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What is a Categorical Model of Intuitionistic Linear Logic?
, 1995
"... . This paper readdresses the old problem of providing a categorical model for Intuitionistic Linear Logic (ILL). In particular we compare the now standard model proposed by Seely to the lesser known one proposed by Benton, Bierman, Hyland and de Paiva. Surprisingly we find that Seely's model is uns ..."
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Cited by 99 (5 self)
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. This paper readdresses the old problem of providing a categorical model for Intuitionistic Linear Logic (ILL). In particular we compare the now standard model proposed by Seely to the lesser known one proposed by Benton, Bierman, Hyland and de Paiva. Surprisingly we find that Seely's model is unsound in that it does not preserve equality of proofs. We shall propose how to adapt Seely's definition so as to correct this problem and consider how this compares with the model due to Benton et al. 1 Intuitionistic Linear Logic For the first part we shall consider only the multiplicative, exponential fragment of Intuitionistic Linear Logic (MELL). Rather than give a detailed description of the logic and associated term calculus we assume that the reader is familiar with other work [15, 5]. The sequent calculus formulation is originally due to Girard [9] and is given below. Identity A \Gamma A \Gamma \Gamma B B; \Delta \Gamma C Cut \Gamma; \Delta \Gamma C \Gamma \Gamma A (I L ) \Gamm...
Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
"... ..."
Syntactic Control of Interference Revisited
, 1995
"... In "Syntactic Control of Interference" (POPL, 1978), J. C. Reynolds proposes three design principles intended to constrain the scope of imperative state effects in Algollike languages. The resulting linguistic framework seems to be a very satisfactory way of combining functional and imperative conc ..."
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Cited by 40 (6 self)
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In "Syntactic Control of Interference" (POPL, 1978), J. C. Reynolds proposes three design principles intended to constrain the scope of imperative state effects in Algollike languages. The resulting linguistic framework seems to be a very satisfactory way of combining functional and imperative concepts, having the desirable attributes of both purely functional languages (such as pcf) and simple imperative languages (such as the language of while programs). However, Reynolds points out that the "obvious" syntax for interference control has the unfortunate property that fireductions do not always preserve typings. Reynolds has subsequently presented a solution to this problem (ICALP, 1989), but it is fairly complicated and requires intersection types in the type system. Here, we present a much simpler solution which does not require intersection types. We first describe a new type system inspired in part by linear logic and verify that reductions preserve typings. We then define a class...
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
Proof Search Issues In Some NonClassical Logics
, 1998
"... This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investig ..."
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Cited by 36 (1 self)
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This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some nonclassical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a prooftheoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cutfree LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 11 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called `permutationfree' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism f...
On Bunched Typing
, 2002
"... We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched ..."
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Cited by 33 (2 self)
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We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most
Linear lambdaCalculus and Categorical Models Revisited
, 1992
"... this paper we shall consider multiplicative exponential linear logic (MELL), i.e. the fragment which has multiplicative conjunction or tensor,\Omega , linear implication, \Gammaffi, and the logical operator "exponential", !. We recall the rules for MELL in a sequent calculus system in Fig. 1. We us ..."
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Cited by 22 (0 self)
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this paper we shall consider multiplicative exponential linear logic (MELL), i.e. the fragment which has multiplicative conjunction or tensor,\Omega , linear implication, \Gammaffi, and the logical operator "exponential", !. We recall the rules for MELL in a sequent calculus system in Fig. 1. We use capital Greek letters \Gamma; \Delta for sequences of formulae and A; B for single formulae. The Exchange rule simply allows the permutation of assumptions. The `! rules' have been given names by other authors. ! L\Gamma1 is called Weakening , ! L\Gamma2 Contraction, ! L\Gamma3 Dereliction and (! R ) Promotion
Soft lambdacalculus: a language for polynomial time computation
 In Proc. FoSSaCS, Springer LNCS 2987
, 2004
"... Abstract. Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambdacalculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time. ..."
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Cited by 19 (2 self)
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Abstract. Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambdacalculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time. We then extend the type system of Soft logic with recursive types. This allows us to consider nonstandard types for representing lists. Using these datatypes we examine the concrete expressiveness of Soft lambdacalculus with the example of the insertion sort algorithm. 1
The LambdaCalculus with Multiplicities
, 1993
"... We introduce a refinement of the λcalculus, where the argument of a function is a bag of resources, that is a multiset of terms, whose multiplicities indicate how many copies of them are available. We show that this "λcalculus with multiplicities" has a natural functionality theory, similar to Cop ..."
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Cited by 17 (2 self)
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We introduce a refinement of the λcalculus, where the argument of a function is a bag of resources, that is a multiset of terms, whose multiplicities indicate how many copies of them are available. We show that this "λcalculus with multiplicities" has a natural functionality theory, similar to Coppo and Dezani's intersection type discipline. In our functionality theory the conjunction is managed in a "multiplicative" manner, according to Girard's terminology. We show that this provides an adequate interpretation of the calculus, by establishing that a term is convergent if and only if it has a nontrivial functional character.
Cut Rules and Explicit Substitutions
, 2000
"... this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the l ..."
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Cited by 15 (0 self)
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this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via leftintroduction and rightintroduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them Lsystems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in Lsystems. The advantage of Nsystems is that they seem closer to actual reasoning, while Lsystems on the other hand seem to have an easier proof theory. Lsystems are often extended with a "cut" rule as part of showing that for a given Lsystem and Nsystem, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider