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

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*

On Sharing, Memoization, and Polynomial Time (Long Version)

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity

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

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