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

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

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 genügen

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

On the scaling of the chemical distance in long-range percolation models

We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\infty}. Our main result is that, for s\in(d,2d), D(x,y)=(\log|x-y|)^{\Delta+o(1)},\qquad x,y\in C_{\infty}, |x-y|\to\infty, where \Delta^{-1} is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x-y|\to\infty. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of ``small-world'' phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.Comment: Published at http://dx.doi.org/10.1214/009117904000000577 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

General Analysis Tool Box for Controlled Perturbation

The implementation of reliable and efficient geometric algorithms is a challenging task. The reason is the following conflict: On the one hand, computing with rounded arithmetic may question the reliability of programs while, on the other hand, computing with exact arithmetic may be too expensive and hence inefficient. One solution is the implementation of controlled perturbation algorithms which combine the speed of floating-point arithmetic with a protection mechanism that guarantees reliability, nonetheless. This paper is concerned with the performance analysis of controlled perturbation algorithms in theory. We answer this question with the presentation of a general analysis tool box. This tool box is separated into independent components which are presented individually with their interfaces. This way, the tool box supports alternative approaches for the derivation of the most crucial bounds. We present three approaches for this task. Furthermore, we have thoroughly reworked the concept of controlled perturbation in order to include rational function based predicates into the theory; polynomial based predicates are included anyway. Even more we introduce object-preserving perturbations. Moreover, the tool box is designed such that it reflects the actual behavior of the controlled perturbation algorithm at hand without any simplifying assumptions.Comment: 90 pages, 30 figure

Polynomial Path Orders

This paper is concerned with the complexity analysis of constructor term rewrite systems and its ramification in implicit computational complexity. We introduce a path order with multiset status, the polynomial path order POP*, that is applicable in two related, but distinct contexts. On the one hand POP* induces polynomial innermost runtime complexity and hence may serve as a syntactic, and fully automatable, method to analyse the innermost runtime complexity of term rewrite systems. On the other hand POP* provides an order-theoretic characterisation of the polytime computable functions: the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP*.Comment: LMCS version. This article supersedes arXiv:1209.379

Effective termination techniques

An important property of term rewriting systems is termination: the guarantee that every rewrite sequence is finite. This thesis is concerned with orderings used for proving termination, in particular the Knuth-Bendix and polynomial orderings. First, two methods for generating termination orderings are enhanced. The Knuth-Bendix ordering algorithm incrementally generates numeric and symbolic constraints that are sufficient for the termination of the rewrite system being constructed. The KB ordering algorithm requires an efficient linear constraint solver that detects the nature of degeneracy in the solution space, and for this a revised method of complete description is presented that eliminates the space redundancy that crippled previous implementations. Polynomial orderings are more powerful than Knuth-Bendix orderings, but are usually much harder to generate. Rewrite systems consisting of only a handful of rules can overwhelm existing search techniques due to the combinatorial complexity. A genetic algorithm is applied with some success. Second, a subset of the family of polynomial orderings is analysed. The polynomial orderings on terms in two unary function symbols are fully resolved into simpler orderings. Thus it is shown that most of the complexity of polynomial orderings is redundant. The order type (logical invariant), either r or A (numeric invariant), and precedence is calculated for each polynomial ordering. The invariants correspond in a natural way to the parameters of the orderings, and so the tabulated results can be used to convert easily between polynomial orderings and more tangible orderings. The orderings of order type are two of the recursive path orderings. All of the other polynomial orderings are of order type w or w2 and each can be expressed as a lexicographic combination of r (weight), A (matrix), and lexicographic (dictionary) orderings. The thesis concludes by showing how the analysis extends to arbitrary monadic terms, and discussing possible developments for the future