Show understanding of how graphs can be used to aid Artificial Intelligence (AI)

AI and Graphs – Cambridge International AS & A Level Computer Science (9618)

Learning Outcome

Show understanding of how graphs can be used to aid Artificial Intelligence (AI) and describe the main machine‑learning techniques that are part of the A‑Level syllabus.

Syllabus‑Map Overview

1. Graphs – The Foundation of Many AI Problems

1.1 Why Graphs are Central to AI

• AI problems can often be expressed as states and the transitions that move between them.
• These naturally form a graph \(G=(V,E)\):

  • \(V\) – vertices = possible states (board positions, robot poses, map locations, etc.)
  • \(E\) – edges = actions or moves that change one state into another.

• Graph models enable systematic exploration, optimisation and reasoning about solutions.

1.2 Graph Representations (exam focus)

2. Graph‑Based Search Algorithms

2.1 Dijkstra’s Algorithm

• Finds the shortest‑cost path from a single source to every other vertex.
• All edge costs must be non‑negative.
• Typical implementation uses a priority queue.

  • Time complexity: O(|E| log |V|)
  • Space complexity: O(|V|)

• Used when no good heuristic is available (e.g., routing in communication networks).

2.2 A* Search

• Combines Dijkstra’s cost‑to‑come g(n) with a heuristic estimate h(n) of the remaining cost.
• Evaluation function: f(n) = g(n) + h(n)
• Admissible heuristic – never over‑estimates the true cheapest cost.
• Consistent (monotonic) heuristic – satisfies h(n) ≤ c(n,n′) + h(n′) for every edge; guarantees each node is expanded at most once.
• Common heuristics:

  • Euclidean distance – straight‑line distance on a grid.
  • Manhattan distance – sum of horizontal and vertical moves.
  • Domain‑specific counts (e.g., minimum moves for a chess piece).

• Complexity depends on heuristic quality:

  • Perfect heuristic → behaves like Dijkstra (explores many nodes).
  • Weak heuristic → approaches breadth‑first search.

2.3 Comparison at a Glance

3. Graph‑Based AI Applications

4. Machine Learning – Enabling AI to Learn from Data

4.1 What is Machine Learning?

ML is a sub‑field of AI that allows a computer to improve its performance on a task through experience (data) rather than by being explicitly programmed.

4.2 Categories (syllabus focus)

5. Artificial Neural Networks (ANNs)

5.1 Basic Structure

• Layers: input, one or more hidden, and output.
• Neurons (nodes) receive weighted inputs, add a bias, and apply an activation function (e.g., sigmoid, ReLU).
• Connections have weights \(w\) that determine signal strength.

5.2 Training – Back‑Propagation (high‑level)

1. Forward pass: compute outputs from the inputs.
2. Compute error: compare with target (e.g., mean‑squared error for regression, cross‑entropy for classification).
3. Backward pass: propagate the error backwards to obtain the gradient \(\frac{\partial E}{\partial w}\) for each weight.
4. Weight update: \(w \leftarrow w - \eta \frac{\partial E}{\partial w}\) where \(\eta\) is the learning rate.
5. Repeat over many epochs until the error stabilises.

5.3 Key Points for the Exam

• Learning rate controls step size – too large → divergence, too small → slow learning.
• Training proceeds by gradient descent; it may get stuck in local minima.
• Only a conceptual description is required – you do not need to derive the full gradient formulas.

6. Deep Learning (DL)

6.1 What Makes a Network “Deep”?

• Contains several hidden layers (often > 5). Each layer learns increasingly abstract features (edges → shapes → objects).
• Requires large training sets and substantial computational resources (GPUs) to train effectively.

6.2 Common Deep Architectures (high‑level)

6.3 Why DL Matters for AI (exam focus)

• Automates feature extraction – no need for hand‑crafted descriptors.
• Achieves state‑of‑the‑art performance in vision, natural language processing and game playing (e.g., AlphaGo).

7. Reinforcement Learning (RL)

7.1 Core Idea

An agent interacts with an environment, receives a reward after each action, and learns a policy that maximises the cumulative reward.

7.2 Key Concepts (syllabus level)

• State (s) – description of the current situation.
• Action (a) – move the agent can take.
• Policy (π) – rule mapping states to actions.
• Reward (r) – immediate feedback.
• Value function (V) / Q‑function – expected total future reward from a state (or state‑action pair).

7.3 Typical Algorithms (overview)

7.4 Link to Graphs

When the set of states and actions is known, they form a directed graph. Planning algorithms such as Dijkstra or A* can be applied if a model of the environment is available. RL, on the other hand, learns the optimal policy when the model is unknown.

8. Regression – Predicting Continuous Values

8.1 Linear Regression (core syllabus idea)

• Model: \(y = \mathbf{w}\cdot\mathbf{x} + b\) where \(\mathbf{x}\) is the input vector.
• Training finds \(\mathbf{w}\) and \(b\) that minimise the sum of squared errors (SSE):

  \[

  \text{SSE} = \sum{i=1}^{n} (yi - \hat{y}_i)^2

  \]

• Used for simple forecasting (e.g., predicting temperature from time of day).

8.2 Non‑Linear Regression (brief)

If data are not well‑fit by a straight line, polynomial or kernel‑based methods can be used. The syllabus expects only the idea that a more complex model can reduce error.

9. Ethical & Societal Considerations in AI

• Bias & fairness – training data may reflect societal prejudices; models can amplify them.
• Privacy – AI systems often require large personal datasets; handling must comply with data‑protection laws.
• Transparency & accountability – black‑box models (deep nets) can be hard to interpret; important for safety‑critical applications.
• Environmental impact – training large deep models consumes significant energy.

10. Summary of Key Points

• Graphs model AI problems as states (vertices) and actions (edges), enabling systematic search.
• Dijkstra finds optimal paths without a heuristic; A* adds an admissible (and preferably consistent) heuristic to guide the search and usually expands fewer nodes.
• Machine learning lets computers improve from data: supervised learning uses labelled examples; unsupervised learning discovers hidden structure.
• Artificial neural networks consist of layers of weighted connections; training uses back‑propagation to reduce prediction error.
• Deep learning extends neural networks with many hidden layers, achieving top performance in vision, language and game playing.
• Reinforcement learning trains agents to maximise cumulative reward; the state‑action space can be viewed as a graph for planning.
• Linear regression fits a straight line to data by minimising the sum of squared errors; more complex regression models handle non‑linear patterns.
• Ethical issues such as bias, privacy and transparency must be considered when deploying AI systems.

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