Limited automata and unary languages

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. When d = 1 these models characterize regular languages. An exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata is proved. Since a similar gap was already known from unary contextfree grammars to finite automata, also the conversion of such grammars into limited automata is investigated. It is proved that from each unary context-free grammar it is possible to obtain an equivalent 1-limited automaton whose description has a size which is polynomial in the size of the grammar. Furthermore, despite the exponential gap between the sizes of limited automata and of equivalent unary finite automata, there are unary regular languages for which d-limited automata cannot be significantly smaller than equivalent finite automata, for any arbitrarily large d

An intervention to support stroke survivors and their carers in the longer term (LoTS2Care): study protocol for a cluster randomised controlled feasibility trial

Background Despite the evidence that many stroke survivors report longer term unmet needs, the provision of longer term care is limited. To address this, we are conducting a programme of research to develop an evidence-based and replicable longer term care strategy. The developed complex intervention (named New Start), which includes needs identification, exploration of social networks and components of problem solving and self-management, was designed to improve quality of life by addressing unmet needs and increasing participation. Methods/Design A multicentre, cluster randomised controlled feasibility trial designed to inform the design of a possible future definitive cluster randomised controlled trial (cRCT) and explore the potential clinical and cost-effectiveness of New Start. Ten stroke services across the UK will be randomised on a 1:1 basis either to implement New Start or continue with usual care only. New Start will be delivered by trained facilitators and will be offered to all stroke survivors within the services allocated to the intervention arm. Stroke survivors will be eligible for the trial if they are 4–6 months post-stroke and residing in the community. Carers (if available) will also be invited to take part. Invitation to participate will be initiated by post and outcome measures will be collected via postal questionnaires at 3, 6 and 9 months after recruitment. Outcome data relating to perceived health and disability, wellbeing and quality of life as well as unmet needs will be collected. A ‘study within a trial’ (SWAT) is planned to determine the most acceptable format in which to provide the postal questionnaires. Details of health and social care service usage will also be collected to inform the economic evaluation. The feasibility of recruiting services and stroke survivors to the trial and of collecting postal outcomes will be assessed and the potential for effectiveness will be investigated. An embedded process evaluation (reported separately) will assess implementation fidelity and explore and clarify causal assumptions regarding implementation. Discussion This feasibility trial with embedded process evaluation will allow us to gather important and detailed data regarding methodological and implementation issues to inform the design of a possible future definitive cRCT of this complex intervention. Trial Registration ISRCTN38920246. Registered 22 June 2016

Linear-time limited automata

The time complexity of 1-limited automata is investigated from a descriptional complexity view point. Though the model recognizes regular languages only, it may use quadratic time in the input length. We show that, with a polynomial increase in size and preserving determinism, each 1-limited automaton can be transformed into an halting linear-time equivalent one. We also obtain polynomial transformations into related models, including weight-reducing Hennie machines, and we show exponential gaps for converse transformations in the deterministic case

Non-self-embedding grammars, constant-height pushdown automata, and limited automata

Non-self-embedding grammars are a restriction of context-free grammars which does not allow to describe recursive structures and, hence, which characterizes only the class of regular languages. A double exponential gap in size from non-self-embedding grammars to deterministic finite automata is known. The same size gap is also known from constant-height pushdown automata and 1-limited automata to deterministic finite automata. Constant-height pushdown automata and 1-limited automata are compared with non-self-embedding grammars. It is proved that non-self-embedding grammars and constant-height pushdown automata are polynomially related in size. Furthermore, a polynomial size simulation by 1-limited automata is presented. However, the converse transformation is proved to cost exponential

Limited Automata: Properties, Complexity and Variants

Limited automata are single-tape Turing machines with severe rewriting restrictions. They have been introduced in 1967 by Thomas Hibbard, who proved that they have the same computational power as pushdown automata. Hence, they provide an alternative characterization of the class of context-free languages in terms of recognizing devices. After that paper, these models have been almost forgotten for many years. Only recently limited automata were reconsidered in a series of papers, where several properties of them and of their variants have been investigated. In this work we present an overview of the most important results obtained in these researches. We also discuss some related models and possible lines for future investigations