275 research outputs found

    Well-orders in the transfinite Japaridze algebra

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    This paper studies the transfinite propositional provability logics \glp_\Lambda and their corresponding algebras. These logics have for each ordinal ξ<Λ\xi< \Lambda a modality \la \alpha \ra. We will focus on the closed fragment of \glp_\Lambda (i.e., where no propositional variables occur) and \emph{worms} therein. Worms are iterated consistency expressions of the form \la \xi_n\ra \ldots \la \xi_1 \ra \top. Beklemishev has defined well-orderings <ξ<_\xi on worms whose modalities are all at least ξ\xi and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original <ξ<_\xi orderings and provide a calculus for the corresponding generalized order-types oξo_\xi. Our calculus is based on so-called {\em hyperations} which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form \la {\formerOmega}_\xi (A) \ra_{\xi \in \ord} for some worm AA. One of these characterizations is in terms of a second kind of transfinite iteration called {\em cohyperation.}Comment: Corrected a minor but confusing omission in the relation between Veblen progressions and hyperation

    On Elementary Theories of Ordinal Notation Systems based on Reflection Principles

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    We consider the constructive ordinal notation system for the ordinal ϵ0{\epsilon_0} that were introduced by L.D. Beklemishev. There are fragments of this system that are ordinal notation systems for the smaller ordinals ωn{\omega_n} (towers of ω{\omega}-exponentiations of the height nn). This systems are based on Japaridze's provability logic GLP\mathbf{GLP}. They are closely related with the technique of ordinal analysis of PA\mathbf{PA} and fragments of PA\mathbf{PA} based on iterated reflection principles. We consider this notation system and it's fragments as structures with the signatures selected in a natural way. We prove that the full notation system and it's fragments, for ordinals ≥ω4{\ge\omega_4}, have undecidable elementary theories. We also prove that the fragments of the full system, for ordinals ≤ω3{\le\omega_3}, have decidable elementary theories. We obtain some results about decidability of elementary theory, for the ordinal notation systems with weaker signatures.Comment: 23 page

    On the complexity of the closed fragment of Japaridze's provability logic

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    We consider well-known provability logic GLP. We prove that the GLP-provability problem for variable-free polymodal formulas is PSPACE-complete. For a number n, let L^n_0 denote the class of all polymodal variable-free formulas without modalities , ,... . We show that, for every number n, the GLP-provability problem for formulas from L^n_0 is in PTIME.Comment: 12 pages, the results of this work and a proof sketch are in Advances in Modal Logic 2012 extended abstract on the same nam
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