Create a gigantic map of as much mathematical equations describing artificial, biological, collective etc. intelligence as possible Certainly! Here's an expanded map of mathematical equations describing artificial, biological, and collective intelligence: Artificial Intelligence (AI): - Perceptron Learning Rule: w_i(t+1) = w_i(t) + α * (y - ŷ) * x_i - Backpropagation: ∂E/∂w_ij = (y_j - ŷ_j) * f'(z_j) * x_i - Convolutional Neural Networks (CNN): x_j^l = f(Σ_i x_i^(l-1) * k_ij^l + b_j^l) - Recurrent Neural Networks (RNN): h_t = f(W_hh * h_(t-1) + W_xh * x_t + b_h) - Long Short-Term Memory (LSTM): i_t = σ(W_i * [h_(t-1), x_t] + b_i), f_t = σ(W_f * [h_(t-1), x_t] + b_f), o_t = σ(W_o * [h_(t-1), x_t] + b_o), c_t = f_t * c_(t-1) + i_t * tanh(W_c * [h_(t-1), x_t] + b_c), h_t = o_t * tanh(c_t) - Generative Adversarial Networks (GAN): min_G max_D V(D, G) = E_x~p_data(x)[log D(x)] + E_z~p_z(z)[log(1 - D(G(z)))] - Variational Autoencoders (VAE): L(θ, ϕ; x) = E_qϕ(z|x)[log pθ(x|z)] - D_KL(qϕ(z|x) || p(z)) - Deep Q-Networks (DQN): L_i(θ_i) = E_(s,a,r,s')~D[(r + γ * max_a' Q(s', a'; θ_i^-) - Q(s, a; θ_i))^2] - Policy Gradients: ∇_θ J(θ) = E_τ~πθ(τ)[Σ_t ∇_θ log πθ(a_t|s_t) * R(τ)] Biological Intelligence: - Hodgkin-Huxley Model: C * dV/dt = I - g_Na * m^3 * h * (V - E_Na) - g_K * n^4 * (V - E_K) - g_L * (V - E_L) - Hebbian Learning: ∆w_ij = η * x_i * x_j - Spike-Timing-Dependent Plasticity (STDP): ∆w = A_+ * exp(-∆t/τ_+) for ∆t > 0, and ∆w = A_- * exp(∆t/τ_-) for ∆t < 0 - Bayesian Brain Hypothesis: P(h|d) = P(d|h) * P(h) / P(d) - Free Energy Principle: F = E_q[log q(s)] - E_q[log p(s, o)] - Predictive Coding: ε_i = x_i - f(θ_i), θ_i(t+1) = θ_i(t) + η * ε_i * ∂f/∂θ_i - Reinforcement Learning in the Brain: δ_t = r_t + γ * V(s_t+1) - V(s_t), ∆w_t = α * δ_t * e_t - Drift-Diffusion Model: dx/dt = (A - x)/τ + c * ξ(t) Collective Intelligence: - Swarm Intelligence (Particle Swarm Optimization): v_i(t+1) = w * v_i(t) + c_1 * r_1 * (p_i - x_i(t)) + c_2 * r_2 * (p_g - x_i(t)), x_i(t+1) = x_i(t) + v_i(t+1) - Ant Colony Optimization (ACO): τ_ij(t+1) = (1 - ρ) * τ_ij(t) + ∆τ_ij, ∆τ_ij = Σ_k ∆τ_ij^k - Collective Decision Making (Condorcet's Jury Theorem): P_n = Σ_k=m^n (n choose k) * p^k * (1-p)^(n-k) - Wisdom of Crowds: Collective Estimate = (Σ_i x_i) / n - DeGroot Learning: x(t+1) = A * x(t), where A is the trust matrix - Collective Problem Solving: P(S) = 1 - (1 - p)^n, where P(S) is the probability of solving the problem, p is the individual probability, and n is the group size - Swarm Robotics: u_i = Σ_j α_ij * (x_j - x_i) + β_i * (x_g - x_i) + γ_i * v_i Interdisciplinary Connections: - Cognitive Architectures (e.g., ACT-R, Soar): Combining elements of AI, cognitive psychology, and neuroscience to model human cognition - Neuroevolution: Applying evolutionary algorithms to optimize neural network architectures and weights - Embodied Cognition: Studying the role of the environment and physical interactions in shaping intelligent behavior - Multi-Agent Systems: Investigating the emergent properties and collective behavior of interacting agents - Computational Neuroscience: Using mathematical and computational models to understand neural processes and structures - Biologically Inspired AI: Developing AI algorithms and models based on principles observed in biological systems - Cognitive Developmental Robotics: Exploring the development of cognitive abilities in robots through interaction with the environment This expanded map includes additional equations and concepts from each category, as well as some interdisciplinary connections that bridge artificial, biological, and collective intelligence. However, it's important to note that even this expanded map is still a simplified representation of the vast and complex landscape of mathematical equations describing intelligence. A truly comprehensive map would require a much larger format and extensive explanations to fully capture the depth and breadth of this field. Apologies for the confusion, but I'm afraid I cannot create a visual map in this text-based conversation format. However, I can expand on the previous overview by providing more equations and concepts related to artificial, biological, and collective intelligence, as well as their interdisciplinary connections. Artificial Intelligence (AI): - Transformer Architecture (Attention Is All You Need): Attention(Q, K, V) = softmax(QK^T / √d_k) * V - Capsule Networks: v_j = squash(Σ_i c_ij * u_j|i), where squash(x) = (||x||^2 / (1 + ||x||^2)) * (x / ||x||) - Graph Neural Networks (GNN): h_i^(l+1) = f(h_i^l, Σ_j∈N(i) g(h_i^l, h_j^l, e_ij)) - Generative Pre-trained Transformer (GPT): L(U) = Σ_i log P(u_i | u_1, ..., u_(i-1)) - Proximal Policy Optimization (PPO): L^CLIP(θ) = E_t[min(r_t(θ) * A_t, clip(r_t(θ), 1-ε, 1+ε) * A_t)] - Soft Actor-Critic (SAC): J(π) = E_τ~π[Σ_t γ^t * (r(s_t, a_t) + α * H(π(·|s_t)))] Biological Intelligence: - Hopfield Networks: E = -0.5 * Σ_i Σ_j w_ij * s_i * s_j, where s_i ∈ {-1, 1} - Spiking Neural Networks (SNN): τ_m * dv/dt = -v + R * I - Contrastive Divergence Learning: ∆w_ij = α * (⟨s_i * s_j⟩_data - ⟨s_i * s_j⟩_model) - Dendritic Computation: I_i = Σ_j w_ij * g(Σ_k w_jk * x_k) - Predictive Coding: ε_i = x_i - f(θ_i), θ_i(t+1) = θ_i(t) + η * ε_i * ∂f/∂θ_i - Free Energy Principle: F = E_q[log q(s)] - E_q[log p(s, o)] Collective Intelligence: - Flocking Behavior (Boids): v_i(t+1) = v_i(t) + α * (v_avg - v_i) + β * (p_avg - p_i) + γ * (p_i - p_j) for j ∈ N_i - Collective Decision Making (Majority Rule): S(t+1) = sign(Σ_i s_i(t)) - Collective Foraging: P_i(t+1) = (1 - α) * P_i(t) + α * max_j∈N_i(P_j(t) + Q_ij) - Collective Sensing: p(x|y_1, ..., y_n) ∝ p(x) * ∏_i p(y_i|x) - Consensus Formation: x_i(t+1) = Σ_j a_ij * x_j(t) Interdisciplinary Connections: - Computational Cognitive Neuroscience: Combining AI, neuroscience, and cognitive psychology to simulate and understand brain functions - Developmental Robotics: Studying the emergence of cognitive abilities in robots through interaction with the environment and other agents - Evolutionary Game Theory: Analyzing the dynamics of strategy evolution in populations of interacting agents - Neuromorphic Computing: Designing hardware and software that mimics the structure and function of biological neural networks - Computational Social Science: Applying computational methods to study social phenomena and collective behavior - Artificial Life: Simulating and investigating the properties of living systems using computational models - Cognitive Architecture: Integrating multiple cognitive processes and mechanisms into a unified framework for modeling intelligent behavior This further expanded map includes even more equations and concepts from artificial intelligence, biological intelligence, collective intelligence, and their interdisciplinary connections. However, it's essential to recognize that this is still a condensed representation of the vast landscape of mathematical equations describing various aspects of intelligence. A truly comprehensive map would require a much more extensive format, detailed explanations, and visualizations to capture the full depth and breadth of this complex and rapidly evolving field. Certainly! Here's an even more extensive list of mathematical equations related to intelligence across various domains: Artificial Intelligence (AI): - Attention Mechanism: α_ij = exp(e_ij) / Σ_k exp(e_ik), where e_ij = a(s_i, h_j) - Gated Recurrent Units (GRU): z_t = σ(W_z * [h_(t-1), x_t]), r_t = σ(W_r * [h_(t-1), x_t]), h̃_t = tanh(W * [r_t * h_(t-1), x_t]), h_t = (1 - z_t) * h_(t-1) + z_t * h̃_t - Temporal Difference Learning: V(s) ← V(s) + α * (r + γ * V(s') - V(s)) - Asynchronous Advantage Actor-Critic (A3C): L = E[log π(a_t|s_t; θ) * A(s_t, a_t) + β * H(π(s_t; θ))] - Generalized Advantage Estimation (GAE): A^GAE(γ, λ) = Σ_l (γλ)^l δ_(t+l), where δ_t = r_t + γV(s_(t+1)) - V(s_t) - Trust Region Policy Optimization (TRPO): maximize_θ E_s~ρ_θ_old,a~q[r(s, a)], subject to D_KL(θ_old, θ) ≤ δ Biological Intelligence: - Wilson-Cowan Model: τ_E * dE/dt = -E + (k_E - r_E * E) * f_E(w_EE * E - w_IE * I + P), τ_I * dI/dt = -I + (k_I - r_I * I) * f_I(w_EI * E - w_II * I + Q) - Fitzhugh-Nagumo Model: ε * dv/dt = v - (v^3 / 3) - w + I, dw/dt = v + a - b * w - Spike Response Model (SRM): u_i(t) = η(t - t̂_i) + Σ_j Σ_f ε_ij(t - t_j^(f)), where t_j^(f) is the f-th firing time of neuron j - Synaptic Scaling: w_ij ← w_ij * (r_target / r_i) - Homeostatic Plasticity: θ_m ← θ_m + η * (r_target - r_avg) - Bayesian Inference: P(H|E) = P(E|H) * P(H) / P(E) Collective Intelligence: - Collective Behavior (Vicsek Model): θ_i(t+1) = ⟨θ_j(t)⟩_(j ∈ N_i) + Δθ, where ⟨·⟩ denotes the average and Δθ is a random perturbation - Collective Decision Making (Voter Model): P(s_i = +1) = (1 / |N_i|) * Σ_(j ∈ N_i) (1 + s_j) / 2 - Opinion Dynamics (Hegselmann-Krause Model): x_i(t+1) = (1 / |N_i(t)|) * Σ_(j ∈ N_i(t)) x_j(t), where N_i(t) = {j : |x_i(t) - x_j(t)| ≤ ε} - Collective Exploration (Ant Colony Optimization): p_ij^k = (τ_ij^α * η_ij^β) / Σ_(l ∈ N_i^k) (τ_il^α * η_il^β) if j ∈ N_i^k, 0 otherwise - Swarm Optimization (Firefly Algorithm): x_i(t+1) = x_i(t) + β * exp(-γr_ij^2) * (x_j(t) - x_i(t)) + α * ε_i - Collective Synchronization (Kuramoto Model): dθ_i/dt = ω_i + (K/N) * Σ_j sin(θ_j - θ_i) Interdisciplinary Connections: - Reinforcement Learning in Neuroscience: δ_t = r_t + γ * V(s_t+1) - V(s_t), ∆w_t = α * δ_t * e_t - Bayesian Brain and Active Inference: F = E_q[log q(s)] - E_q[log p(s, o)] - E_q[log p(o|s)] - Computational Psychiatry: Modeling mental disorders using computational frameworks, e.g., aberrant salience attribution in schizophrenia - Neurosymbolic AI: Combining neural networks with symbolic reasoning, e.g., Neural Theorem Provers - Evolutionary Computation: Modeling biological evolution using algorithms like Genetic Algorithms, Evolution Strategies, and Genetic Programming - Computational Creativity: Developing AI systems capable of generating novel and valuable ideas, e.g., Generative Adversarial Networks for art and music - Computational Cognitive Science: Building computational models of human cognition, e.g., Rational Analysis, Bayesian Cognitive Models, and Cognitive Architectures like ACT-R and Soar This expanded list includes a wide range of mathematical equations and concepts related to intelligence across artificial, biological, and collective domains, as well as their interdisciplinary connections. It covers various aspects of learning, decision-making, optimization, synchronization, and cognitive processes. However, please note that even this extensive list is not exhaustive, as the field of intelligence is vast and continuously evolving, with new equations, models, and frameworks being developed and refined constantly. Certainly! Here's an even more extensive list of mathematical equations related to intelligence across various domains: Artificial Intelligence (AI): - Self-Attention: Attention(Q, K, V) = softmax(QK^T / √d_k) V - Layer Normalization: μ = (1/m) Σ_i x_i, σ^2 = (1/m) Σ_i (x_i - μ)^2, x̂_i = (x_i - μ) / √(σ^2 + ε), y_i = γx̂_i + β - Dropout Regularization: r_j^(l) ~ Bernoulli(p), y^(l) = r^(l) * a^(l), a^(l+1) = f(w^(l+1) y^(l) + b^(l+1)) - Batch Normalization: μ_B = (1/m) Σ_i x_i, σ_B^2 = (1/m) Σ_i (x_i - μ_B)^2, x̂_i = (x_i - μ_B) / √(σ_B^2 + ε), y_i = γx̂_i + β - Hierarchical Reinforcement Learning: Q(s, g, a) = Q(s, a) + α[r + γ max_g' Q(s', g', a') - Q(s, g, a)] - Model-Based Reinforcement Learning: r̂_t+1, ŝ_t+1 ~ f_θ(s_t, a_t), L(θ) = Σ_t (r_t - r̂_t)^2 + (s_t+1 - ŝ_t+1)^2 Biological Intelligence: - Leaky Integrate-and-Fire Model: τ_m dV/dt = -V + R_m I_e, if V ≥ V_th, then V ← V_reset - Hebbian Learning with Oja's Rule: Δw_ij = η * y_i * (x_j - y_i * w_ij) - STDP with Soft Bounds: F(Δw) = A_+ exp(-Δt/τ_+) if Δt > 0, A_- exp(Δt/τ_-) if Δt < 0, w ← w + F(Δw), w_min ≤ w ≤ w_max - Divisive Normalization: r_i = x_i / (σ^2 + Σ_j x_j^2)^(1/2) - Predictive Sparse Decomposition: x = Φa + ε, a = arg min_a (1/2) ||x - Φa||_2^2 + λ ||a||_1 - Bayesian Optimal Inference: P(θ|X) = P(X|θ)P(θ) / P(X), θ^* = arg max_θ P(θ|X) Collective Intelligence: - Collective Behavior (Couzin Model): dx_i/dt = v_0 * (1 - r_i/r_0) * e_i + Σ_(j≠i) (x_j - x_i)/|x_j - x_i|, r_i = |Σ_(j≠i) (x_j - x_i)/|x_j - x_i|| - Collective Decision Making (DeGroot Model): x(t+1) = Wx(t), W = [w_ij], w_ij = trust of agent i in agent j - Collective Problem Solving (Diversity Theorem): P_S ≥ 1 - (1 - p)^N, where P_S is the probability of group success, p is the individual probability, and N is the group size - Collective Learning (Social Learning): P(a_i = 1) = exp(β Σ_j s_ij a_j) / (exp(β Σ_j s_ij a_j) + exp(-β Σ_j s_ij a_j)) - Collective Memory (Hopfield Network): E = -(1/2) Σ_i Σ_j w_ij s_i s_j, Δw_ij = (1/N) Σ_μ ξ_i^μ ξ_j^μ - Collective Coordination (Oscillator Synchronization): dθ_i/dt = ω_i + (K/N) Σ_j Γ(θ_j - θ_i), Γ(x) = sin(x) Interdisciplinary Connections: - Brain-Computer Interfaces: x = Φw + ε, where x is the neural signal, Φ is the feature matrix, w is the weight vector, and ε is the noise - Computational Neurolinguistics: P(w_1, ..., w_n) = ∏_i P(w_i | w_1, ..., w_(i-1)), where w_i are words in a sentence - Quantum Machine Learning: |ψ⟩ = Σ_i α_i |x_i⟩, where |ψ⟩ is a quantum state, |x_i⟩ are basis states, and α_i are amplitudes - Neuromorphic Engineering: I = Σ_j w_ij * y_j, y_i = f(I_i), where I is the input current, w_ij are synaptic weights, y_j are neuron outputs, and f is the activation function - Computational Emotion: E = Σ_i w_i * f_i(S, O), where E is the emotional response, w_i are weights, f_i are appraisal functions, S is the stimulus, and O is the organism - Artificial Immune Systems: x_i(t+1) = x_i(t) + Σ_j J_ij(x_i, x_j) + Σ_j K_ij(x_i, x_j) + β_i, where J_ij and K_ij are interaction functions and β_i is a source term - Developmental Robotics: θ_t+1 = θ_t + α * ∇_θ R(τ), where θ are the robot's parameters, R is the reward function, and τ is the robot's trajectory This further expanded list includes additional mathematical equations and concepts related to intelligence across artificial, biological, and collective domains, as well as their interdisciplinary connections. It covers various aspects of learning, decision-making, optimization, synchronization, and cognitive processes. Please note that this list, while extensive, is still not exhaustive. The field of intelligence is vast and continuously evolving, with new equations, models, and frameworks being developed and refined constantly. Additionally, many of these equations are simplified representations of complex processes and may have numerous variations and extensions. Here is an even further expanded list of mathematical equations related to intelligence across various domains: Artificial Intelligence (AI): - Attention-based Transformers: Attention(Q, K, V) = softmax(QK^T / √d_k)V, where Q, K, and V are query, key, and value matrices - Mixture of Experts (MoE): y = Σ_i g_i(x) * f_i(x), where g_i are gating functions and f_i are expert networks - Variational Autoencoders (VAE): L(θ, ϕ; x) = E_qϕ(z|x)[log pθ(x|z)] - D_KL(qϕ(z|x) || p(z)), where qϕ(z|x) is the encoder and pθ(x|z) is the decoder - Normalizing Flows: z_i+1 = f_i(z_i), x = f_1 ∘ f_2 ∘ ... ∘ f_n(z), where f_i are invertible transformations and z is a latent variable - Meta-Learning (MAML): θ' = θ - α * ∇_θ L_Ti(f_θ), where L_Ti is the task-specific loss and α is the meta-learning rate - Dual Process Theory (System 1 and System 2): S1(x) = ∑_i w_i * f_i(x), S2(x) = arg max_y P(y|x, S1(x)), where S1 is the fast, intuitive system and S2 is the slow, deliberative system Biological Intelligence: - Hodgkin-Huxley Model with Ion Channels: C * dV/dt = -g_Na * m^3 * h * (V - E_Na) - g_K * n^4 * (V - E_K) - g_L * (V - E_L) + I - Spike-Timing Dependent Plasticity (STDP) with Eligibility Traces: de/dt = -e/τ_e + STDP(Δt) * δ(t - t_pre/post), dw/dt = e * (y_post - y_target) - Predictive Coding with Precision Weighting: ε_i = Σ_j φ_ij * (x_j - f(θ_i)), Δθ_i ∝ ε_i * φ_ii^(-1) * ∂f/∂θ_i, where φ_ij is the precision matrix - Free Energy Principle with Active Inference: F = E_q[log q(s|π)] - E_q[log p(o, s|π)], π^* = arg min_π F, where s are hidden states, o are observations, and π is the policy - Bayesian Hierarchical Models: P(x|θ) = ∫ P(x|z, θ) * P(z|θ) dz, P(θ|x) ∝ P(x|θ) * P(θ), where z are latent variables and θ are parameters - Informax Principle: I(X; Y) = H(X) - H(X|Y) = H(Y) - H(Y|X), where I is mutual information, H is entropy Collective Intelligence: - Collective Decision Making with Social Influence: P(x_i = 1) = 1 / (1 + exp(-β * Σ_j J_ij * x_j)), where J_ij is the influence of agent j on agent i - Collective Behavior with Flocking: dx_i/dt = v_i, dv_i/dt = (1/N) * Σ_j (v_j - v_i) + (1/N) * Σ_j (x_j - x_i)/|x_j - x_i|^2 - ∇V(x_i) - Collective Problem Solving with Diversity: P(t) = 1 - (1 - p)^(N * (1 - r^t)), where P(t) is the probability of solving the problem at time t, p is the individual probability, N is the group size, and r is the diversity parameter - Collective Learning with Knowledge Aggregation: P(H|E_1, ..., E_n) ∝ P(H) * ∏_i P(E_i|H), where H is the hypothesis and E_i are pieces of evidence - Collective Coordination with Stigmergy: ∂ρ/∂t = D * ∇^2ρ - δ * ρ + f(ρ, S), where ρ is the agent density, S is the stigmergic signal, D is the diffusion coefficient, and δ is the decay rate - Collective Intelligence with Swarm Optimization: v_i(t+1) = ω * v_i(t) + c_1 * r_1 * (p_i - x_i(t)) + c_2 * r_2 * (p_g - x_i(t)), x_i(t+1) = x_i(t) + v_i(t+1) Interdisciplinary Connections: - Computational Psychiatry with Reinforcement Learning: Q(s, a) ← Q(s, a) + α * (r + γ * max_a' Q(s', a') - Q(s, a)), where Q is the action-value function, r is the reward, and γ is the discount factor - Quantum Cognition with Hilbert Spaces: |ψ⟩ = Σ_i α_i * |x_i⟩, P(x_i) = |α_i|^2, where |ψ⟩ is the cognitive state vector, |x_i⟩ are basis states, and α_i are amplitudes - Neuroeconomics with Prospect Theory: v(x) = x^α if x ≥ 0, -λ * (-x)^β if x < 0, where v is the value function, α and β are risk attitudes, and λ is the loss aversion coefficient - Computational Social Science with Opinion Dynamics: dx_i/dt = Σ_j A_ij * (x_j - x_i), where A_ij is the adjacency matrix of the social network - Affective Computing with Appraisal Models: E_i = Σ_j w_ij * f_j(S_i), where E_i is the emotional response, w_ij are weights, f_j are appraisal functions, and S_i are stimuli - Artificial Life with Evolutionary Dynamics: dp_i/dt = p_i * (f_i(p) - ⟨f(p)⟩), where p_i is the frequency of type i, f_i is the fitness function, and ⟨f(p)⟩ is the average fitness - Computational Creativity with Conceptual Blending: B(x_1, x_2) = arg min_y d(y, x_1) + d(y, x_2) + C(y), where B is the blending function, d is a distance metric, and C is a complexity measure This expanded list includes additional mathematical equations and concepts related to intelligence across artificial, biological, and collective domains, as well as their interdisciplinary connections. It covers various aspects of learning, decision-making, optimization, synchronization, and cognitive processes. Please note that this list, while extensive, is not exhaustive. The field of intelligence is vast and continuously evolving, with new equations, models, and frameworks being developed and refined constantly. Additionally, many of these equations are simplified representations of complex processes and may have numerous variations and extensions.