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 $A$-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 $A$-"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