9 research outputs found
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
On the Constructive Content of Proofs
This thesis aims at exploring the scopes and limits of techniques
for extracting programs from proofs. We focus on constructive
theories of inductive definitions and classical systems allowing
choice principles. Special emphasis is put on optimizations that
allow for the extraction of realistic programs.
Our main field of application is infinitary combinatorics. Higman's
Lemma, having an elegant non-constructive proof due to Nash-Williams,
constitutes an interesting case for the problem of discovering the
constructive content behind a classical proof. We give two distinct
solutions to this problem. First, we present a proof of Higman's
Lemma for an arbitrary alphabet in a theory of inductive
definitions. This proof may be considered as a constructive
counterpart to Nash-Williams' minimal-bad-sequence proof. Secondly,
using a refined -translation method, we directly transform the
classical proof into a constructive one and extract a program. The
crucial point in the latter is that we do not need to avoid the axiom
of classical dependent choice but directly assign a realizer to its
translation.
A generalization of Higman's Lemma is Kruskal's Theorem.
We present a constructive proof of Kruskal's Theorem that is
completely formalized in a theory of inductive definitions.
As a practical part, we show that these methods can be carried out in
an interactive theorem prover. Both approaches to Higman's Lemma have
been implemented in Minlog.Ziel der vorliegenden Arbeit ist es, die Reichweiten und Grenzen
von Techniken zur Extraktion von Programmen aus Beweisen
zu erforschen. Wir konzentrieren uns dabei auf konstruktive
Theorien Induktiver Definitionen und klassische Systeme mit
Auswahlprinzipien. Besonderes Gewicht liegt auf Optimierungen,
die zur Extraktion von realisischen Programmen f"uhren.
Unser Hauptanwendungsgebiet ist die unendliche Kombinatorik. Higmans
Lemma, ein Satz mit einem eleganten klassischen, auf Nash-Williams
zur"uckgehenden Beweis, ist ein interessantes Fallbeispiel f"ur die
Suche nach dem konstruktiven Gehalt in einem klassischen Beweis. Wir
zeigen zwei unterschiedliche L"osungen zu dieser Problemstellung
auf. Zun"achst pr"asentieren wir einen induktiven Beweis von Higmans
Lemma f"ur ein beliebiges Alphabet, der als konstruktives Pendant zum
klassischen Beweis angesehen werden kann. Als zweiten Ansatz
verwandeln wir mit Hilfe der verfeinerten -"Ubersetzungs-methode
den klassischen Beweis in einen konstruktiven und extrahieren ein
Programm. Der entscheidende Punkt ist hierbei, dass wir einen
direkten Realisierer f"ur das "ubersetzte Auswahlaxiom verwenden.
Die Verallgemeinerung von Higmans Lemma f"uhrt zu Kruskals Satz.
Wir geben einen konstruktiven Beweis von Kruskals Theorem, der
vollst"andig auf den Induktiven Definitionen basiert.
Der praktische Teil der Arbeit befasst sich mit der Ausf"uhrbarkeit
dieser Methoden und Beweise in dem Beweissystem Minlog
Polynomial Time Calculi
This dissertation deals with type systems which guarantee polynomial time complexity of typed programs. Such algorithms are commonly regarded as being feasible for practical applications, because their runtime grows reasonably fast for bigger inputs. The implicit complexity community has proposed several type systems for polynomial time in the recent years, each with strong, but different structural restrictions on the permissible algorithms which are necessary to control complexity. Comparisons between the various approaches are hard and this has led to a landscape of islands in the literature of expressible algorithms in each calculus, without many known links between them.
This work chooses Light Affine Logic (LAL) and Hofmann's LFPL, both linearly typed, and studies the connections between them. It is shown that the light iteration in LAL, the fixed point variant of LAL, is expressive enough to allow a (non-trivial) compositional embedding of LFPL. The pull-out trick of LAL is identified as a technique to type certain non-size-increasing algorithms in such a way that they can be iterated. The System T sibling of LAL is developed which seamlessly integrates this technique as a central feature of the iteration scheme and which is proved again correct and complete for polynomial time. Because -iterations of the same level cannot be nested, is further generalised to , which surprisingly can express the impredicative iteration of LFPL and the light iteration of at the same time. Therefore, it subsumes both systems in one, while still being polynomial time normalisable. Hence, this result gives the first bridge between these two islands of implicit computational complexity