Well-orders in the transfinite Japaridze algebra

This paper studies the transfinite propositional provability logics \glp_\Lambda and their corresponding algebras. These logics have for each ordinal $\xi< \Lambda$ 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 $<_\xi$ on worms whose modalities are all at least $\xi$ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original $<_\xi$ orderings and provide a calculus for the corresponding generalized order-types $o_\xi$. 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 $A$. 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

We consider the constructive ordinal notation system for the ordinal ${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of this system that are ordinal notation systems for the smaller ordinals ${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This systems are based on Japaridze's provability logic $\mathbf{GLP}$. They are closely related with the technique of ordinal analysis of $\mathbf{PA}$ and fragments of $\mathbf{PA}$ 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 ${\ge\omega_4}$, have undecidable elementary theories. We also prove that the fragments of the full system, for ordinals ${\le\omega_3}$, 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

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