Abstract intelligence is the human enquiry of both natural and artificial intelligence at the neural, cognitive, functional, and logical levels reductively from the bottom up. According to the abstract intelligence theory, a cognitive robot is an autonomous robot that is capable of thought, perception, and learning based on the three-level computational intelligence known as the imperative, autonomic, and cognitive intelligence. This paper presents the theoretical foundations of cognitive robots based on the latest advances in abstract intelligence, cognitive informatics, and denotational mathematics. A formal model of intelligence known as the Generic Abstract Intelligence Mode (GAIM) is developed, which provides a foundation to explain the mechanisms of advanced natural intelligence such as thinking, learning, and inference. A set of denotational mathematics is introduced for rigorously modeling and manipulating the behaviors of cognitive robots. A case study on applications of a denotational mathematics, visual semantic algebra (VSA), is presented in architectural and behavioral modeling of cognitive robots based on the theory of abstract intelligence.