Ranking Functions for Vector Addition Systems

Vector addition systems are an important model in theoretical computer science and have been used for the analysis of systems in a variety of areas. Termination is a crucial property of vector addition systems and has received considerable interest in the literature. In this paper we give a complete method for the construction of ranking functions for vector addition systems with states. The interest in ranking functions is motivated by the fact that ranking functions provide valuable additional information in case of termination: They provide an explanation for the progress of the vector addition system, which can be reported to the user of a verification tool, and can be used as certificates for termination. Moreover, we show how ranking functions can be used for the computational complexity analysis of vector addition systems (here complexity refers to the number of steps the vector addition system under analysis can take in terms of the given initial vector)

Cross-sectional associations between movement behaviors and cardio-metabolic risk markers in Danish children.

<p>Abbreviations: PA, physical activity; MVPA, moderate-to-vigorous physical activity; CSHQ, Children’s Sleep Habits Questionnaire; WC, waist circumference; MAP, mean arterial blood pressure; HOMA_IR, homeostatic model assessment of central insulin resistance; HDL-C, high density lipoprotein cholesterol; MetS-score, metabolic syndrome score.</p><p>Data are presented as unstandardized regression coefficients (β) with 95% confidence intervals (CI) using a linear mixed model with school and subject as random effects. The five cardio-metabolic risk markers were adjusted for baseline age, sex, pubertal status, and sex-pubertal status interaction (<b>Model 1</b>).</p><p>Coefficients represent the change in the outcome for a 100-cpm change in total PA, a 10-minute change in time spent in MVPA, a 1% change in sedentary time, a 60-minute change in screen time and sleep duration and a 1-point change in CSHQ. *, P<0.05; **, P<0.001.</p>¥<p>P<0.05 in <b>Model 2</b>: Model 1+ mutual adjustments between MVPA, sedentary time and sleep duration. Total PA was only adjusted for sleep duration.</p>£<p>P<0.05 in <b>Model 3</b>: Model 2+ fat mass index.</p>1<p>MAP was also adjusted for height.</p>2<p>HOMA_IR and triglycerides were log transformed.</p>3<p>MetS-score = (z-scores by baseline age, sex, pubertal status and sex-pubertal status interaction of) WC + MAP + HOMA_IR + triglycerides – HDL-C.</p