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ABSTRACT: In this paper we consider transfinite provability logics where for each
ordinal in some recursive well-order we have a corresponding modal provability
operator. The modality [xi] will be interpreted as "provable in ACA_0 together
with at most xi nested applications of the omega rule". We show how to
formalize this in in second order number theory. Next we prove both soundness
and completeness under this interpretation. We conclude by showing how one can
lower the base theory ACA_0 to theories below RCA_0.
02/2013;
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ABSTRACT: In the generalized Russian cards problem, the three players Alice, Bob and
Cath draw a,b and c cards, respectively, from a deck of a+b+c cards. Players
only know their own cards and what the deck of cards is. Alice and Bob are then
required to communicate their hand of cards to each other by way of public
messages. The communication is said to be safe if Cath does not learn the
ownership of any specific card; in this paper we consider a strengthened notion
of safety introduced by Swanson and Stinson which we call k-safety.
An elegant solution by Atkinson views the cards as points in a finite
projective plane. We propose a general solution in the spirit of Atkinson's,
although based on finite vector spaces rather than projective planes, and call
it the `geometric protocol'. Given arbitrary c,k>0, this protocol gives an
informative and k-safe solution to the generalized Russian cards problem for
infinitely many values of (a,b,c) with b=O(ac). This improves on the collection
of parameters for which solutions are known. In particular, it is the first
solution which guarantees $k$-safety when Cath has more than one card.
01/2013;
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ABSTRACT: We introduce the logics GLP(\Lambda), a generalization of Japaridze's
polymodal provability logic GLP(\omega) where \Lambda is any linearly ordered
set representing a hierarchy of provability operators of increasing strength.
We shall provide a reduction of these logics to GLP(\omega) yielding among
other things a finitary proof of the normal form theorem for the variable-free
fragment of GLP(\Lambda) and the decidability of GLP(\Lambda) for recursive
orderings \Lambda. Further, we give a restricted axiomatization of the
variable-free fragment of GLP(\Lambda).
10/2012;
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ABSTRACT: In the Russian cards problem, Alice, Bob and Cath draw three, three and one
cards, respectively, from a deck of seven. Alice and Bob must then communicate
their entire hand to each other, without Cath learning the owner of a single
card. Unlike many traditional problems in cryptography, however, they are not
allowed to hide or codify the messages they exchange from Cath. The problem is
then to find methods through which they can achieve this. One elegant solution,
due to Atkinson, considers the cards as points in a finite projective plane.
In this paper we consider the generalized Russian cards problem, where the
number of cards that each player draws is a,b and c, respectively, from a deck
of $a+b+c$ cards. We propose a general solution in the spirit of Atkinson's,
although based on finite vector spaces, and call it the "colouring protocol",
as it involves colourings of affine subsets.
Our main results show that the colouring protocol provides a solution to the
generalized Russian cards problem in cases where $a$ is a power of a prime,
c=O(a^2) and b=O(c^2).
This improves substantially on the collection of parameters for which
solutions are known. In particular, it is the first solution which allows the
eavesdropper to have more cards than one of the players.
07/2012;
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ABSTRACT: In this paper we introduce hyperations and cohyperations, which are forms of
transfinite iteration of ordinal functions.
Hyperations are iterations of normal functions. Unlike iteration by pointwise
convergence, hyperation preserves normality. The hyperation of a normal
function f is a sequence of normal functions so that f^0= id, f^1 = f and for
all ordinals \alpha, \beta we have that f^(\alpha + \beta) = f^\alpha f^\beta.
These conditions do not determine f^\alpha uniquely; in addition, we require
that the functions be minimal in an appropriate sense. We study hyperations
systematically and show that they are a natural refinement of Veblen
progressions.
Next, we define cohyperations, very similar to hyperations except that they
are left-additive: given \alpha, \beta, f^(\alpha + \beta)= f^\beta f^\alpha.
Cohyperations iterate initial functions which are functions that map initial
segments to initial segments. We systematically study cohyperations and see how
they can be employed to define left inverses to hyperations.
Hyperations provide an alternative presentation of Veblen progressions and
can be useful where a more fine-grained analysis of such sequences is called
for. They are very amenable to algebraic manipulation and hence are convenient
to work with. Cohyperations, meanwhile, give a novel way to describe slowly
increasing functions as often appear, for example, in proof theory.
05/2012;
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ABSTRACT: 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.
04/2012;
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ABSTRACT: The logic GLP is a polymodal logic that has for each ordinal \alpha an
operator [\alpha ], whose intended interpretation is a provability predicate in
a hierarchy of theories of increasing strength. Its corresponding algebra is
called the (transfinite) Japaridze algebra. There are various natural orders in
this algebra that are based on comparing consistency strength of its elements.
In particular, for each \alpha we define A <_{\alpha} iff over GLP, B implies
<\alpha> A.
In this paper we shall consider worms, which are formulas of the form
<\alpha_0>...<\alpha_n>T, and the partial orders <_\alpha on their images in
the Japaridze algebra. Given a worm A and an ordinal \alpha, our goal is to
show how one computes the order type that is naturally associated to
\Omega_\alpha(A):={B:B<_\alpha A}.
Our main results show how the sequences <\Omega_\alpha(A)> can be computed
via hyperations and cohyperations, which are forms of transfinite iterations of
ordinal functions closely related to Veblen hierarchies.
04/2012;
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Electr. Notes Theor. Comput. Sci. 01/2011; 278:1-2.
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International Symposium on Distributed Computing and Artificial Intelligence, DCAI 2011, Salamanca, Spain, 6-8 April 2011; 01/2011
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Computational Logic in Multi-Agent Systems - 12th International Workshop, CLIMA XII, Barcelona, Spain, July 17-18, 2011. Proceedings; 01/2011
