Simon Kramer
Abstract
We produce a decidable super-intuitionistic normal modal logic of internalised intuitionistic (and thus disjunctive and monotonic) interactive proofs (LIiP) from an existing classical counterpart of classical monotonic non-disjunctive interactive proofs (LiP). Intuitionistic interactive proofs effect a durable epistemic impact in the possibly adversarial communication medium CM (which is imagined as a distinguished agent) and only in that, that consists in the permanent induction of the perfect and thus disjunctive knowledge of their proof goal by means of CM's knowledge of the proof: If CM knew my proof then CM would persistently and also disjunctively know that my proof goal is true. So intuitionistic interactive proofs effect a lasting transfer of disjunctive propositional knowledge (disjunctively knowable facts) in the communication medium of multi-agent distributed systems via the transmission of certain individual knowledge (knowable intuitionistic proofs). Our (necessarily) CM-centred notion of proof is also a disjunctive explicit refinement of KD45-belief, and yields also such a refinement of standard S5-knowledge. Monotonicity but not communality is a commonality of LiP, LIiP, and their internalised notions of proof. As a side-effect, we offer a short internalised proof of the Disjunction Property of Intuitionistic Logic (originally proved by Gödel).
- R. Anderson. 2008. Security Engineering: A Guide to Building Dependable Distributed Systems (2nd ed.). Wiley, Hoboken, NJ. Google Scholar
Digital Library
- C. Areces and B. Ten Cate. 2007. Hybrid Logics. In Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3, P. Blackburn, J. van Benthem, and F. Wolters (Eds.). Elsevier, 821--868.Google Scholar
- S. Artemov. 2008. The logic of justifications. The Review of Symbolic Logic 1, 4.Google ScholarCross Ref
- S. Artemov and R. Iemhoff. 2007. The basic intuitionistic logic of proofs. The Journal of Symbolic Logic 72, 2.Google ScholarCross Ref
- A. Baskar, R. Ramanujam, and S. Suresh. 2010. A DEXPTIME-complete Dolev-Yao theory with distributive encryption. In Proceedings of MFCS. LNCS, vol. 6281. Springer. Google ScholarDigital Library
- P. Blackburn and J. van Benthem. 2007. Modal Logic: A Semantic Perspective. In Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3, P. Blackburn, J. van Benthem, and F. Wolters (Eds.). Elsevier, 1--84.Google Scholar
- P. Blackburn, J. van Benthem, and F. Wolter, Eds. 2007. Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3. Elsevier. Google ScholarDigital Library
- T. Braüner and S. Ghilardi. 2007. Chapter First-Order Modal Logic. In Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3, P. Blackburn, J. van Benthem, and F. Wolters (Eds.). Elsevier, 549--620.Google Scholar
- V. de Paiva and E. Ritter. 2011. Basic Constructive Modality. In Logic without Frontiers: Festschrift for Walter Alexandre Carnielli on the occasion of his 60th birthday, Jean-Yves Beziau and Marcelo Esteban Coniglio (Eds.). Tribute Series, vol. 17. College Publications, London.Google Scholar
- K. Došen. 1984. Intuitionistic double negation as a necessity operator. Publications de l'Institut Mathématique (Beograd) 35, 49.Google Scholar
- R. Fagin, J. Halpern, Y. Moses, and M. Vardi. 1995. Reasoning about Knowledge. MIT Press, Cambridge, MA. Google ScholarDigital Library
- S. Feferman. 1964 (1989). The Number Systems: Foundations of Algebra and Analysis (2nd ed.). AMS Chelsea Publishing. Reprinted by the American Mathematical Society, 2003.Google Scholar
- N. Ferguson, B. Schneier, and T. Kohno. 2010. Cryptography Engineering: Design Principles and Practical Applications. Wiley, Hoboken, NJ. Google ScholarDigital Library
- G. Fischer Servi. 1984. Axiomatisations for some intuitionistic modal logics. Rendiconti del seminario matematico del Politecnico di Torino 42, 3.Google Scholar
- M. Fitting. 2007. Modal Proof Theory. In Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3, P. Blackburn, J. van Benthem, and F. Wolters (Eds.). Elsevier, 85--138.Google Scholar
- D. Gabbay, Ed. 1995. What Is a Logical System? Number 4 in Studies in Logic and Computation. Oxford University Press, New York, NY. Google ScholarDigital Library
- D. Gollmann. 2011. Computer Security (3rd ed). Wiley, Hoboken, NJ. Google ScholarDigital Library
- V. Goranko and M. Otto. 2007. Model Theory of Modal Logic. In Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3, P. Blackburn, J. van Benthem, and F. Wolters (Eds.). Elsevier, 249--329.Google Scholar
- V. Hendricks and O. Roy, Eds. 2010. Epistemic Logic: 5 Questions. Automatic Press, New York.Google Scholar
- J. Hindley and J. Seldin. 2008. Lambda-Calculus and Combinators (2nd ed.). Cambridge University Press, New York, NY. Google ScholarDigital Library
- P. Hrubeš. 2007. A lower bound for intuitionistic logic. Annals of Pure and Applied Logic 146.Google Scholar
- E. Jeřábek. 2008. Independent bases of admissible rules. Logic Journal of the Interest Group in Pure and Applied Logic 16, 3.Google Scholar
- S. Kramer. 2012a. A logic of interactive proofs (formal theory of knowledge transfer). Technical Report 1201.3667, arXiv. http://arxiv.org/abs/1201.3667.Google Scholar
- S. Kramer. 2012b. Logic of negation-complete interactive proofs (formal theory of epistemic deciders). Technical Report 1208.5913, arXiv. Retrieved August 30, 2015 from http://arxiv.org/abs/1208.5913.Google Scholar
- S. Kramer. 2012c. Logic of non-monotonic interactive proofs (formal theory of temporary knowledge transfer). Technical Report 1208.1842, arXiv. Retrieved August 30, 2015 from http://arxiv.org/abs/1208.1842.Google Scholar
- S. Kramer. 2013a. Logic of intuitionistic interactive proofs (formal theory of disjunctive knowledge transfer). Short paper presented at the Congress on Logic and Philosophy of Science, Ghent.Google Scholar
- S. Kramer. 2013b. Logic of intuitionistic interactive proofs (formal theory of perfect knowledge transfer). Technical Report 1309.1328, arXiv. Retrieved August 30, 2015 from http://arxiv.org/abs/1309.1328.Google ScholarDigital Library
- S. Kramer. 2013c. Logic of non-monotonic interactive proofs. In Proceedings of ICLA. Lecture Notes in Computer Science, vol. 7750. Springer.Google Scholar
- S. Kramer. 2014. Logic of negation-complete interactive proofs (formal theory of epistemic deciders). In Proceedings of IMLA. Electronic Notes in Theoretical Computer Science, vol. 300. Elsevier. Google ScholarDigital Library
- S. Kripke. 1965. Semantical Analysis of Intuitionistic Logic I. In Formal Systems and Recursive Functions. Studies in Logic and the Foundations of Mathematics, vol. 40. Elsevier.Google Scholar
- J.-J. Meyer and F. Veltman. 2007. Handbook of Modal Logic, Chapter Intelligent Agents and Common Sense Reasoning. In Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3, P. Blackburn, J. van Benthem, and F. Wolters (Eds.). Elsevier, 991--1029.Google Scholar
- J. Moschovakis. 2010. Intuitionistic logic. In The Stanford Encyclopedia of Philosophy, Summer 2010 ed., E. N. Zalta (Ed.).Google Scholar
- Y. Moschovakis. 2006. Notes on Set Theory (2nd ed.). Springer.Google Scholar
- G. Plotkin and C. Stirling. 1986. A framework for intuitionistic modal logics. In Proceedings of the Conference on Theoretical Aspects of Rationality and Knowledge. Morgan Kaufmann Publishers Inc., Burlington, MA. Google ScholarDigital Library
- K. Pouliasis and G. Primiero. 2014. J-Calc: A typed lambda calculus for Intuitionistic Justification Logic. In Proceedings of IMLA. Electronic Notes in Theoretical Computer Science, vol. 300. Elsevier. Google ScholarDigital Library
- K. Ranalter. 2010. Embedding constructive K into intuitionistic K. Electronic Notes in Theoretical Computer Science 262. Google ScholarDigital Library
- A. Simpson. 1994. The proof-theory and semantics of intuitionistic modal logic. Ph.D. thesis, University of Edinburgh, Edinburgh, UK.Google Scholar
- R. Statman. 1979. Intuitionistic propositional logic is polynomial-space complete. Theoretical Computer Science 9.Google Scholar
- G. Steren and E. Bonelli. 2014. Intuitionistic hypothetical logic of proofs. In Proceedings of IMLA. Electronic Notes in Theoretical Computer Science, vol. 300. Elsevier. Google ScholarDigital Library
- P. Taylor. 1999. Practical Foundations of Mathematics. Cambridge University Press, New York, NY.Google Scholar
- A. Tiu, R. Goré, and J. Dawson. 2010. A proof theoretic analysis of intruder theories. Logical Methods in Computer Science 6, 3.Google ScholarCross Ref
- J. van Benthem. 1997. Modal Logic as a Theory of Information. In Logic and Reality: Essays on the Legacy of Arthur Prior, B. J. Copeland (Ed.). Clarendon Press, Oxford, UK.Google Scholar
- J. van Benthem. 2009. The information in intuitionistic logic. Synthese 167.Google Scholar
- D. Wijesekera. 1990. Constructive modal logic I. Annals of Pure and Applied Logic 50.Google ScholarCross Ref
Index Terms
Logic of Intuitionistic Interactive Proofs (Formal Theory of Perfect Knowledge Transfer)
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