Answer:The Traveling Salesperson Problem (TSP) involves finding the shortest possible route that visits each city exactly once and returns to the origin city. Let's break down the components you've asked for:
1. **Set of States and Operators / State Space:**
- **States:** A state in the TSP can be represented as a tuple `(current_city, visited_cities)`, where:
- `current_city` is the city where the salesperson currently is.
- `visited_cities` is a set of cities that the salesperson has already visited.
- **Operators:** Operators define how you can transition between states. In TSP, an operator can be moving from one city to another that hasn't been visited yet.
- **State Space:** The state space is the set of all possible states that can be reached from the initial state using valid operators. It grows exponentially with the number of cities due to the combinatorial nature of the problem.
Example:
- Suppose you have 4 cities (A, B, C, D). The initial state could be `(A, {})`, meaning the salesperson starts at city A and has visited no other cities yet. Operators would allow transitions like `(A, {}) -> (B, {A})`, `(B, {A}) -> (C, {A, B})`, and so on.
2. **Goal Test Function:**
The goal test function checks if a given state is a goal state, i.e., if all cities have been visited exactly once and the salesperson returns to the starting city. In terms of the state representation `(current_city, visited_cities)`, the goal test would be:
```python
def is_goal_state(state, num_cities):
current_city, visited_cities = state
return len(visited_cities) == num_cities - 1 and current_city == START_CITY
```
- `num_cities` is the total number of cities in the problem.
- `START_CITY` is the city where the salesperson starts and ends the journey.
3. **Path Cost:**
The path cost is the total distance (or any other metric) accumulated while traveling through the cities in the state space.
- If each city has a distance matrix `distances`, the cost function for a path (sequence of transitions from start to end) can be computed as:
```python
def path_cost(path, distances):
cost = 0
for i in range(len(path) - 1):
current_city = path[i]
next_city = path[i + 1]
cost += distances[current_city][next_city]
return cost
```
- `path` is a sequence of cities (including the start city repeated at the end to form a loop).
- `distances` is a matrix where `distances[i][j]` gives the distance from city `i` to city `j`.
In summary, constructing the state space involves defining states and operators, the goal test checks if all cities are visited in the correct order, and path cost computes the total distance traveled in the optimal route found.
Explanation: