2,543 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
On Elementary Theories of Ordinal Notation Systems based on Reflection Principles
We consider the constructive ordinal notation system for the ordinal
that were introduced by L.D. Beklemishev. There are fragments of
this system that are ordinal notation systems for the smaller ordinals
(towers of -exponentiations of the height ). This
systems are based on Japaridze's provability logic . They are
closely related with the technique of ordinal analysis of and
fragments of 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 , have undecidable elementary theories.
We also prove that the fragments of the full system, for ordinals
, have decidable elementary theories. We obtain some results
about decidability of elementary theory, for the ordinal notation systems with
weaker signatures.Comment: 23 page
Short Proofs for Slow Consistency
Let denote the finite
consistency statement "there are no proofs of contradiction in with
symbols". For a large class of natural theories , Pudl\'ak
has shown that the lengths of the shortest proofs of
in the theory
itself are bounded by a polynomial in . At the same time he conjectures that
does not have polynomial proofs of the finite consistency
statements . In contrast we show that Peano arithmetic
() has polynomial proofs of
,
where is the slow consistency statement for
Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also
obtain a new proof of the result that the usual consistency statement
is equivalent to iterations
of slow consistency. Our argument is proof-theoretic, while previous
investigations of slow consistency relied on non-standard models of arithmetic
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