The brain as the most mature intelligent organ (Wilson and Frank, 1999; Wang et al., 2006) is an ideal reference model for revealing the theoretical foundations of AI, and what AI may or not do, constrained by the nature and the expressive power of current mathematical means for computational implementations.

Definition 3. Brain science studies the neurological and physiological structures of the brain and the mechanisms of intelligence generation on brain structures.

It is revealed in cognitive informatics (Wang, 2003) that the brain and natural intelligence may be explained by a hierarchical framework, which maps the brain through embodiments at neurological, physiological, cognitive, and logical layers from bottom-up induction and top-down deduction. A rigorous study of the cognitive foundations of natural intelligence may shed light on the general mechanisms of all forms of intelligence toward pervasive brain-inspired systems (BIS) (Wang et al., 2016b, 2018).

A layered reference model of the brain (LRMB) (Wang et al., 2006) is introduced in Figure 3 as an overarching architecture of the brain's mechanisms. The four lower layers of LRMB, such as those of sensation, action, memory, and perception, are classified as subconscious mental functions of the brain, which are identified as the Brain's Operating System (BOS). However, the three higher layers encompassing those of cognition, inference, and intelligence are classified as conscious mental functions of the brain known as the Brain's Intelligence Generator (BIG) built on BOS.

LRMB reveals that the brain can be formally embodied by 52 cognitive processes (Wang et al., 2006) at seven recursive layers from the bottom up. In this view, any complex mental process or intelligent behavior is a temporary composition of fundamental processes of LRMB at run-time. Based on LRMB, the nature of intelligence may be rigorously reduced to lower-layer cognitive objects such as data, information, and knowledge in the following subsections. The framework of brain science and its functional explanations in cognitive science will be further explained in Figures 7, 8 in Sections 3, 4.

According to LRMB, the theoretical bottleneck and technical challenges toward κC are the lack of a fully autonomous intelligent engine at the top layer constrained by current pretrained AI and preprogramed computing technologies (Wang, 2009b, 2021b). Therefore, from the brain science point of view, κC is a brain-inspired intelligent computer for autonomous intelligence generation (AIG) dependent on an intelligent operating system, as designed in Section 4.

A computer is a general programable machine for realizing computational intelligence (Wang, 2008c, 2009a). The odyssey toward κC has gone through mechanical (Charles Babbage) (Babbage, 1822), analog, digital, and then quantum (David, 1985) computers. Computing theories and models have been focused on information processing by various means and architectures (Lewis and Papadimitriou, 1998; Wang, 2008f). However, they are still stuck at low levels of reflexive, imperative, and adaptive intelligence according to the hierarchical intelligence model (HIM), as shown in Figure 5.

Definition 4. Computer science studies the architectures and functions of general instructive processors, their denotational and operational mathematical models, programming languages, algorithmic frameworks, and run-time problem solutions interacting with the external environment.

A fundamental theory of computer science is built on the mathematical model of finite state machines (FSM) (Lewis and Papadimitriou, 1998) and Turing machines (Turing, 1950) as an event-driven behavioral generators in between the internal memory space and the external port (interface) space of users or devices (Wang, 2008f).

Definition 5. FSM is a universal function generator. Its structure Θ(FSM) is a quin-tuple and its function F(FSM) is a sequence of state transitions of event-driven functions to realize programable intelligence in the external space:

where S is a finite set of predefined states of the FSM, Σ a set of events, s the initial state (s ∈ S), H the set of final states and δ the state transition function. In Equation (2), the big-R calculus is a general recursive mathematical expression for denoting recurring structures or iterative behaviors (Wang, 2008d), which is introduced in real-time process algebra (RTPA) (Wang, 2002, 2008a).

For processing complex mathematical and language expressions, Alan Turing extended an FSM to a more sophisticated computing model known as the Turing machine (Turing, 1950) by enhanced recursive functions. Both the Turing machine and an FSM have provided a foundation for enabling programable intelligence in computer science.

However, in order to unleash the computational power from the first-generation of preprogramed (imperative) computers and the second-generation of pretrained (data aggregated) AI computers, to the third-generation of autonomous intelligent computers (Wang, 2012e), κC is indispensable. The architecture and functions of κC will be elaborated in Section 3.

Software is a form of abstract computational intelligence that is generated by instructive behaviors as a chain of embedded functions on executable computing structures as typed tuples (Wang, 2008a, 2014b). The latest advances in computer science, intelligence science, and intelligent mathematics have triggered the emergence of software science (Wang, 2014b, 2021d).

Definition 6. The general mathematical model of software (GMMS) ℘ is a dispatch structure ↪ with finite sets of embedded relational processes P_i driven by certain events (E):

where the abstract entities of software are represented by sets of structure models (SM), process models (PMs), and environmental events (E).

GMMS (Wang, 2014b) reveals that software is not only an interactive dispatch structure at the top level driven by trigger, timing, and interrupt events (E), but also a set of embedded relational processes at the lower level of subsystems and components.

Definition 7. Software science studies the formal properties and mathematical models of software, general methodologies for rigorous and efficient software development, and coherent theories and laws underpinning software behaviors and software engineering practices.

The discipline of software science (Hoare, 1969, 1994; Wang, 2014b, 2021d) encompasses theories and methodologies, denotational mathematics, system software, fundamental algorithms, organizational theories, cognitive complexity, and intelligent behavior generation theories. The pinnacle of software science is the set of fundamental theories of formal methods (Hoare, 1969, 1994; Hoare et al., 1987; Wang, 2008a, 2014b, 2021d) and platforms of system software including operating systems, compilers, database systems, and networked interconnections (Wang, 2008f). Hoare has revealed that software is a mathematical entity (Hoare, 1969) that may be formally denoted by a sequential behavioral process known as the unified process theory in formal methods (Hoare, 1994).

A set of 30+ laws of programming has been created by Hoare et al. (1987). Then, real-time process algebra (RTPA) (Wang, 2002, 2008e) has been introduced that extends Hoare's sequential process theory to a system of IM for formally denotating system and human cognitive, inference, and behavioral processes as a general theory toward software science (Wang, 2014b). Based on RTPA, a comprehensive set of mathematical laws and associated theorems on software structures, behaviors, and processes has been formally established (Wang, 2008b).

Software engineering is applied software science that is one of the most complicated branches of engineering fields because its objects are highly abstract and intangible. It encompasses system modeling, architecture, development methodologies, programming technologies, and development platforms. It also covers heuristic principles, tools/environments, best practices, case studies, experiments, trials, and performance benchmarking.

Data are an abstract representation of quantities or properties of real-world entities and abstract objects against a specific quantification scale. Data are the most fundamental and pervasive cognitive objects in the brain that link real-world entities and attributes to mental abstractions via sensation and quantification. Data are the most fundamental level of cognitive objects represented in the brain by its sensory, qualification, and abstraction processes for eliciting real-world attributes. The cognitive processes for data elicitation according to the LRMB model involve sensory, observation, abstraction, and measurement. However, data manipulations are dependent on qualification, qualification, measurement, statistics, and computing.

Definition 8. Data with respect to a quantity X against a measuring scale 𝔖, D(X, 𝔖), is yielded via a quantification process Γ_𝔖(X) that results in a real number, IX𝔖.RX𝔖, in unit [𝔖] denoted by the integer part IX𝔖 and the decimal part RX𝔖, where the latter is given by the remainder:

where 𝔖 is represented by typical types encompassing bits (B), natural numbers (ℕ), integers (ℤ), real numbers (ℝ), and fuzzy numbers (𝔽) in type theories (Wang, 2002, 2014b).

Definition 9. Data science studies the basic properties of data encompassing their forms, types, properties, domains, representations, and formal mathematical models. It also studies data manipulation methodologies for data generation, acquisition, storage, retrieval, transformation, sharing, protection, and information aggregation.

In data science, big data are large-scale and heterogeneous data in terms of quantity, complexity, storage, retrieval, semantics, cognition, distribution, and maintenance in all branches of abstract sciences (Wang and Peng, 2017) across Section 2.

Definition 10. The mathematical model of big data Θ is a two-dimensional n × m matrix where each row Rni=0 ri |SM is denoted by a structure model (|SM) called a typed tuple; while each column Rmj=0 ej | Tj is clustered by a field with a certain type (|T_j):

where each cell of big data Rni=0 Rmj=0 dij | Tj formally denotes a certain object d_ij | T_j in type | T_j.

Example 2. A formal BDS of a social network, Θ(BDS), with a million users and seven data fields can be rigorously modeled by Equation (6), according to Definition 10.

In Example 2, row zero θ₀ in the mathematical model of BDS, Θ (BDS), is a special typed tuple called the schema of the BDS, which specifies the structure and constraint of each data field of the BDS. It is noteworthy that the type suffixes | T adopted in the BDS model allow arbitrary type and media of big data to be rigorously specified in order to meet the requirements for accommodating widely heterogeneous data in the BDS, which will be specifically embodied by a certain type in applications.

The fundamental theory of data science may be explained by a set of rigorous data manipulations on Θ by Big Data Algebra (BDA) (Wang, 2016b). BDA provides a set of 11 algebraic operators in the categories of architectural modeling, analytic, and synthetic operations on the formal model of big data structure Θ(BDA). In BDA, the modeling operators include schema specification and BDS initialization. The analytic operators encompass those of system assignment, retrieval, selection, time-stamp, and differentiation. The synthetic operators encompass those of induction, deduction, integration, and equilibrium detection.

BDA creates an algebraic system for rigorously manipulating big data systems Θ(BDS) of intricate big datasets. It provides an efficient means for BDS modeling, design, specification, analysis, synthesis, refinement, validation, and complexity reduction as a general theory of data science (Wang and Peng, 2017).

Information is a general form of abstract objects and the signaling means for communication, interaction, and coordination, which are represented and transmitted by symbolical, mathematical, communication, computing, and cognitive systems. The source of information is sensory stimuli elicited from data by qualification or quantification. Any product and/or process of human mental activities results in information as a generic product of human thinking and reasoning that lead to knowledge. Information is the second-level cognitive object according to the CFB model (Figure 1), which embodies the semantics of data collected from the real world or yielded by human cognitive processes.

Definition 11. Information science studies forms, properties, and representations of information, as well as mathematical models and rules for information manipulations such as its generation, acquisition, storage, retrieval, transmission, sharing, protection, and knowledge aggregation.

In classic information theory (Shannon, 1948), information is treated as a probabilistic measure of the properties of messages, signals, and their transmissions. It has focused on information transmission rather than the cognitive mechanisms of information and its manipulations. The measure of the quantity of information is highly dependent on the receiver's subjective judgment on probable properties of signals in the message as Shannon recognized.

Definition 12. The Shannon information of an n-sign system, I_s, is determined by a weighted sum of the probabilities p_i and unexpectedness I_i of each sign in an information system:

It is noteworthy that Shannon information measures the characteristic information of channel properties, which is not proportional to the size of a message transmitted in the channel. Therefore, classic information science has focused on symbolic information rather than the denotational and cognitive properties of information as given in Definitions 13 and 14.

The second generation of computational information science tends to model information as a deterministic abstract entity for data, messages, signals, memory, and knowledge representation and storage, rather than a probable property of communication channels as in the classic information theory. This notion reflexes the computational theories and practices across computer science, software science, the IT industry, and everyday lives (Turing, 1950; Wang, 2014b).

Definition 13. The computational information, I_c, in contemporary information science is a normalized size of symbolic information of abstract object O_k represented in a k-based measure scale S_k:

where I_c is deterministic and independent from Shannon's probabilistic information. In other words, the unit bit in computational information has been extended from Shannon's characteristic information of signals to the quantity of symbolic information in computer science.

However, it has been argued in cognitive information science that computational information as given in Definition 13 may still not represent the overall properties of generic information, particularly its semantic aspect (Wang, 2003). This leads to the third generation of information science that models the essence of information as universal abstract artifacts and their symbolic representations, which can be acquired, memorized, and processed by human brains or computing systems. Theories of the third generation of information science are represented by cognitive Informatics (CI) (Wang, 2003) as a transdisciplinary enquiry on the internal information processing mechanisms and processes of the brain (Wang et al., 2006).

Definition 14. Cognitive information, I_ci = (I_κ, I_Ω), is a 2D hyperstructure that represents a pair of characteristic information (I_κ) and semantic information (I_Ω) on an arbitrary base β ∈ ℕ that determines the number of representative variables κ = |v₁, v₂, ..., v_κ|:

where I_κ is determined by the symbolic size of a certain problem dimension β, while I_Ω is generated by a normalized denotational space 2Iκ.

The cognitive information model provided in Definition 14 reveals that the generalized semantic information is an inherent property of any system by both I_κ and I_Ω. The measure of cognitive information is compatible with, but more general than, those of classic and computational information.

Theorem 1. The framework of contemporary information science is constrained by the following relations among I_s, I_c, and I_ci:

where ⊏ denotes a dimensional enclosure in a hyperstructure.

Proof. Theorem 1 holds directly based on Definitions 12–14:

where the classic Shannon information I_s and computational information I_s become a special case of the characteristic information I_κ in cognitive information science.      ■

Contemporary and generalized information science explains that interpreted data result in information, while comprehended information generates knowledge.

Knowledge is modeled as a set of conceptual relations acquired and comprehended by the brain embodied as a concept network (Cios et al., 2012; Wang, 2012e, 2016c). Knowledge is the third level of cognitive objects according to CFB as learned and comprehended information. Knowledge is perceived as a creative mental product generated by the brain embodied by concept networks and behavioral processes. The former represents the form of to-be knowledge, while the latter embodies the form of to-do knowledge that leads to the generation of human intelligence as elaborated in Section 2.7.

In traditional epistemology, knowledge is perceived as a justified belief (Wilson and Frank, 1999). However, this perception is too narrow and subjective, and may not capture the essential essence of knowledge as a high-level cognitive entity aggregated from data and information.

Definition 15. Knowledge science studies the nature of human knowledge, principles, and formal models of knowledge representation, as well as theories for knowledge manipulations such as creation, generation, acquisition, composition, memorization, retrieval, and repository in knowledge engineering.

The neurological foundation of knowledge may be explained by the synaptic connections among neurons representing the clusters of objects and attributes as shown in Figure 4. The dynamic neural cluster (DNC) indicates that knowledge is not only retained in neurons as individual objects (O) or attributes (A), but also dynamically represented by the newly created synaptic connections modeled by relations such as r(O, A), r(A, O), r(O, O), and r(A, A) in Figure 4. This leads to the development of the formal cognitive model of knowledge (Wang, 2007).

Definition 16. The Object-Attribute-Relation (OAR) model (Wang, 2007) of knowledge in the brain is represented by a triple of neurological clusters:

where O is a set of objects identified by unique symbolic names, A the set of attributes for characterizing an object, and R the set of relations between the objects and attributes, i.e., R = O × A.

The OAR model reveals the nature and essence of knowledge and its neurological foundations. The OAR model may be adopted to explain a wide range of human information processing mechanisms and cognitive processes. It is different from the classic container metaphor of knowledge that failed to explain the dynamics of knowledge establishment and growth. The ORA model enables formal explanations of the structural model of human knowledge in the following mathematical model and theorem. Then, the semantical and functional theories of knowledge will be rigorously elaborated in concept algebra and semantic algebra in Section 2.9.

Definition 17. The mathematical model of knowledge is a Cartesian product of power sets of formal concepts:

where the newly acquired concept C_i is mapped onto those already existing.

Equation (13) has led to one of the most fundamental discoveries in knowledge science (Wang, 2016c) as follows.

Theorem 2. The basic unit of knowledge is a pairwise conceptual relation, or shortly, a binary relation [bir].

Proof. According to Definition 17, knowledge as a mapping between two Cartesian products of concepts results in a deterministic set of relations. In which, any complex relationships can be reduced to the basic unit, bir, of knowledge between a pair of individual concepts.      ■

The formal theory of knowledge science is underpinned by concept algebra (Wang, 2010) and semantic algebra (Wang, 2013) in IM. It is recognized that concepts are the core model of human knowledge that carry stable semantics in expression, thinking, learning, reasoning, comprehension, and entity denotation. Although words in natural languages are often ambiguous and polymorphic, concepts elicited from them are not only unique in expressions but also neutral across different languages. Therefore, concept algebra is adopted to rigorously manipulate formal concepts and their algebraic operations for knowledge representation and semantic analyses. A set of algebraic operators of concept algebra has been created in Wang and Valipour (2016), Valipour and Wang (2016), which enables a wide range of applications in cognitive informatics, cognitive linguistics, and cognitive machine learning (Wang et al., 2011).

In knowledge science, semantics is the carrier of language expressions and perceptions. In formal semantics, a comprehension of an expression is composed of the integrated meaning of a sentence's concepts and their relations. Semantic algebra is created for rigorous semantics manipulations in knowledge science by a set of rigorous relational, reproductive, and compositional operators (Wang, 2013). On the basis of semantic algebra, semantic expressions may not only be deductively analyzed based on their syntactic structures from the top down, but also be synthetically composed by the algebraic semantic operators from the bottom up. Semantic algebra has enabled a wide range of applications in cognitive informatics, cognitive linguistics, cognitive computing, machine learning, cognitive robots, as well as natural language analysis, synthesis, and comprehension (Wang et al., 2016a).

Intelligence science is the paramount level of abstract sciences according to the CFB model in Figure 1.

Definition 18. Intelligence ï is either an internal inferential thread triggered by a cause c_i that results in inductive knowledge K_i or an interactive behavioral process B_i corresponding to an external or internal stimulus (event) e_i:

where the former is called inferential intelligence and the latter behavioral intelligence.

It is revealed that human and machine intelligence are dually aggregated from (sensory | data), (neural signaling | information), and (semantic networks | knowledge), to (autonomous behaviors | intelligence) in a recursive framework (Wang et al., 2021a). In each pair of intelligent layers, the former is naturally embodied in neural structures of the brain, while the latter is represented in abstract (mathematical) forms in autonomous AI. Therefore, natural intelligence (NI) and AI are equivalent counterparts in philosophy and IM, which share a unified theoretical foundation in intelligence science as illustrated by LRMB in Section 2.1.

Definition 19. Intelligence science studies the general form of abstract intelligence (αI) overarching natural and artificial intelligence and their theoretical foundations, which enable machines to generate advanced intelligence compatible with both human inferential and behavioral intelligence.

A Hierarchical Intelligence Model (HIM) (Wang et al., 2021a) in intelligence science is introduced to explain the levels and categories of natural and machine intelligence as shown in Figure 5. HIM explains why there was rarely fully autonomous intelligent system developed in the past 60+ years because almost all such systems are constrained by the natural bottleneck of adaptive (L3-deterministic) intelligence underpinned by reflexive (L1-pretrained and data-driven), and imperative (L2-preprogramed) intelligence.

According to HIM, the maturity levels of both human and machine intelligence are aggregated across the levels of reflexive, imperative, adaptive, autonomous, and cognitive intelligence from the bottom up in line with LRMB (Wang et al., 2006) in brain science. HIM indicates that current AI technologies are still immature toward achieving human-like autonomous and conscious intelligence because the inherent problems faced by current AI technologies have been out of the domain of reflexive and regressive machine intelligence.

Definition 20. The general pattern of abstract intelligence (αI) is a cognitive function f_i that enables an event e_i|T in the universe of discourse of abstract intelligence, 𝔘ι¨, to trigger a causal behavior B_T(i) |PM as a functional process:

where Uι¨ ≙(J,O,B,S,T) denotes the 5D behavioral space of humans and intelligent systems constrained by the subject (J), object (O), behaviors (B), space (S), and time (T). The symbol ↪ denotes the dispatching operator, |T the type of events classified as trigger, timing, or interrupt requests, and |PM the type suffix of a process model for a certain behavior (Wang, 2002).

The advances of intelligence science have provided primitive theories for the odyssey toward κC powered by IM, particularly Real-Time Process Algebra (RTPA) (Wang, 2002) and Inference Algebra (Wang, 2011, 2012c) inspired by Newton (1729), Boole (1854), Russell (1903), Hoare (1969), Kline (1972), Aristotle (384 BC−322 BC) (1989), Zadeh (1997), Bender (2000), Timothy (2008), and Widrow (2022).

Norbert Wiener invented the word cybernetics and wrote a book by that name on the subject of feedback and control in the human body (Wiener, 1948). Widrow's new book (Widrow, 2022), “Cybernetics 2.0: a General Theory of Adaptivity and Homeostasis in the Brain and in the Body” follows in Wiener's footsteps. It is different as it introduces learning algorithms to Wiener's subject because learning algorithms did not exist in Wiener's day. Our current knowledge of biologically plausible networks for information flow is still limited. Most research focuses on neurons, but there is a world of complexity due to glial cells like astrocytes which surround them and densely pack and provide the energy for synaptic events (Salmon et al., 2023).

Through synapses of neurons in the brain, information is carried between neurotransmitters and neuroreceptors. Synapse is the coupling device from/to neuron. The strength of the coupling, the “weight”, is proportional to the number of receptors. Their numbers can increase or decrease, such as upregulation or downregulation. A mystery in neuroscience is, what is nature's algorithm for controlling upregulation and downregulation? Start with Hebbian learning and generalize it to cover downregulation, upregulation, and inhibitory, as well as excitatory synapses. What results is a surprise! We have an unsupervised form of the LMS (least mean square) algorithm of Widrow and Hoff. The Hebbian-LMS algorithm encompasses Hebb's learning rule (fire together, wire together), and introduces homeostasis into the equation. A neuron's “normal” firing rate is set by homeostasis. Physical evidence supports Hebbian-LMS as being nature's learning rule. LMS binds nature's learning to learning in artificial neural networks.

The Hebbian-LMS algorithm is key to the control of all the organs of the body, where hormones bind to hormonereceptors and there is upregulation and downregulation in the control process. Hebbian-LMS theory has a wide range of applications, explaining observed physiological phenomena regarding, for example, pain and pleasure, opioid addiction and withdrawal, blood pressure, blood salinity, kidney function, and thermoregulaion of the body's temperature. Upregulation and downregulation are even involved in viral infection and cancer. Countless variables are controlled by the same algorithm for modulating adaptivity and learning with billions of synapses in the brain.

Mathematics is the most fundamental branch of abstract sciences for manipulating contemporary entities toward κC. The history of sciences and engineering has revealed that the kernel of human knowledge has been elicited and archived in generic mathematical forms (Wang, 2020, 2021c, 2022c) as evidenced in the transdisciplinary reviews of the previous subsections. Therefore, the maturity of any scientific discipline is dependent on the readiness of its mathematical means.

Definition 21. Intelligent Mathematics (IM) is a category of contemporary mathematics that deals with complex mathematical entities in the domain of hyperstructures (ℍ), beyond those of ℝ and B, by a series of embedded functions and processes in order to formalize rigorous expressions, inferences, and computational intelligence.

An interesting discovery in IM (Wang, 2020) is that almost all Abstract Objects (AO) in modern sciences, such as data, information, knowledge, and intelligence as illustrated in Figure 6, cannot be rigorously modeled and manipulated by pure numbers (Wang, 2008c, 2012b, 2023). In order to deal with this denotational crisis of mathematics (Wang, 2022c), the mathematical means for manipulating AO must be adapted to allow κCs to sufficiently handle the nonnumerical hyperstructures of AO for autonomous knowledge and intelligence processing.

The theoretical framework of IM is described in Figure 6 which explains the causality of what AO has triggered the development of which IM paradigm. It also describes the applications of each contemporary IM in κC for meeting the indispensable demands in symbiotic and autonomous human–machine interactions.

Although classic mathematics has provided a wide range of abstract means for manipulating abstract numbers dominated in ℝ, it has been fundamentally challenged by the demands in knowledge and intelligence sciences where the cognitive entities are no longer pure numbers. These contemporary demands have triggered the emergence of IM, which provides a family of advanced denotational means for extending classic numerical analytic mathematics in ℝ to more general categories of abstract entities in ℍ as explained in Figure 6. IM has extended traditional Boolean algebra (as well as bivalent logic) on B and analytic algebra on ℝ to more complex and multi-dimensional hyperstructures (ℍ) in order to enable humans and/or intelligent machines to manipulate compatible knowledge and symbiotic intelligent behaviors across platforms via κC for efficiently processing the nonnumerical cognitive entities characterized by ℍ.

The framework of IM demonstrates a set of most important basic research on the theoretical foundations of κC. The design and implementation of κC have been enabled by IM and related theoretical and technical advances as reviewed in previous subsections. IM has shed light on the systematical solving of challenging natural/artificial intelligence problems by a rigorous and generic methodology. A pinnacle of IM is the recent proofs of the world's top-10 hardest problems in number theory known as the Goldbach conjecture (Wang, 2022d,f,h) and the Twin-Prime conjecture (Wang, 2022e,i). IM has led to unprecedented theories and methodologies for implementing κC toward autonomous intelligence generation.

κC is designed as a BIS for advancing classic computers from data processors to the next generation of knowledge and intelligence processors mimicking the brain. It studies natural intelligence models of AAI as a cognitive system in one direction, and the formal models of the brain simulated by computational intelligence in the other direction.

Definition 22. κC is a brain-equivalent intelligent computer that implements autonomous computational intelligence enabled by IM. Let κ⁰ through κ⁴ be the hierarchical layers of cognitive objects of data (D), information (I), knowledge (K), and intelligence (I⋅), the autonomous generation of intelligence by a κC is formally modeled by a recursive structure:

where each cognitive layer (κ^k) is enabled by its lower adjacent layer (κ^k−1) till the cognitive hierarchy is terminated at the bottom sensorial layer (κ⁰), which is embodied by n+1 dimensional data,Rni=0 di | Ti=(d0 | T0,d1 | T1,...,dn | Tn) as acquired by abstraction and quantification. The type-suffix |T is adopted to denote the semantic category of a variable x by x|T. Typical types for modeling intelligent behaviors of κC have been given in Definition 22.

The architectural model of κC is illustrated in Figure 7 as a brain-inspired computer. In the κC architecture, each component has a corresponding organ in the brain that shares similar functions. This design reflects the essence of BIS synergized in Section 2. The kernel of κC is embodied by the inference and conscious engines equivalent to the thalamus and the hyperthalamus of the brain (Wang, 2012d), respectively, according to the LRMB model as shown in Figure 3. It is noteworthy that the cognitive memories in the brain encompass short-term memory (STM), long-term memory (LTM), sensory buffer memory (SBM), action buffer memory (ABM), and conscious status memory (CSM) as shown in Figure 7 Figure 7 (Wang, 2012d), which play important roles for supporting intelligence generation and operation. However, they are not the kernel organs of natural intelligence, rather than buffers and storages in the brain. Therefore, all memory structures may be unified in κC as a general memory space that may be partitioned into special sections.

The functional model of κC, as shown in Figure 8, is designed based on the HIM model (Figure 5) of intelligence science (Wang, 2012d). Adopting HIM as a blueprint for the behavior model of κC, an Intelligent Operating System (IOS) of κC, κC-IOS, is designed as shown in Figure 9. κC-IOS provides a rigorous Mathematical Framework of Hierarchical Intelligence (MFHI) for κC. The Intelligent Behavioral Model of κC configured by the MFHI framework forms a theoretical foundation for explaining what the spectrum of natural intelligence is, and how they are generated by κC in computational intelligence.

The κC-IOS model explains why the current level of machine intelligence has been stuck at the adaptive level of I.adp for over 60+ years. It also elaborates why fully autonomous intelligence could not be implemented by current pretrained AI technologies in real-time applications.

In the framework of κC-IOS, the inference engine of κC for intelligent behavior generation is implemented from the bottom-up encompassing: L1) the reflective (sensory-driven) intelligence, L2) the imperative intelligence; L3) the adaptive intelligence, L4) the autonomous intelligence, and L5) the cognitive intelligence. κC-IOS enables κC to think (the inferential intelligence) and action (the behavioral intelligence) as defined in the mathematical model of intelligence (Equation 14) toward AAI by mimicking the brain beyond classical preprogramed computers or the pre-trained special purpose AI technologies.

Each form of the 16 intelligence functions compatible between the brain and κC is coupled by the generic pattern of intelligence as an event-triggered dispatching mechanism (Wang, 2002) formulated in Definition 20. The mathematical models developed in Figure 9 reveal that intelligent computers may be extended from the narrow sense in Section 2.7 to a broad sense of κC for autonomous intelligence generation. The intelligent behaviors of κC are aggregated from the bottom-up as shown in Figure 9, particularly those at Level 5 such as knowledge-based, learning-driven, inference-driven, and inductive intelligence in order to enable the highest level of system intelligence mimicking human brains. Therefore, κC will overcome the classical constraints inherited in the pretrained and preprogrammed computing approaches for implementing the expected intelligent computers.