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Term Assignment for Intuitionistic Linear Logic
, 1992
"... In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. Th ..."
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Cited by 53 (9 self)
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In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is welltyped). We define a simple (but more general than previous proposals) categorical model for Intuitionistic Linear Logic and show how this can be used to derive the term assignment system. We also consider term reduction arising from cutelimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these, as well as with the equations which follow from our categorical model.
A Classical Linear λcalculus
, 1997
"... This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natu ..."
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Cited by 8 (0 self)
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This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. This formulation is compared in detail to the sequent calculus formulation. In an appendix I shall also demonstrate a somewhat hidden connexion with the paradigm of control operators for functional languages which gives a new computational interpretation of Parigot's techniques.
A Short Note on Intuitionistic Propositional Logic with Multiple Conclusions
"... It is a common misconception among logicians to think that intuitionism is necessarily tiedup with single conclusion (sequent or Natural Deduction) calculi. Single conclusion calculi can be used and are convenient, but they are by no means necessary, as shown by such influential authors as Kleene, ..."
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Cited by 2 (0 self)
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It is a common misconception among logicians to think that intuitionism is necessarily tiedup with single conclusion (sequent or Natural Deduction) calculi. Single conclusion calculi can be used and are convenient, but they are by no means necessary, as shown by such influential authors as Kleene, Takeuti and Dummett, to cite only three. If single conclusions are not necessary, how do we guarantee that only intuitionistic derivations are allowed? Traditionally one insists on restrictions on particular rules: implication right, negation right and universal quantification right are required to be single conclusion rules. In this note we show that instead of a cardinality restrictionm such as one conclusion only, we can use a notion of dependency between formulae to enforce the constructive character of derivations. Since Gentzenâ€™s pioneering work it has been traditional to associate intuitionism with a singleconclusion sequent calculus or Natural Deduction system.
Mechanizing linear logic in Isabelle
 In 10th International Congress of Logic, Philosophy and Methodology of Science
, 1995
"... We present an implementation of propositional Linear Logic in the Isabelle proof system. Previous implementations of Linear Logic have often been geared to studies of efficiency of proof search; ours provides an environment for users to describe problems and to develop proofs interactively. Isabelle ..."
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Cited by 2 (0 self)
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We present an implementation of propositional Linear Logic in the Isabelle proof system. Previous implementations of Linear Logic have often been geared to studies of efficiency of proof search; ours provides an environment for users to describe problems and to develop proofs interactively. Isabelle provides many facilities for developing a useful specification and verification environment from the basic formulation of logical systems. We briefly introduce the logic and Isabelle, and discuss some of the issues in automatic theorem proving in Linear Logic. We then describe the system we have built for executing proofs in Isabelle, and illustrate its use. 1