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

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

The Hydra Battle and Cichon’s Principle

Abstract In rewriting the Hydra battle refers to a term rewrite system H proposed by Dershowitz and Jouannaud. To date, H withstands any attempt to prove its termination automatically. This motivates our interest in term rewrite systems encoding the Hydra battle, as a careful study of such systems may prove useful in the design of automatic termination tools. Moreover it has been an open problem, whether any termination order compatible with H has to have the Howard-Bachmann ordinal as its order type, i.e., the proof theoretic ordinal of the theory of one inductive denition. We answer this question in the negative, by providing a reduction order compatible with H, whose order type is at most 0 , the proof theoretic ordinal of Peano arithmetic

The Derivational Complexity Induced by the Dependency Pair Method

We study the derivational complexity induced by the dependency pair method, enhanced with standard refinements. We obtain upper bounds on the derivational complexity induced by the dependency pair method in terms of the derivational complexity of the base techniques employed. In particular we show that the derivational complexity induced by the dependency pair method based on some direct technique, possibly refined by argument filtering, the usable rules criterion, or dependency graphs, is primitive recursive in the derivational complexity induced by the direct method. This implies that the derivational complexity induced by a standard application of the dependency pair method based on traditional termination orders like KBO, LPO, and MPO is exactly the same as if those orders were applied as the only termination technique