Compositional bisimulation metric reasoning with Probabilistic Process Calculi

We study which standard operators of probabilistic process calculi allow for compositional reasoning with respect to bisimulation metric semantics. We argue that uniform continuity (generalizing the earlier proposed property of non-expansiveness) captures the essential nature of compositional reasoning and allows now also to reason compositionally about recursive processes. We characterize the distance between probabilistic processes composed by standard process algebra operators. Combining these results, we demonstrate how compositional reasoning about systems specified by continuous process algebra operators allows for metric assume-guarantee like performance validation

Approximate reasoning for real-time probabilistic processes

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -- such as probabilities and time -- in the model. We give a fixed point characterization of the pseudo-metric. This makes available coinductive reasoning principles for reasoning about distances. We demonstrate that our approach is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity. We provide a logical characterization of this uniformity using a real-valued modal logic. We show that several quantitative properties of interest are continuous with respect to the pseudo-metric. Thus, if two processes are metrically close, then observable quantitative properties of interest are indeed close.Comment: Preliminary version appeared in QEST 0

The integrated use of enterprise and system dynamics modelling techniques in support of business decisions

Enterprise modelling techniques support business process re-engineering by capturing existing processes and based on perceived outputs, support the design of future process models capable of meeting enterprise requirements. System dynamics modelling tools on the other hand are used extensively for policy analysis and modelling aspects of dynamics which impact on businesses. In this paper, the use of enterprise and system dynamics modelling techniques has been integrated to facilitate qualitative and quantitative reasoning about the structures and behaviours of processes and resource systems used by a Manufacturing Enterprise during the production of composite bearings. The case study testing reported has led to the specification of a new modelling methodology for analysing and managing dynamics and complexities in production systems. This methodology is based on a systematic transformation process, which synergises the use of a selection of public domain enterprise modelling, causal loop and continuous simulationmodelling techniques. The success of the modelling process defined relies on the creation of useful CIMOSA process models which are then converted to causal loops. The causal loop models are then structured and translated to equivalent dynamic simulation models using the proprietary continuous simulation modelling tool iThink

The Development of Spatial-Temporal, Probability, and Covariation Information to Infer Continuous Causal Processes

This paper considers how 5- to 11-year-olds’ verbal reasoning about the causality underlying extended, dynamic natural processes links to various facets of their statistical thinking. Such continuous processes typically do not provide perceptually distinct causes and effect, and previous work suggests that spatial–temporal analysis, the ability to analyze spatial configurations that change over time, is a crucial predictor of reasoning about causal mechanism in such situations. Work in the Humean tradition to causality has long emphasized on the importance of statistical thinking for inferring causal links between distinct cause and effect events, but here we assess whether this is also viable for causal thinking about continuous processes. Controlling for verbal and non-verbal ability, two studies (N = 107; N = 124) administered a battery of covariation, probability, spatial–temporal, and causal measures. Results indicated that spatial–temporal analysis was the best predictor of causal thinking across both studies, but statistical thinking supported and informed spatial–temporal analysis: covariation assessment potentially assists with the identification of variables, while simple probability judgment potentially assists with thinking about unseen mechanisms. We conclude that the ability to find out patterns in data is even more widely important for causal analysis than commonly assumed, from childhood, having a role to play not just when causally linking already distinct events but also when analyzing the causal process underlying extended dynamic events without perceptually distinct components

Reasoning Continuously: A Formal Construction of Continuous Proofs

Funder: University of CambridgeAbstract: We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism

Children's reasoning about continuous causal processes: The role of verbal and non-verbal ability

BACKGROUND: Causes produce effects via underlying mechanisms that must be inferred from observable and unobservable structures. Preschoolers show sensitivity to mechanisms in machine-like systems with perceptually distinct causes and effects, but little is known about how children extend causal reasoning to the natural continuous processes studied in elementary school science, or how other abilities impact on this. AIMS: We investigated the development of children's ability to predict, observe, and explain three causal processes, relevant to physics, biology, and chemistry, taking into account their verbal and non-verbal ability. SAMPLE: Children aged 5-11 years (N = 107) from London and Oxford, with wide ethnic/linguistic variation, drawn from the middle/upper socioeconomic status (SES) range. METHODS: Children were tested individually on causal tasks focused on sinking, absorption, and dissolving, using a novel approach in which they observed contrasting instances of each, to promote attention to mechanism. Further tasks assessed verbal (expressive vocabulary) and non-verbal (block design) ability. RESULTS: Reports improved with age, though with differences between tasks. Even young participants gave good descriptions of what they observed. Causal explanations were more strongly related to observation than to prediction from prior knowledge, but developed more slowly. Non-verbal but not generic verbal ability predicted performance. CONCLUSIONS: Reasoning about continuous processes is within the capacity of children from school entry, even using verbal reports, though they find it easier to address more rapid processes. Mechanism inference is uncommon, with non-verbal ability an important influence on progress. Our research is the first to highlight this key factor in children's progress towards thinking about scientific phenomena

The graphing calculator in the promotion of mathematical writing

Through writing, students express many of their processes and ways of thinking. Since at high school level some of the activities are carried out with the graphing calculator, we intend to investigate the contribution of this resource to promote the mathematical writing in the learning of continuous nonlinear models at 11th grade. Adopting a qualitative methodology, we collected and analyzed the students’ writing productions. What they write when using the calculator gives evidence about the information valued (when they sketch graphics without any justification); about the strategies used (when they define the viewing window and relate different menus on the graphing calculator); and about the reasoning developed (when they justify the information given by the calculator and the formulation of generalizations and conjectures validation).info:eu-repo/semantics/publishedVersio