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Mechanizing proofs with logical relations – Kripke-style

ANDREW CAVE, BRIGITTE PIENTKA
2018 Mathematical Structures in Computer Science  
Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize.  ...  how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms.  ...  using Kripke-style logical relations.  ... 
doi:10.1017/s0960129518000154 fatcat:uhyfqoacjfdido2zdg3pijayha

Kripke-style models for typed lambda calculus

John C. Mitchell, Eugenio Moggi
1991 Annals of Pure and Applied Logic  
Moggi, Kripke-style models for typed lambda calculus, Annals of Pure and Applied Logic 51 (1991) 99-124.  ...  To those familiar with Kripke models of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately 'semantic'.  ...  Acknowledgments We are grateful to Edmund Robinson and Pino Rosolini for very helpful suggestions regarding the relationship between Cartesian closed categories and Kripke lambda models.  ... 
doi:10.1016/0168-0072(91)90067-v fatcat:dolao3mfczbaho62ibnfl36opq

Forcing-Based Cut-Elimination for Gentzen-Style Intuitionistic Sequent Calculus [chapter]

Hugo Herbelin, Gyesik Lee
2009 Lecture Notes in Computer Science  
We give a simple intuitionistic completeness proof of Kripke semantics with constant domain for intuitionistic logic with implication and universal quantification.  ...  The proof, which has been formalised in the Coq proof assistant, easily extends to the case of the absurdity connective using Kripke models with exploding nodes.  ...  Coquand's idea to the intuitionistic first-order predicate logic with implication and universal quantification as logical symbols. The predicate system used here is a Gentzen-style sequent calculus.  ... 
doi:10.1007/978-3-642-02261-6_17 fatcat:xzdtlwbstbgntbdgm7v4fgnxbq

Ancestral Kripke models and nonhereditary Kripke models for the Heyting propositional calculus

Kosta Došen
1991 Notre Dame Journal of Formal Logic  
This is a sequel to two previous papers, where it was shown that, for the Heyting propositional calculus H, we can give Kripke-style models whose accessibility relation R need not be a quasi-ordering relation  ...  The resulting Kripke-style models are called "nonhereditary Kripke models". These models are inspired by some particular embeddings of H into 54 and a somewhat weaker normal modal logic.  ...  Acknowledgment I would like to thank the Alexander von Humboldt Foundation for supporting my work on this paper with a research scholarship.  ... 
doi:10.1305/ndjfl/1093635930 fatcat:7kzofnvl5vhkbk23uorogtnrpi

Page 3151 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
ordered by the submodel relation.  ...  The obvious question is then: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA?”  ... 

Proof Search and Counter-Model Construction for Bi-intuitionistic Propositional Logic with Labelled Sequents [chapter]

Luís Pinto, Tarmo Uustalu
2009 Lecture Notes in Computer Science  
Bi-intuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion.  ...  Negri's general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Kripke semantics, it is does not directly support proof search.  ...  In Sect. 2, we introduce BiInt with its Kripke semantics and the translation to KtT4. We also show its Dragalin-style sequent calculus and why cut elimination fails.  ... 
doi:10.1007/978-3-642-02716-1_22 fatcat:2trnrno37nb2pdjsrqfa6gj3sy

Page 2384 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews  
The embedding also suggests a completeness proof for H with respect to Kripke-style models in which the “accessibility” relation is not a quasi-ordering, but satisfies weaker conditions.  ...  Finally, the complete- ness of nS4* with respect to suitable Kripke semantics is shown by means of Henkin’s proof. Wojciech Dzik (PL-SILS) 92e:03025 03B45 03B65 Slater, B.  ... 

Continuation-passing style models complete for intuitionistic logic

Danko Ilik
2013 Annals of Pure and Applied Logic  
with sum types.  ...  The connection between ours and Kripke models is made through a strengthening of the Double-negation Shift schema.  ...  predicate logic with disjunction (and, by analogy, with ∃) with respect to Kripke models.  ... 
doi:10.1016/j.apal.2012.05.003 fatcat:purfbc4ny5b4pebtmvvk7t2l2q

Belief Semantics of Authorization Logic [article]

Andrew K. Hirsch, Michael R. Clarkson
2013 arXiv   pre-print
A proof system is given for the logic; that system is proved sound with respect to the belief and Kripke semantics.  ...  The soundness proof for the belief semantics, and for a variant of the Kripke semantics, is mechanized in Coq.  ...  Schneider consulted on the design of the proof system and the Kripke semantics.  ... 
arXiv:1302.2123v3 fatcat:h2rbqglozjecdmmqght2h3yjpm

Categorical and Kripke Semantics for Constructive S4 Modal Logic [chapter]

Natasha Alechina, Michael Mendler, Valeria de Paiva, Eike Ritter
2001 Lecture Notes in Computer Science  
Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.  ...  We consider two systems of constructive modal logic which are computationally motivated.  ...  Introduction This paper is about relating traditional Kripke-style semantics for constructive modal logics to their corresponding categorical semantics.  ... 
doi:10.1007/3-540-44802-0_21 fatcat:r5r4zdyv7ffsbj55tnrdf5wuxy

Page 5464 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
Further, the authors present the proofs of soundness and completeness for GL and its Kripke semantics. In conclusion, relational deduction for GL is also developed.  ...  Math. 74 (1952), 127-162; MR 13, 524g] are use- ful for developing representational mechanisms of some abstract algebraic structures related to uncertainty.”  ... 

A Henkin-style completeness proof for the modal logic S5 [article]

Bruno Bentzen
2019 arXiv   pre-print
This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover.  ...  Here the proof is based on a Hilbert-style presentation better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as rule of inference.  ...  Henkin-style completeness proofs for modal logics have been around for over five decades [12] but the formal verification of completeness with respect to Kripke semantics is comparatively recent.  ... 
arXiv:1910.01697v1 fatcat:h7ogr2oxczfohfifdf4c7rz7qy

The Right Tools for the Job: Correctness of Cone of Influence Reduction Proved Using ACL2 and HOL4

Michael J. C. Gordon, Matt Kaufmann, Sandip Ray
2010 Journal of automated reasoning  
Our proof exploits a link between the HOL4 and ACL2 proof systems to show correctness of a cone of influence reduction algorithm, implemented in ACL2, with respect to the classical semantics of linear  ...  temporal logic, formalized in HOL4.  ...  This work has been funded in part by the National Science Foundation under Grants IIS-0417413, CNS-0429591 (with DARPA), CCF-0945316, CCF-0916772, and CNS-0910913, and by the Semiconductor Research Corporation  ... 
doi:10.1007/s10817-010-9169-y fatcat:rblnzquszzatfogsbdrexvtbyy

Characterizing Frame Definability in Team Semantics via the Universal Modality [chapter]

Katsuhiko Sano, Jonni Virtema
2015 Lecture Notes in Computer Science  
Thus we obtain Goldblatt-Thomason-style theorems for each of the logics listed above.  ...  We show that, with respect to frame definability, the logics ML( u + ), modal logic with intuitionistic disjunction, and (extended) modal dependence logic all coincide.  ...  Below we assume only that the logics L and L are such that the global satisfaction relation for Kripke models (i.e., M ϕ) is defined.  ... 
doi:10.1007/978-3-662-47709-0_11 fatcat:zjpe35s3lnf5hocluyb345s5xa

Rudimentary Beth Models and Conditionally Rudimentary Kripke Models for the Heyting Propositional Calculus

KOSTA DOÈ EN
1991 Journal of Logic and Computation  
This paper continues the investigation of rudimentary Kripke models, i.e. non-quasiordered Kripke-style models for the Heyting propositional calculus H.  ...  It is shown that, if we don't want to change the usual semantic clauses for the connectives, conditionally rudimentary Kripke models make the largest class of Kripkestyle models with respect to which we  ...  Acknowledgments I would like to thank the Alexander von Humboldt Foundation for supporting my work on this paper with a research scholarship.  ... 
doi:10.1093/logcom/1.5.613 fatcat:oyxo5hkt6bf65a53q2nrkha4da
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