A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for a fresh name quantifier (cf. Nominal Logic [13] A formula in our logic describes a property of a particular part of a concurrent system (a world) at a particular time; therefore it is modal in space as well as in time. In our sequents, formulas are indexed by the worlds they predicate over [17], so a sequent can talk about many distinct worlds at once. Each sequent incorporates also a finite set of constraints over the worlds, including process reduction and congruence constraints. In general, the constraint structure can be fashioned as an algebra [18] which in our case is a ....

....v satisfies formula A (i.e, that A holds at world v, written v : A) and that t satisfies B, and if we can show from the constraints in S that u = vjt, then we can conclude that u satisfies AjB. Hence, the reading of this logical rules incorporates much of the intended satisfaction semantics [17]. The (jL) rule features the assumption X and Y not free in the conclusion (of the rule) This assumption means, in particular, that X and Y are completely generic and unconstrained variables. A reading is: to show that u : AjB entails , we must show that for an arbitrary decomposition of u as ....

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A. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. Ph.D. thesis, Dept. of Computer Science, Edingburgh University, 1994.

....for a fresh name quantifier (cf. Nominal Logic [18] A formula in our logic describes a property of a particular part of a concurrent system (a world) at a particular time; therefore it is modal in space as well as in time. In our sequents, formulas are indexed by the worlds they predicate over [22], so a sequent can talk about many distinct worlds at once. Each sequent incorporates also a finite set of constraints over the worlds, including process reduction and congruence constraints. In general, the constraint structure can be fashioned as an algebra [24] which in our case is a ....

....(i.e, that A holds at world v, written v : A) and that t satisfies B, and if we can show from the constraints in S that u is structurally congruent to vjt, then we can conclude that u satisfies AjB. Hence, the reading of this logical rules incorporates much of the intended satisfaction semantics [22]. The (jL) rule features the assumption X and Y not free in the conclusion (of the rule) This assumption means, in particular, that X and Y are completely generic and unconstrained variables. A reading is: to show that u : AjB entails , we must show that for an arbitrary decomposition of u as ....

[Article contains additional citation context not shown here]

A. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. Ph.D. thesis, Dept. of Computer Science, Edinburgh University, 1994.

....proof calculus for dynamic logic (we shall not be concerned with deduction in quanti ed dynamic logic here) The necessitation rule is subject to the usual constraints on the variables x. The rst ve axioms are the standard axioms of the K fragment of intuitionistic modal logic as given e.g. in [12, 19], with a slight variation of the third axiom the usual form :hpi is a special case of the second of the two dual forms given. All intuitionistic propositional tautologies are implicitly included here. Moreover, there are four axioms concerning composition of program sequences. The last three ....

A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

....ponens) while the remainder of the calculus is given in the form of axioms (more precisely, axiom schemes) The necessitation rule is subject to the usual constraints on the variables x. The rst ve axioms are the standard axioms of the K fragment of intuitionistic modal logic as given e.g. in [16, 24], with a slight variation of the third axiom the usual form :hpi is a special case of the second of the two dual forms given. All intuitionistic propositional tautologies are implicitly included here. Moreover, there are four axioms concerning sequential composition of program sequences. The ....

A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

....and the other modal R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the # modality. The mainstream approach as exemplified by Ewald [Ewa86] Fischer Servi [FS80] Plotkin and Stirling [PS86] Simpson [Sim94] is based on the analogy of # with and of # with # quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to one arrives at the semantic interpretation w #v. ##u. v R u A for necessity, and #u. w R u u A (1) for ....

A.K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic.PhD thesis, University of Edinburgh, 1994.

....a more complete denotation would be IS4 KV#,I . Much work exist on different models for intuitionistic modal logic. The classical approach is based on Kripke s possible worlds models, where both modal logic and intuitionistic logic have a natural translation. Such semantics are described in [13]. Other approach include categorical ones [3, 4] computational ones [3, 11, 9] and others [8] 3.3 Relating IS4 KV and description models We now show that there is a close relation between the logic IS4 KV#,I and the description models of L(#, I) For this, we define two notions of truth ....

....exactly those provable in IS4 KV (that is those which verify IS4 KV #) The proof of this theorem is given more precisely in Appendix B. This theorem provides a simple and general class of model for the modal logic IS4 KV. While many classes of model exist, either based on Kripke structures [13], on categories [3, 4] or on adaptations of # calculus [9, 11] the present model originates from approximation techniques and its application to information flow formalisms [6] offering new possibilities in the logical study of complex systems and knowledge representation. 4 Conclusion In this ....

A. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logics. PhD thesis, University of Edinburgh, 1993.

....and Technology Agency (STA) Japan. one fixed variable) of intuitionistic predicate logic can be interpreted in MIPC (see [1] 11] We also note that MIPC is an extension of FS (Fisher Servi s logic) introduced in [8] For recent surveys on intuitionistic modal logics we refer the reader to [13], 15] The finite model property of a logic L ensures the decidability of L when L is finitely axiomatizable, and hence has been one of the central focuses in the study of logics. The finite model property of MIPC was shown in [7] 10] and that of FS in [5] Several general results on the ....

A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD Thesis, University of Ediburgh, 1994.

....S4 (IS4) is a modal logic with a modality like that of S4, but built on intuitionistic logic rather than classical logic. The two sided single succedent calculus with a single modality that we deal with here can be found in the appendix. More details on IS4 can be found in [BdP96] and [Sim94]. As for S4, we are faced with an immediate problem the context is not increasing (owing to the (2R ) rule) For S4 this wasn t problematic as we only needed to check for looping owing to the modalities the propositional classical logic needs no history. The modal context was increasing: ....

A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

....recent years, several researches have addressed the problem of computer assisted proof search in the context of modal logics. For instance, Fitting, Simpson, Wallen have built new systems for modal logics which could be used as the basis of proof development environments, or even, theorem provers [10, 11, 29, 32]. Basin et.al. Coen, Merz have developed packages for using Modal Logics and Temporal Logics within existing Logical Frameworks (namely, Isabelle) 5, 6, 22] These approaches either utilize special formats or they are based on representations of Kripke semantics, or they do not address ....

....[5, 6, 22] These approaches either utilize special formats or they are based on representations of Kripke semantics, or they do not address explicitly ND style presentations. Truly sequentlike formats or tableaux formats are used in [6, 11, 32] while accessibility relations are exploited in [5, 29]. For instance, a thorough treatment of modal logics based on semantics is the one carried out by Basin and his co authors in [5] In this paper, Kripke semantics is built in the calculus from the outset with great ingenuity: worlds are reified, and a first order proposition R over worlds is ....

A. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1993.

....and the other modal R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the # modality. The mainstream approach as exemplified by Ewald [Ewa86] Fischer Servi [FS80] Plotkin and Stirling [PS86] Simpson [Sim94] is based on the analogy of # with 8 and of # with 9 quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to , one arrives at the semantic interpretation w j= #A iff 8v: w v ) 8u: v R u ) u j= A for necessity, and w j= #A iff 9u: w R u u j= A (1) ....

A.K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

....system suitable for proof theoretical results is to reflect in the system a semantic notion, like the transition relation of the underlying model. This approach has been successfully adopted by Simpson in the construction of strong normalizing Natural Deduction style proof systems for modal logics [26]. Another future work stemming from this research is the development of a user friendly, mouse oriented environment which adopts our Coq formalization as proof kernel. In such an environment, the user could carry out interactively formal verifications based on the modal calculus. APPENDIX: Coq ....

A. Simpson. The proof theory and semantics of intuitionistic modal logic. PhD thesis, LFCS, Univ. of Edinburgh, 1994.

.... and the other modal R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the 3 modality. The main stream approach as exemplified by Ewald [Ewa86] Fischer Servi [FS80] Plotkin and Stirling [PS86] Simpson [Sim94] is based on the analogy of 2 with 8 and of 3 with 9 quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to , one arrives at the semantic interpretation w j= 2A iff 8v: w v ) 8u: w R u ) u j= A for necessity, and w j= 3A iff 9u: w R u u j= A ....

A.K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

....worlds , an idea originally due to Kripke. Kripke also provided a possible worlds semantics for intuitionistic logic, and so it is natural to consider intuitionistic logic extended with intensional modalities. Indeed much work already exists on this topic (a good account is given by Simpson [37]) In fact there is an almost bewildering number of intuitionistic modal logics most of which consist of extensions of intuitionistic logic with some selection of modal rules and axioms. Much of this work is justified by reference to a Kripke model the number of choices arises from the large ....

.... of providing natural deduction formulations of (intuitionistic) modal logics (some in response to our earlier work [6] These include Benevides and Maibaum [2] Bull and Segerberg [9, pages 29 30] Davies and Pfenning [13] Martini and Masini [26] Mints [27, Pages 221 294] and Simpson [37]. However they all use extensions of one form or another to the nature of natural deduction (for example, by indexing formulae with possible worlds information) Again we reiterate the conceptual simplicity of our proposal we use no new features of natural deduction. The techniques used in this ....

A. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logics. PhD thesis, University of Edinburgh, December 1993.

....provide solutions to the above problems. Let us first briefly summarize the approach and some of the results that we previously developed. In [3] we formalized natural deduction proof systems for propositional modal logics, based on the view of a logic as a Labelled Deductive System [9] see [7,24] for similar approaches. We decomposed a modal logic into two interacting parts: a base logic, fixed for all modal logics, and a relational theory, different for each modal logic. In the base logic, we reason about formulae paired with labels; i.e. instead of A, we prove w:A, where w:A is a ....

A. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, Edinburgh, 1993.

....system suitable for proof theoretical results is to reflect in the system a semantic notion, like the transition relation of the underlying model. This approach has been successfully adopted by Simpson in the construction of strong normalizing Natural Deduction style proof systems for modal logics [26]. Another future work stemming from this research is the development of a user friendly, mouse oriented environment which adopts our Coq formalization as proof kernel. In such an environment, the user could carry out interactively formal verifications based on the modal calculus. APPENDIX: Coq ....

A. Simpson. The proof theory and semantics of intuitionistic modal logic. PhD thesis, LFCS, Univ. of Edinburgh, 1994.

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A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

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Simpson, A. K. 1994. The proof theory and semantics of intuitionistic modal logic. Ph.D. thesis, University of Edinburgh.

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A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, Department of Philosophy, University of Edinburgh, 1994. A Proofs of the properties of ### Proposition A.1.

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A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, Department of Philosophy, University of Edinburgh, 1994.

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A.K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

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A.K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

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A. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

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A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

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A.K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

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A.K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.

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