Part 3 — AGI; An Advanced Mathematical Perspective

Following up on Part 2, which explored the potential nature of AGI and its comparison with human cognition, this blog delves into the ‘how’ of building the AGI. We’ll examine how AGI can develop the ability to perform economically valuable work, focusing on its mathematical foundations.

Consider this an exploration of the journey towards achieving AGI. The mathematics discussed here is both contemporary and applicable, yet the full spectrum of AGI’s capabilities remains a goal not yet fully realized.

I have written in the past on how mathematical models helps us get closer to AGI.

• The Mathematical Odyssey to Artificial General Intelligence
• Making Machines Think Like Us: Mixing Randomness with Brain-Like Models

Graph Neural Networks (GNNs) in AGI

AGI can leverage sophisticated GNNs, encapsulating complex interactions within graph structures:

where,

• h_v(l+1)​: This represents the feature vector of a node v in the graph at layer l+1. In a GNN, each node has a feature vector that is updated at each layer of the network.
• σ: This denotes a non-linear activation function, like sigmoid or ReLU. It’s applied to the linear combination of feature vectors.
• ∑_u∈N(v)​: This is a summation over all nodes u that are in the neighborhood N(v) of node v. In other words, it aggregates information from the feature vectors of the nodes adjacent to v.
• ReLU: This is the Rectified Linear Unit function, a popular activation function in neural networks. It’s used here to introduce non-linearity and help the network learn complex patterns.
• W_1(l)​ and W_2(l)​: These are weight matrices at layer l. They are learned parameters of the model that transform the feature vectors of the nodes.
• h_u(l)​ and h_v(l)​: These are the feature vectors of the neighboring node u and the current node v, respectively, at layer l.
• b(l): This is the bias vector at layer l, another set of parameters learned by the model.

This equation’s relevance to potential AGI lies in its ability to update and refine a node’s feature vector within a Graph Neural Network (GNN). By gathering and transforming feature vectors from adjacent nodes and implementing a non-linear activation function, the equation enhances each node’s representation at successive layers, guided by the graph’s structure and the characteristics of neighboring nodes.

This methodology enables GNNs to adeptly interpret and leverage the inherent structure of graph-based data. In the context of AGI, this capability is crucial, as it equips AGI systems with the advanced computational tools needed to analyze and understand complex, interconnected data systems, a fundamental aspect of replicating human-like intelligence and adaptability in diverse learning scenarios.

Transformers in AGI:

Transformers aiding to build AGI can utilize complex self-attention mechanism:

where,

• Q,K,V are the query, key, and value matrices, respectively.
• dk​ is the dimension of the keys (and queries), which is used to scale the dot products for stability. The division by dk​​ is particularly important to prevent the dot products from growing too large in magnitude, leading to vanishing gradients during training.

I have written earlier about the relevance of Transformers in AGI:

• Part 1 — Fractional Elliptic Problems and Artificial General Intelligence.
• Part 2 — Fractional Laplacian and Cognitive Modeling

The Mathematical Landscape of Human Cognition

Higher-Order Neural Dynamics Modeling human cognition involves higher-order differential equations that describe neural dynamics:

The provided equations represent a system of nonlinear differential equations, often used in models of neural dynamics and other complex systems. Here’s a detailed explanation of each component:

First Equation — Neuron Dynamics:

• d²v/dt²​: This is the second derivative of the neuron membrane potential v with respect to time. It represents the acceleration of the change in membrane potential, indicating how the rate of change of membrane potential itself changes over time.
• αdt/dv​: This term, where α is a constant, represents a damping force proportional to the first derivative of v (the rate of change of membrane potential). It can be related to the resistive properties of the neuron’s membrane.
• γ(v−v³/3​+w): This represents the driving force on the membrane potential. Here, γ is a scaling factor, v represents the linear component of membrane potential, −v³/3​ introduces nonlinearity (often important for modeling action potentials or spikes in neurons), and w is a variable representing recovery or adaptation mechanisms.

Second Equation — Recovery or Adaptation Dynamics:

• dt/dw​: This is the rate of change of the recovery variable w over time. The recovery variable can represent processes like ion channel recovery or adaptation effects in the neuron.
• 1/γ​(v−δw+β): In this driving force, 1/γ1​ scales the whole expression, v represents the influence of the membrane potential on the recovery process, −δw (with δ being a constant) is a term that typically models the recovery process itself, and β is a constant that can represent external influences or a baseline recovery rate.

Together, these equations form a coupled system describing the interaction between the neuron’s membrane potential and its recovery mechanisms. The nonlinearity in these equations is crucial for capturing the complex behaviors observed in real neurons, such as their ability to generate action potentials (spikes) and exhibit various patterns of activity. This kind of model is widely used in theoretical neuroscience to study neural dynamics and can be a component of larger neural network models.

AGI Algorithms vs. Human Cognitive Algorithms

Enhanced Reinforcement Learning in AGI:

On the path to AGI, models can employ advanced reinforcement learning algorithms like Proximal Policy Optimization (PPO), formulated as:

where,

Objective Function in Reinforcement Learning:

• The equation L(θ) defines the objective function that the PPO algorithm tries to maximize. It is used to update the policy in a way that improves the expected future rewards.

Policy Ratio:

• rt​(θ) represents the ratio of the new policy to the old policy. It’s a way of measuring how much the current policy θ has deviated from the old policy.

Advantage Estimate:

• A^t​ is the advantage estimate at time t. It indicates how much better or worse an action is compared to the policy’s average action. The advantage function helps in reducing the variance of the policy gradient estimate and speeds up learning.

Clipping Mechanism:

• The term clip(rt​(θ),1−ϵ,1+ϵ) is a clipping mechanism that limits the ratio rt​(θ) to the range [1−ϵ,1+ϵ]. This prevents the policy from changing too drastically, which can lead to more stable and reliable learning.

Relation to AGI:

• In AGI, one of the goals is to develop systems that can learn a wide range of tasks in various environments. Reinforcement learning, and specifically algorithms like PPO, are pivotal in this area. They can enable an AGI system to learn optimal behaviors through interaction with the environment, adapting and improving based on feedback (rewards or penalties).
• AGI systems using PPO can continuously update their policies (strategies) to perform tasks more effectively, making them suitable for complex, dynamically changing environments, and closer to achieving human-like adaptability and decision-making.

Integration of GNNs and Transformers

The integration can be mathematically represented by combining the feature transformations of GNNs and the attention mechanisms of Transformers, followed by a nonlinear transformation function:

Here, Λ and Ξ are advanced nonlinear transformation functions that map the outputs of GNNs and Transformers into a unified feature space. ⨁ denotes a high-dimensional fusion operation, such as concatenation or more complex manifold embedding, which combines the distinct representations from GNNs and Transformers into a coherent, integrated representation.

This comprehensive formulation captures the depth and complexity of integrating GNNs and Transformers, leveraging their respective strengths in processing graph-structured data and sequential data, and enabling sophisticated representation learning in AGI systems.

Cognitive Neuroscience and Its Mathematical Underpinnings

Graph Theoretical Models in Neuroscience:

Advanced graph theoretical models in neuroscience use spectral clustering techniques:

• L is the normalized Laplacian matrix.
• D is the degree matrix, a diagonal matrix where each diagonal element Dii represents the degree of node i (the number of edges connected to node i).
• A is the adjacency matrix of the graph, where Aij​ is 1 if there is an edge between nodes i and j, and 0 otherwise.
• D^(1/2) ​ is the matrix obtained by taking the inverse square root of each non-zero element on the diagonal of the degree matrix D.

In this context:

• L is the Laplacian matrix of the graph.
• Tr denotes the trace of a matrix, which is the sum of all the diagonal elements.
• H is the matrix of indicators for the cluster assignments of the nodes in the graph.
• n×k represents the dimension of the matrix H, where n is the number of nodes and k is the number of clusters.
• argmin is the argument of the minimum, indicating that we are looking for the matrix H that minimizes the trace of H^T LH.

Bridging AGI and Human Cognition: A Mathematical Synthesis

Exploring Learning Mechanisms Through Chaos Theory:

AGI’s learning processes can be explored through chaos theory, involving dynamic systems exhibiting sensitive dependence on initial conditions, contrasting with the stochastic nature of human cognitive processes.

Consciousness and Subjective Experience in Mathematical Terms: Theoretical models to describe consciousness in mathematical terms within AGI remain elusive, emphasizing the philosophical and experiential chasm between AGI and human cognition.

Quantum Computing’s Impact on AGI

Quantum algorithms could revolutionize AGI’s computational capabilities:

• Q represents a quantum operation.
• ∣ψ⟩ is a quantum state, often referred to as a ‘ket’ in Dirac notation.
• The summation ∑x​ indicates that we sum over all possible states x.
• e^(iθ)x​ is a complex exponential representing the phase factor for state x, where θx​ is the angle in radians.
• ∣x⟩ is the state vector (ket) corresponding to state x.
• ⟨x∣U∣ψ⟩ represents the inner product (or amplitude) of state x after the unitary operation U has been applied to state ∣ψ⟩, often referred to as a ‘bra’ in Dirac notation. This term captures the probability amplitude of the state x resulting from applying the operation U to ∣ψ⟩.

This equation embodies a quantum system’s evolution and is relevant to the concept of Artificial General Intelligence (AGI) in several advanced and theoretical contexts:

• Quantum Computing and AI: Quantum computing holds potential for revolutionizing the field of AI due to its ability to perform certain computations much faster than classical computers. AGI systems could, in theory, leverage quantum algorithms to solve complex problems more efficiently.
• Superposition and Parallelism: The equation reflects the principle of superposition, where the quantum state ∣ψ⟩ is a combination of all possible states ∣x⟩. For AGI, this could translate into an ability to simultaneously consider and process a vast number of possible solutions or scenarios, a feature that could dramatically enhance problem-solving and decision-making processes.
• Probabilistic Reasoning: The term ⟨x∣U∣ψ⟩ represents the probability amplitude of transitioning to state ∣x⟩ from ∣ψ⟩ under the unitary operation U. This aspect of quantum mechanics can inspire probabilistic models in AGI, which are used to handle uncertainty and make predictions.
• Quantum-enhanced Learning: AGI systems could potentially incorporate quantum-enhanced machine learning algorithms. The quantum operation in the equation suggests an analogy to neural network transformations in AGI, where U represents an operation analogous to a layer in a neural network, and the resulting superposition captures the complex transformations of input data.
• Phase Factors and Interference: The phase factor e^(iθ)x​ can cause constructive or destructive interference, a unique property of quantum systems. In AGI, leveraging such interference could lead to novel ways of filtering and converging upon optimal solutions.
• Exploration of Quantum Cognition: Some researchers hypothesize that human cognition could exploit quantum computational principles. If true, an AGI aspiring to mimic human cognitive processes might need to consider quantum computational frameworks.
• Theoretical Models of Consciousness: There are speculative theories, like Orchestrated Objective Reduction (Orch-OR), suggesting that quantum processes may play a role in consciousness. Though highly controversial and not widely accepted, such theories raise the question of whether a truly conscious AGI would require some form of quantum processing.

While currently, the practical implementation of quantum computing in AGI is largely theoretical and remains a subject of research, the principles underlying the equation provide inspiration for the future direction of AGI development.

Open Discussion

This mathematical exposition of AGI delves into the complexities that set it apart from human cognition. While AGI’s computational models exhibit mathematical sophistication, they fall short of replicating the full spectrum of human consciousness and experiential learning.

We have navigated the topical layers of AGI, touching on the integration of advanced neural architectures and speculating on the quantum frontiers. As we ponder AGI’s trajectory, we must consider the ethical dimensions, the potential for human enhancement, and the philosophical questions surrounding consciousness.

Your perspectives are crucial in shaping this dialogue. What do you foresee for the future of AGI? Share your thoughts and join the conversation as we look ahead to the horizon of artificial general intelligence.