State Space and Expansion Rule

The problem defines a state space starting at state 1. The successor function generates two states, $n+1$ and $n+2$, from state $n$. The goal state is 6. A crucial rule states that 'states in the unexpanded state list are expanded in the ascending order of numbers'. This implies that the algorithm always selects the state with the numerically smallest value from the set of available unexpanded states, regardless of whether it's BFS or DFS.

Additionally, 'previously expanded states are not added to the unexpanded state list', meaning visited states are tracked to avoid cycles and redundant exploration.

BFS State Expansion Analysis

Breadth-First Search (BFS) typically uses a queue. However, the overriding rule here is the ascending order expansion. This forces the frontier (queue) to be treated as a min-priority queue.

Tracing the expansion:

• Start: Frontier = {1}, Expanded = {}
• Expand 1 (smallest): Frontier = {2, 3}, Expanded = {1}
• Expand 2 (smallest): Frontier = {3, 4}, Expanded = {1, 2} (Children of 2 are 3, 4. 3 is in frontier, 4 is new)
• Expand 3 (smallest): Frontier = {4, 5}, Expanded = {1, 2, 3} (Children of 3 are 4, 5. 4 is in frontier, 5 is new)
• Expand 4 (smallest): Frontier = {5, 6}, Expanded = {1, 2, 3, 4} (Children of 4 are 5, 6. 5 is in frontier, 6 is new)
• Expand 5 (smallest): Frontier = {6, 7}, Expanded = {1, 2, 3, 4, 5} (Children of 5 are 6, 7. 6 is in frontier, 7 is new)
• Expand 6 (smallest): Goal reached. Expanded = {1, 2, 3, 4, 5, 6}

BFS expands 6 states: {1, 2, 3, 4, 5, 6}.

DFS State Expansion Analysis

Depth-First Search (DFS) typically uses a stack. However, the same critical rule applies: 'states [...] are expanded in the ascending order of numbers'. This means DFS must also select the numerically smallest state from its frontier.

Therefore, the expansion process for DFS under this specific constraint is identical to that of BFS.

• Start: Frontier = {1}, Expanded = {}
• Expand 1: Frontier = {2, 3}, Expanded = {1}
• Expand 2: Frontier = {3, 4}, Expanded = {1, 2}
• Expand 3: Frontier = {4, 5}, Expanded = {1, 2, 3}
• Expand 4: Frontier = {5, 6}, Expanded = {1, 2, 3, 4}
• Expand 5: Frontier = {6, 7}, Expanded = {1, 2, 3, 4, 5}
• Expand 6: Goal reached. Expanded = {1, 2, 3, 4, 5, 6}

DFS also expands 6 states: {1, 2, 3, 4, 5, 6}.

Conclusion

Both BFS and DFS, when operating under the constraint that states are expanded strictly in ascending numerical order from the frontier, explore the exact same sequence of states. Consequently, they expand an equal number of states to reach the goal state 6.