P.W. O'Hearn. Resource interpretations, bunched implications and the ff-calculus (preliminary version). In Typed -calculus and Applications, J-Y Girard editor, LNCS 1581, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the relevantists resource semantics, disturbed by the impact of Girard and modified by two further factors. One of these factors is a spatial view of possible worlds semantics, which can, for example, be seen in the semantics of storage, where sharing rather than duplication is the major concern [11, 12, 13, 14]. The demands of the specific models there require a logic different from extant linear or relevant logics. A similar view, from a type theoretic, or logical frameworks, perspective, can be seen in [15, 16] In this paper, the spatial view of semantics arises in the more abstract setting of ....

....that a linear function cannot use its argument twice. However, if one discards this perspective and thinks declaratively, using the reading of formul advanced in this section, then the truth of BI s judgements is straightforward. In BI, the proof of the last judgement, when viewed as an ff term [1, 14, 3], does indeed use its argument twice. Indeed, in [1] we advanced a resource reading of proofs to justify this judgement; the declarative justification is much more immediate. All of the true judgements we have claimed in this section, and non judgements, will hold in the semantics of this paper. ....

P.W. O'Hearn. Resource interpretations, bunched implications and the ff-calculus (preliminary version). In Typed -calculus and Applications, J-Y Girard editor, LNCS 1581, 1999.

....of linear logic s . This implication exists naturally in the Petri net model; taking it as primitive allows the discord between logic and model observed by Engberg and Winskel to be largely avoided. The Petri net semantics also illustrates well the resource interpretation of BI advanced in [24, 23, 25]. The presentation in this paper is arranged by first giving the net semantics, and then exhibiting the properties of it which lead to BI. This essentially parallels, in abbreviated and concrete fashion, the derivation of BI described in [24] the route from the net model is so direct that one ....

....0 or M 6j= A but M j= p 1 Gamma 0. The proposition can be proven using net with two places and no transitions. Notice that in an atomic net, A is equal to I A. Finally, one might wonder if there is a different interpretation of , which one might use to decompose in terms of Gamma . In [24, 23], it has already been shown that there is no functor from a certain model of BI to itself such that E Gamma F = E F . A counter example can be made to show the same result for the net model. Proposition 5. There is a net N and interpretation i such that for some propositional letters p and ....

P.W. O'Hearn. Resource interpretations, bunched implications and the ff- calculus. In preparation. Preliminary version to appear in Typed -calculus and Applications, Lecture Notes in Computer Science, 1999.

....of linear logic s . This implication exists naturally in the Petri net model; taking it as primitive allows the discord between logic and model observed by Engberg and Winskel to be largely avoided. The Petri net semantics also illustrates well the resource interpretation of BI advanced in [27, 26, 28]. The presentation in this paper is arranged by first giving the net semantics, and then exhibiting the properties of it which lead to BI. This essentially parallels, in abbreviated and concrete fashion, the derivation of BI described in [27] the route from the net model is so direct that one ....

....satisfy the lemma, it can t be expressed by any formula without . End of Proof ) Notice that in an atomic net, A is equal to I A. b a Fig. 2. counter example Finally, one might wonder if there is a different interpretation of , which one might use to decompose in terms of Gamma . In [27, 26], it has already been shown that there is no functor from a certain model of BI to itself such that E Gamma F = E F . A counter example can be made to show the same result for the net model. Proposition 5. There is a net N and interpretation i such that for some propositional letters p and ....

P.W. O'Hearn. Resource interpretations, bunched implications and the ff- calculus. In preparation. Preliminary version to appear in Typed -calculus and Applications, Lecture Notes in Computer Science, 1999.

....such as this. Secondly, we hope that this style of presentation will help to expose some of BI s main features and properties in a direct fashion; many of these features can, we feel, be appreciated without becoming embroiled in too many formal details and proofs. Companion papers by each of us [31, 33, 34] explore technical aspects of BI s semantics, proof theory and applications in more detail. 1.2 A Proof theoretic Introduction Recall that implication is inextricably bound up with conjunction, or at least with antecedent forming operations used to formulate sequents. This connection goes so far ....

....operations they use. Because ; admits Weakening and Contraction, rules where additive maintenance is explicit are admissible: Gamma Gamma Gamma Gamma Gamma Gamma The question of whether these rules can replace the more explicit ones above is delicate [31, 33]. Lemma 1 The Cut rule is admissible: Gamma( Delta Gamma( Delta) Cut Notice that this formulation of cut covers both the ; and , cases in the construction of bunches. A presentation of BI as a sequent calculus, in which elimination rules are replaced by left rules, can be ....

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P.W. O'Hearn. Resource interpretations, bunched implications and the ff-calculus. In preparation. Preliminary version to appear in Typed -calculus and Applications, Lecture Notes in Computer Science, 1999.

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