4,033 research outputs found
Separating Bounded Arithmetics by Herbrand Consistency
The problem of separating the hierarchy of bounded arithmetic has
been studied in the paper. It is shown that the notion of Herbrand Consistency,
in its full generality, cannot separate the theory from ; though it can
separate from . This extends a
result of L. A. Ko{\l}odziejczyk (2006), by showing the unprovability of the
Herbrand Consistency of in the theory .Comment: Published by Oxford University Press. arXiv admin note: text overlap
with arXiv:1005.265
Herbrand Consistency of Some Arithmetical Theories
G\"odel's second incompleteness theorem is proved for Herbrand consistency of
some arithmetical theories with bounded induction, by using a technique of
logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz
[Herbrand consistency and bounded arithmetic, \textit{Fundamenta Mathematicae}
171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink
the witness of a bounded formula logarithmically, but in the presence of
Herbrand consistency, for theories with , any witness for any bounded formula can be shortened logarithmically. This
immediately implies the unprovability of Herbrand consistency of a theory
in itself.
In this paper, the above results are generalized for . Also after tailoring the definition of Herbrand
consistency for we prove the corresponding theorems for . Thus the Herbrand version of G\"odel's second incompleteness
theorem follows for the theories and
An elementary way to rigorously estimate convergence to equilibrium and escape rates
We show an elementary method to have (finite time and asymptotic) computer
assisted explicit upper bounds on convergence to equilibrium (decay of
correlations) and escape rate for systems satisfying a Lasota Yorke inequality.
The bounds are deduced by the ones of suitable approximations of the system's
transfer operator. We also present some rigorous experiment showing the
approach and some concrete result.Comment: 14 pages, 6 figure
Polylogarithmic Cuts in Models of V^0
We study initial cuts of models of weak two-sorted Bounded Arithmetics with
respect to the strength of their theories and show that these theories are
stronger than the original one. More explicitly we will see that
polylogarithmic cuts of models of are models of
by formalizing a proof of Nepomnjascij's Theorem in such cuts. This is a
strengthening of a result by Paris and Wilkie. We can then exploit our result
in Proof Complexity to observe that Frege proof systems can be sub
exponentially simulated by bounded depth Frege proof systems. This result has
recently been obtained by Filmus, Pitassi and Santhanam in a direct proof. As
an interesting observation we also obtain an average case separation of
Resolution from AC0-Frege by applying a recent result with Tzameret.Comment: 16 page
- …