Tijn Borghuis. Coming to Term with Modal Logic: On the Interpretation of Modalities in Typed -calculus. PhD thesis, Eindhoven University of Technology, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of intensional notions in functional programming. 6 Appendix: Variations on the same Theme We now discuss the two other solutions for the problem of Natural Deduction for system K. These other systems also pose some problems of their own. 8 6. 1 Fitch style Modal Natural Deduction Borghuis ([B98]) work is based on Pure Type Systems, Barendregt s elegant systematization of higher order lambda calculi. This framework is very expressive, but since we are not interested (at least in this note) on higher order systems, some of the complications of Borghuis system can be avoided. The main idea ....

T. J. Borghuis. Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus. PhD thesis, Technical University of Eindhoven, 1994.

....Since modal logic S4 and all standard term constructors can be represented by proof polynomials, the Logic of Proofs can also emulate modal calculi. As it was shown in [8] 11] the intuitionistic version of LP naturally realizes the modal calculus for IS4 ( 23] 68] 84] cf. also [27]) and thus supplies modal terms with standard provability semantics. This EXPLICIT PROVABILITY 31 result may be considered as a more general abstract version of the CurryHoward isomorphism which relates terms types with proofs formulas. x11. First order case. Theories based on the first order ....

V.A.J. Borghuis, Coming to terms with modal logic: On the interpretation of modalities in typed -calculus, Ph.D. thesis, Technische Universiteit Eindhoven, 1994.

....ILPG y : Gamma ) p( y) B. Since both modal logic S4 and all standard term constructors can be emulated by proof polynomials, the Logic of Proofs can also emulate modal calculi. As it was shown in [6] 7] ILPG naturally realizes the modal calculus for IS4 ( 10] 45] 60] cf. also [15]) and thus supplies modal terms with standard provability semantics. This result may be considered as a more general abstract version of the well known Curry Howard isomorphism which relates terms types with proofs formulas. 10 Discussion Roughly speaking, LP is an advanced system of combinatory ....

V. A. J. Borghuis, Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus, Ph.D. Thesis, Technische Universiteit Eindhoven, 1994

....holds also for ILP instead of LP and IS42 instead of S4. In other words, the intuitionistic logic of proofs is an explicit version of IS42 in the same sense that LP is an explicit version of S4. We will show how ILPG naturally emulates the modal calculus for IS42 ( 8] 30] 42] cf. also [11]) and thus supplies modal terms with standard provability semantics. 9.9 Theorem. Realization of modal calculus) There is an effective step by step realization r of any derivation x : Gamma ) t( x) A in the term calculus for IS42 as a derivation of x : Gamma r ) t r ( x) A r ....

V. A. J. Borghuis, Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus, Ph.D. Thesis, Technische Universiteit Eindhoven, 1994

....realized as admissible rules in LPGi (cf. 2] 3] Since both modal logic IS4 and all standard term constructors can be emulated by proof polynomials, LPi can also emulate modal calculi. As it was shown in [2] 3] LPGi naturally realizes the modal calculus for IS4 ( 4] 6] 7] cf. also [5]) 6 Deep realization of modalities by combinatory ( terms Realization algorithm from Section 4 recovers combinatory terms for every occurrence of modalities in any IS4 derivation. Natural fragments of S4 may be be now regarded as implicit description of the corresponding subsystems of LPi: ....

V. A. J. Borghuis, Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus, Ph.D. Thesis, Technische Universiteit Eindhoven, 1994

....for Int) can be realized in a small fragment of ILPN consisting of pure derivations only. We already have enough ingredients to demonstrate that the Logic of Proofs can emulate modal calculi. We will show how ILPG naturally emulates the modal calculus for IS4 ( 7] 16] 21] cf. also [10]) and thus supplies modal terms with standard provability semantics. 6.9 Theorem. Realization of modal calculus) There is an effective step by step realization r of any derivation x : Gamma ) t( x) A in the term calculus for IS4 as a derivation of x : Gamma r ) t r ( x) A r in ....

V. A. J. Borghuis, Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus, Ph.D. Thesis, Technische Universiteit Eindhoven, 1994

.... from Mini ML 2 to Mini ML 2 e is inspired by one direction of the proof of equivalence between the two calculi given in [PW95] Systems similar to the implicit modal calculus of [PW95] have been proposed by Martini and Masini [MM94] who introduce a simple reduction semantics, and Bourghuis [Bor94], who considers modal pure type systems. None of the prior work on modal calculi has considered the relationship to computation staging. Partly motivated by a previous version of the current paper [DP96] Goubault Larrecq [GL96a, GL96b, GL96c, GL97] has proposed a formulation of modal calculi ....

Tijn Borghuis. Coming to Term with Modal Logic: On the Interpretation of Modalities in Typed -calculus. PhD thesis, Eindhoven University of Technology, 1994.

....say that the notion of group knowledge is distributed if the group knowledge (apprehended as a set of formulae) equals the set of formulae that can be derived from the union of the knowledge of the agents that together constitute the group. The following quote is taken from a recent dissertation ([1]) Implicit knowledge is of interest in connection with information dialogues: if we think of the dialog participants as agents with information states represented by epistemic formulae, then implicit knowledge precisely defines the propositions the participants could conclude to during an ....

....in connection with information dialogues: if we think of the dialog participants as agents with information states represented by epistemic formulae, then implicit knowledge precisely defines the propositions the participants could conclude to during an information dialogue . Borghuis ([1]) means with implicit knowledge what we call group knowledge . We will see in this section, that using standard epistemic logic, one cannot guarantee that group knowledge is precisely that what can be concluded during an information dialogue. To do so, we will first formalize the notion of ....

T. Borghuis. Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus. PhD thesis, Technical University Eindhoven, 1994.

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Tijn Borghuis. Coming to Term with Modal Logic: On the Interpretation of Modalities in Typed -calculus. PhD thesis, Eindhoven University of Technology, 1994.

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Borghuis, T. 1994. Coming to Terms with Modal Logic: On the Interpretation of Modalities in Typed -calculus. Ph. D. thesis, Eindhoven University of Technology.

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V. A. J. Borghuis, Coming to terms with modal logic: On the interpretation of modalities in typed #-calculus, Ph.D. thesis, Technische Universiteit Eindhoven, 1994.

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Tijn Borghuis. Coming to Term with Modal Logic: On the Interpretation of Modalities in Typed -calculus. PhD thesis, Eindhoven University of Technology, 1994.

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T. J. Borghuis. Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus. PhD thesis, Technical University of Eindhoven, 1994.

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Borghuis, T.: 1994, Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus, PhD thesis, Technische Universiteit Eindhoven.

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