Understanding AI Search Algorithms and Evaluation Functions

Different search algorithms used in Artificial Intelligence employ various strategies to explore the state space and find a solution path. These strategies are often guided by an evaluation function that determines the priority of expanding a node.

Let's analyze each algorithm and its corresponding evaluation function:

Greedy Best-First Search Evaluation

Greedy best-first search is an informed search algorithm. It expands the node that is estimated to be closest to the goal. This estimation is provided by a heuristic function, denoted as \(h(p)\), which estimates the cost from the current node \(p\) to the nearest goal state. The algorithm's priority is based solely on this heuristic value.

• Algorithm: Greedy best-first search
• Evaluation Function: Minimal \(h(p)\)

Lowest Cost-First Search Evaluation (Uniform Cost Search)

Lowest cost-first search, also known as Uniform Cost Search, is an uninformed search algorithm. It expands the node with the lowest path cost from the start node. The path cost to reach a node \(p\) from the start node is denoted as \(g(p)\). This algorithm prioritizes nodes based on the accumulated cost to reach them.

• Algorithm: Lowest cost-first search
• Evaluation Function: Minimal \(g(p)\)

A* Algorithm Evaluation

The A* algorithm is an informed search algorithm that combines aspects of both uniform cost search and greedy best-first search. It evaluates nodes based on the sum of the cost to reach the node from the start node (\(g(p)\)) and the estimated cost from the node to the goal state (\(h(p)\)). The evaluation function for A* is \(f(p) = g(p) + h(p)\), and the algorithm expands the node with the lowest \(f(p)\) value.

• Algorithm: A* algorithm
• Evaluation Function: Minimal \(g(p) + h(p)\)

Matching List-I with List-II

Based on the evaluation functions discussed above, we can match the algorithms in List-I with the criteria in List-II:

• (a) Greedy best-first: Evaluates based on minimal \(h(p)\). This matches (ii) Minimal \(h(p)\).
• (b) Lowest cost-first: Evaluates based on minimal \(g(p)\). This matches (iii) Minimal cost \(g(p)\).
• (c) A* algorithm: Evaluates based on minimal \(g(p) + h(p)\). This matches (i) Minimal cost \(g(p)\) + \(h(p)\).

Therefore, the correct matching is (a)-(ii), (b)-(iii), (c)-(i).

The final correct option is the one that provides the mapping (a)-(ii); (b)-(iii); (c)-(i).

Revision Table: AI Search Algorithm Evaluation

Additional Information on Search Algorithms

Understanding the evaluation function is key to understanding how these search algorithms behave:

• Completeness: An algorithm is complete if it guarantees finding a solution whenever one exists. Uniform Cost Search is complete (for non-negative step costs). Greedy Best-First is not complete in infinite state spaces or with zero step costs. A* is complete if the heuristic is admissible and the branching factor is finite.
• Optimality: An algorithm is optimal if it guarantees finding the lowest-cost solution. Uniform Cost Search is optimal (for non-negative step costs). Greedy Best-First is not optimal. A* is optimal if the heuristic is admissible.
• Admissible Heuristic: A heuristic \(h(p)\) is admissible if it never overestimates the actual cost to reach the goal from node \(p\). \(h(p) \le \text{actual_cost}(p, \text{goal})\) for all \(p\). Admissibility is crucial for the optimality of A*.
• Consistent Heuristic: A heuristic \(h(p)\) is consistent if for every node \(p\) and every successor \(p'\) of \(p\) generated by action \(a\), \(h(p) \le \text{cost}(p, a, p') + h(p')\). Consistency implies admissibility and is required for A* to work correctly on graphs (preventing revisiting nodes with higher costs).

These properties highlight why the evaluation function choice is critical for the performance, completeness, and optimality of a search algorithm.