1,994 research outputs found
A uniform approach to fundamental sequences and hierarchies
In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one
Proof Theory at Work: Complexity Analysis of Term Rewrite Systems
This thesis is concerned with investigations into the "complexity of term
rewriting systems". Moreover the majority of the presented work deals with the
"automation" of such a complexity analysis. The aim of this introduction is to
present the main ideas in an easily accessible fashion to make the result
presented accessible to the general public. Necessarily some technical points
are stated in an over-simplified way.Comment: Cumulative Habilitation Thesis, submitted to the University of
Innsbruc
Complexity Bounds for Ordinal-Based Termination
`What more than its truth do we know if we have a proof of a theorem in a
given formal system?' We examine Kreisel's question in the particular context
of program termination proofs, with an eye to deriving complexity bounds on
program running times.
Our main tool for this are length function theorems, which provide complexity
bounds on the use of well quasi orders. We illustrate how to prove such
theorems in the simple yet until now untreated case of ordinals. We show how to
apply this new theorem to derive complexity bounds on programs when they are
proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability
Problems (RP 2014, 22-24 September 2014, Oxford
Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations
Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of
order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n))
(Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and
lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor,
1989). Our first result is an improvement of the upper-bound technique of
Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for
even s up to lower-order terms in the exponent. More importantly, we also
present a new technique for deriving upper bounds for lambda_s(n). With this
new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) +
O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive our own new
upper bounds for general s; and (3) obtain improved upper bounds for the
generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and
Valtr (1992). Regarding lower bounds, we show that lambda_3(n) >= 2n alpha(n) -
O(n), and therefore, the coefficient 2 is tight. We also present a simpler
version of the construction of Agarwal, Sharir, and Shor that achieves the
known lower bounds for even s>=4.Comment: To appear in Journal of the ACM. 48 pages, 3 figure
Multiply-Recursive Upper Bounds with Higman's Lemma
We develop a new analysis for the length of controlled bad sequences in
well-quasi-orderings based on Higman's Lemma. This leads to tight
multiply-recursive upper bounds that readily apply to several verification
algorithms for well-structured systems
Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model
Pattern formation in biological, chemical and physical problems has received
considerable attention, with much attention paid to dissipative systems. For
example, the Ginzburg--Landau equation is a normal form that describes pattern
formation due to the appearance of a single mode of instability in a wide
variety of dissipative problems. In a similar vein, a certain "single-wave
model" arises in many physical contexts that share common pattern forming
behavior. These systems have Hamiltonian structure, and the single-wave model
is a kind of Hamiltonian mean-field theory describing the patterns that form in
phase space. The single-wave model was originally derived in the context of
nonlinear plasma theory, where it describes the behavior near threshold and
subsequent nonlinear evolution of unstable plasma waves. However, the
single-wave model also arises in fluid mechanics, specifically shear-flow and
vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter
physics, and other Hamiltonian theories characterized by mean field
interaction. We demonstrate, by a suitable asymptotic analysis, how the
single-wave model emerges from a large class of nonlinear advection-transport
theories. An essential ingredient for the reduction is that the Hamiltonian
system has a continuous spectrum in the linear stability problem, arising not
from an infinite spatial domain but from singular resonances along curves in
phase space whereat wavespeeds match material speeds (wave-particle resonances
in the plasma problem, or critical levels in fluid problems). The dynamics of
the continuous spectrum is manifest as the phenomenon of Landau damping when
the system is ... Such dynamical phenomena have been rediscovered in different
contexts, which is unsurprising in view of the normal-form character of the
single-wave model
Simplification orders in term rewriting
Thema der Arbeit ist die Anwendung von Methoden der Beweistheorie
auf Termersetzungssysteme, deren Termination mittels einer
Simplifikationsordnung beweisbar ist. Es werden optimale
Schranken für Herleitungslängen im allgemeinen Fall und im
Fall der Termination mittels einer Knuth-Bendix-Ordnung (KBO)
angegeben. Zudem werden die Ordnungstypen von KBOs vollständig
klassifiziert und die unter KBO berechenbaren Funktionen
vorgestellt. Einen weiteren Schwerpunkt bildet die Untersuchung
der Löngen von Reduktionsketten, die bei einfach terminierenden
Termersetzungssysteme auftreten und bestimmten Wachstumsbedingungen
genĂĽgen
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