4 research outputs found
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
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
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 of Acyclic Term Graph Rewriting
Term rewriting has been used as a formal model to reason about the
complexity of logic, functional, and imperative programs. In contrast
to term rewriting, term graph rewriting permits sharing of
common sub-expressions, and consequently is able to capture more
closely reasonable implementations of rule based languages. However,
the automated complexity analysis of term graph rewriting has received
little to no attention.
With this work, we provide first steps towards overcoming this
situation. We present adaptions of two prominent complexity techniques
from term rewriting, viz, the interpretation method and
dependency tuples. Our adaptions are non-trivial, in the sense
that they can observe not only term but also graph structures, i.e.
take sharing into account. In turn, the developed methods allow us to
more precisely estimate the runtime complexity of programs where
sharing of sub-expressions is essential