44 research outputs found
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
Models of transfinite provability logic
For any ordinal \Lambda, we can define a polymodal logic GLP(\Lambda), with a
modality [\xi] for each \xi<\Lambda. These represent provability predicates of
increasing strength. Although GLP(\Lambda) has no Kripke models, Ignatiev
showed that indeed one can construct a Kripke model of the variable-free
fragment with natural number modalities. Later, Icard defined a topological
model for the same fragment which is very closely related to Ignatiev's.
In this paper we show how to extend these constructions for arbitrary
\Lambda. More generally, for each \Theta,\Lambda we build a Kripke model
I(\Theta,\Lambda) and a topological model T(\Theta,\Lambda), and show that the
closed fragment of GLP(\Lambda) is sound for both of these structures, as well
as complete, provided \Theta is large enough
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 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 on worms whose modalities are all at least and
presented a calculus to compute the respective order-types.
In the current paper we present a generalization of the original
orderings and provide a calculus for the corresponding generalized order-types
. 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 . 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
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness