Termination of Rewriting with Right-Flat Rules Modulo Permutative Theories

We present decidability results for termination of classes of term rewriting systems modulo permutative theories. Termination and innermost termination modulo permutative theories are shown to be decidable for term rewrite systems (TRS) whose right-hand side terms are restricted to be shallow (variables occur at depth at most one) and linear (each variable occurs at most once). Innermost termination modulo permutative theories is also shown to be decidable for shallow TRS. We first show that a shallow TRS can be transformed into a flat (only variables and constants occur at depth one) TRS while preserving termination and innermost termination. The decidability results are then proved by showing that (a) for right-flat right-linear (flat) TRS, non-termination (respectively, innermost non-termination) implies non-termination starting from flat terms, and (b) for right-flat TRS, the existence of non-terminating derivations starting from a given term is decidable. On the negative side, we show PSPACE-hardness of termination and innermost termination for shallow right-linear TRS, and undecidability of termination for flat TRS.Comment: 20 page

Termination of Rewriting with Right-Flat Rules

Abstract. Termination and innermost termination are shown to be decidable for rewrite systems whose right-hand side terms are restricted to be shallow (variables occur at depth at most one) and linear. Innermost termination is also shown to be decidable for shallow rewrite systems. In all cases, we show that nontermination implies nontermination starting from flat terms. The proof is completed by using the useful enabling result that, for right shallow rewrite systems, existence of nonterminating derivations starting from a given term is decidable. We also show that termination is undecidable for shallow rewrite systems. For right-shallow systems, general and innermost termination are both undecidable

Climate, Ticks and Disease

This book brings together expert opinions from scientists to consider the evidence for climate change and its impacts on ticks and tick-borne infections. It considers what is meant by 'climate change', how effective climate models are in relation to ecosystems, and provides predictions for changes in climate at global, regional and local scales relevant for ticks and tick-borne infections. It examines changes to tick distribution and the evidence that climate change is responsible. The effect of climate on the physiology and behaviour of ticks is stressed, including potentially critical impacts on the tick microbiome. Given that the notoriety of ticks derives from pathogens they transmit, the book considers whether changes in climate affect vector capacity. Ticks transmit a remarkable range of micro- and macro-parasites many of which are pathogens of humans and domesticated animals. The intimacy between a tick-borne agent and a tick vector means that any impacts of climate on a tick vector will impact tick-borne pathogens. Most obviously, such impacts will be apparent as changes in disease incidence and prevalence. The evidence that climate change is affecting diseases caused by tick-borne pathogens is considered, along with the potential to make robust predictions of future events. This book contains: Expert opinions and predictions. Global coverage of trends in ticks and disease. In-depth examination of climate change and tick distribution links.This book is suitable for researchers and students studying zoology, biological sciences, medical entomology, animal health, veterinary medicine, epidemiology, parasitology, and climate change impacts; and for those concerned with public health planning or livestock management where ticks and tick-borne pathogens pose a threat