Polynomial Path Orders: A Maximal Model

This paper is concerned with the automated complexity analysis of term rewrite systems (TRSs for short) and the ramification of these in implicit computational complexity theory (ICC for short). We introduce a novel path order with multiset status, the polynomial path order POP*. Essentially relying on the principle of predicative recursion as proposed by Bellantoni and Cook, its distinct feature is the tight control of resources on compatible TRSs: The (innermost) runtime complexity of compatible TRSs is polynomially bounded. We have implemented the technique, as underpinned by our experimental evidence our approach to the automated runtime complexity analysis is not only feasible, but compared to existing methods incredibly fast. As an application in the context of ICC we provide an order-theoretic characterisation of the polytime computable functions. To be precise, the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP*

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

Complexity Analysis of Precedence Terminating Infinite Graph Rewrite Systems

The general form of safe recursion (or ramified recurrence) can be expressed by an infinite graph rewrite system including unfolding graph rewrite rules introduced by Dal Lago, Martini and Zorzi, in which the size of every normal form by innermost rewriting is polynomially bounded. Every unfolding graph rewrite rule is precedence terminating in the sense of Middeldorp, Ohsaki and Zantema. Although precedence terminating infinite rewrite systems cover all the primitive recursive functions, in this paper we consider graph rewrite systems precedence terminating with argument separation, which form a subclass of precedence terminating graph rewrite systems. We show that for any precedence terminating infinite graph rewrite system G with a specific argument separation, both the runtime complexity of G and the size of every normal form in G can be polynomially bounded. As a corollary, we obtain an alternative proof of the original result by Dal Lago et al.Comment: In Proceedings TERMGRAPH 2014, arXiv:1505.06818. arXiv admin note: text overlap with arXiv:1404.619

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

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