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  1. Lev D. Beklemishev, David Fernández-Duque & Joost J. Joosten (2014). On Provability Logics with Linearly Ordered Modalities. Studia Logica 102 (3):541-566.
    We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...)
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  2. Georges Gonthier, Martin Ziegler, Steve Awodey, George Barmpalias & Lev D. Beklemishev (2012). Barcelona, Catalonia, Spain July 11–16, 2011. Bulletin of Symbolic Logic 18 (3).
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  3. Lev D. Beklemishev (2010). Kripke Semantics for Provability Logic GLP. Annals of Pure and Applied Logic 161 (6):756-774.
    A well-known polymodal provability logic inlMMLBox due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of inlMMLBox is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for inlMMLBox . First, we isolate a certain subsystem inlMMLBox (...)
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  4. T. Franzen & Lev D. Beklemishev (2008). REVIEWS-Inexhaustibility: A Non-Exhaustive Treatment and a Survey on Transfinite Progressions. Bulletin of Symbolic Logic 14 (2).
     
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  5. I. Franzen & Lev D. Beklemishev (2007). REVIEWS-Godel's Theorem: An Incomplete Guide to its Use and Abuse. Bulletin of Symbolic Logic 13 (2):241.
     
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  6. Lev D. Beklemishev & Albert Visser (2005). On the Limit Existence Principles in Elementary Arithmetic and -Consequences of Theories. Annals of Pure and Applied Logic 136 (1-2):56-74.
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  7. Lev D. Beklemishev (2004). Provability Algebras and Proof-Theoretic Ordinals, I. Annals of Pure and Applied Logic 128 (1-3):103-123.
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  8. Lev D. Beklemishev (2003). On the Induction Schema for Decidable Predicates. Journal of Symbolic Logic 68 (1):17-34.
    We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta_1$ . We show that $I\Delta_1$ is independent from the set of all true arithmetical $\Pi_2-sentences$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta_1-induction$ . An open problem formulated by J. (...)
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  9. Lev D. Beklemishev (2003). Proof-Theoretic Analysis by Iterated Reflection. Archive for Mathematical Logic 42 (6):515-552.
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  10. Lev D. Beklemishev, Stephen Cook, Olivier Lessmann, Simon Thomas, Jeremy Avigad, Arnold Beckmann, Tim Carlson, Robert L. Constable & Kosta Došen (2003). 2002 European Summer Meeting of the Association for Symbolic Logic Logic Colloquium'02. Bulletin of Symbolic Logic 9 (1):71.
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  11. Lev Dmitrievich Beklemishev (1999). Provability, Complexity, Grammars. American Mathematical Society.
    (2) Vol., Classification of Propositional Provability Logics LD Beklemishev Introduction Overview. The idea of an axiomatic approach to the study of ...
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  12. Lev D. Beklemishev (1998). A Proof-Theoretic Analysis of Collection. Archive for Mathematical Logic 37 (5-6):275-296.
    By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions left open by Sieg (...)
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  13. Lev D. Beklemishev (1997). Induction Rules, Reflection Principles, and Provably Recursive Functions. Annals of Pure and Applied Logic 85 (3):193-242.
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  14. Lev D. Beklemishev (1996). Bimodal Logics for Extensions of Arithmetical Theories. Journal of Symbolic Logic 61 (1):91-124.
    We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ 0 + EXP, PRA); (PRA, IΣ 1 ); (IΣ m , IΣ n ) for $1 \leq m etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.
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  15. Lev D. Beklemishev (1994). On Bimodal Logics of Provability. Annals of Pure and Applied Logic 68 (2):115-159.
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  16. Sergei N. Artemov & Lev D. Beklemishev (1993). On Propositional Quantifiers in Provability Logic. Notre Dame Journal of Formal Logic 34 (3):401-419.
  17. Lev D. Beklemishev (1993). [Omnibus Review]. Journal of Symbolic Logic 58 (2):715-717.
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  18. Lev D. Beklemishev (1993). On the Complexity of Arithmetical Interpretations of Modal Formulae. Archive for Mathematical Logic 32 (3):229-238.
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