24 research outputs found
Solovay’s Completeness Without Fixed Points
In this paper we present a new proof of Solovay's theorem on arithmetical
completeness of G\"odel-L\"ob provability logic GL. Originally, completeness of
GL with respect to interpretation of as provability in PA was proved by
R. Solovay in 1976. The key part of Solovay's proof was his construction of an
arithmetical evaluation for a given modal formula that made the formula
unprovable PA if it were unprovable in GL. The arithmetical sentences for the
evaluations were constructed using certain arithmetical fixed points. The
method developed by Solovay have been used for establishing similar semantics
for many other logics. In our proof we develop new more explicit construction
of required evaluations that doesn't use any fixed points in their definitions.
To our knowledge, it is the first alternative proof of the theorem that is
essentially different from Solovay's proof in this key part.Comment: 13 pages, accepted to WoLLIC 2017 conferenc
Intercell Interference Coordination for Control Channels in LTE and LTE-A: An Optimization Scheme Based on Evolutionary Algorithms
Typicality and the Role of the Lebesgue Measure in Statistical Mechanics
Consider a finite collection of marbles. The statement "half the marbles are white" is about counting, and not about the probability of drawing a white marble from the collection. The question is whether nonprobabilistic counting notions such as half, or vast majority can make sense, and preserve their meaning when extended to the realm of the continuum. In this paper we argue that the Lebesgue measure provides the proper non-probabilistic extension, which is as natural, and in a sense uniquely forced, as the extension of the concept of cardinal number to infinite sets by Cantor. To accomplish this a different way of constructing the Lebesgue measure is applied. One important example of a non-probabilistic counting concept is typicality, introduced to statistical physics to explain the approach to equilibrium. A typical property is shared by a vast majority of cases. Typicality is not probabilistic, at least in the sense that it is robust and not dependent on any precise assumptions about the probability distribution. A few dynamica