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Big data play an indispensable role not only in the cognitive mechanisms of human sensation, quantification, qualification, estimation, memory, and reasoning, but also in a wide range of engineering applications. A basic study on the theoretical foundations of big data science is presented with a coherent set of general principles and analytic methodologies for big data systems. Cognitive foundations of big data are explored in order to formally explain the origination and nature of big data. A set of mathematical models of big data are created that rigorously elicit the general essences and patterns of big data across pervasive domains in sciences, engineering, and societies. A significant finding towards big data science is that big data systems in nature are a recursive -dimensional-typed hyperstructure (RNTHS) rather than pure numbers. The fundamental topological property of big data reveals a set of denotational mathematical solutions for dealing with inherited complexities and unprecedented challenges in big data engineering. [ABSTRACT FROM AUTHOR]

Abstract: One of the fundamental human cognitive processes is problem solving. As a higher-layer cognitive process, problem solving interacts with many other cognitive processes such as abstraction, searching, learning, decision making, inference, analysis, and synthesis on the basis of internal knowledge representation by the object–attribute-relation (OAR) model. Problem solving is a cognitive process of the brain that searches a solution for a given problem or finds a path to reach a given goal. When a problem object is identified, problem solving can be perceived as a search process in the memory space for finding a relationship between a set of solution goals and a set of alternative paths. This paper presents both a cognitive model and a mathematical model of the problem solving process. The cognitive structures of the brain and the mechanisms of internal knowledge representation behind the cognitive process of problem solving are explained. The cognitive process is formally described using real-time process algebra (RTPA) and concept algebra. This work is a part of the cognitive computing project that designed to reveal and simulate the fundamental mechanisms and processes of the brain according to Wang’s layered reference model of the brain (LRMB), which is expected to lead to the development of future generation methodologies for cognitive computing and novel cognitive computers that are capable of think, learn, and perceive. [Copyright &y& Elsevier]