Cognitive robotics and mathematical engineering

It is recognized in cognitive informatics [10, 12-14, 20, 25, 29, 30] that the core scientific knowledge of the mankind is mainly archived in mathematical forms [1, 2, 5-9, 10-39]. The entire set o f fundamental and long-lasting problems in contemporary disciplines, such as, inter alia, intelligence science, robotics, knowledge science, information science, brain science, system science, software science, data science, neuroinformatics, cognitive linguistics, and computational intelligence, indicate that the aforementioned problems in nature are a hard mathematical problem where there is a lack of suitable mathematical means [3, 4, 8, 9, 12-14, 22, 25, 31, 39]. The current forms of analytic mathematics are inadequate to solve the complex problems in modern sciences and engineering when brain, mind, semantics, knowledge, intelligence, and systems become the objects, because none of them is in the domain of any type of numbers. This notion leads to the generic methodology known as mathematical engineering and the ge neric solution coined as denotational mathematics [15, 16, 21, 24, 25, 27, 30, 31]. Denotational mathematics (DM) is a categ ory of novel mathematical structures as function of functions on hyperstructures () [24, 27, 31], beyond those of real numbers () and bits (), in order to formalize rigorous expressions and inferences [15, 16, 21, 24, 25, 27, 30]. In DM, the mathematical entities in hyperstructures, such as ab stract objects, complex relations, neural clusters, big data, information, concepts, semantics, truth, knowledge, behavioral processes, causations, patterns, perceptions, memories, inferences, decisions, intelligence, and systems, are generally modeled as a typed n-tuple [31, 38]. The mathematical operators of DM are denoted by hyper functions, such as relational, reproductive, and compositional operators, on the hyperstructures. The framework of DM is shown in Fig. 1 where its paradigm s are such as concept algebra [18], behavioral process algebra (RTPA) [11, 17], system algebra [38], semantic algebra [28], inference algebra [23], granular algebra [32], big data algebra [36], fuzzy truth algebra [37], and fuzzy probability algebra [34]. This keynote lecture pres ents the DM system for mathematical engineering and its applications in cognitive computing and computational intelligence particularly cognitive robotics. A cognitive robot is an autonomous robot that is capa ble of perception, inference, and learning mimicking the cognitive mechanisms of the brain [22, 35, 37]. Cognitive robots emerge from basic studies in both natural intelligence in b rain/cognitive sciences and artificial/ab stract intelligence in co mputer/intelligence sciences. In cognitive robotics, intelligence is perceived as an ability that transforms information to behavior. Therefore, abstract intelligence (I) [16, 19, 35] is the kernel and formal embodiment of general intelligence shared by both humans and cognitive systems. A reference model of cogn itive robots (RMCR) [22 ] is elaborated for how a cognitive robot is formally modeled at the imperative, autonomic, and cognitive layers fro m the bottom up. It will b e demonstrated that co gnitive robotics is a typical field of contem porary science and engineering where all fundamental theories and solutions are highly dependent on DM. The development on cognitive robots based on mathematical engineering methodologies reveals a wide range of applications of DM in complex system modeling, formal inference, big data processing, knowledge manipulation, machine learning, abstract intelligence, artificial intelligence, brain science, cognitive computers, computational linguistics, and computational intelligence.