9,671 research outputs found
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
Hyperations, Veblen progressions and transfinite iterations of ordinal functions
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
The Second Order Traffic Fine: Temporal Reasoning in European Transport Regulations
We argue that European transport regulations can be formalized within the Sigma^1_1 fragment of monadic second order logic, and possibly weaker fragments including linear temporal logic. We consider several articles in the regulation to verify these claims
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