmatic interpretation of reasoning with the lin...