To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's define:

- $x_1$ as the number of students employed.
- $x_2$ as the number of full-time employees.

The objective is to minimize the total wage bill. Given that students earn $200 a week and full-time employees earn $500 a week, the objective function can be represented as:

\[ \text{Minimize} \quad 200x_1 + 500x_2 \]

The constraints are:
1. The company needs at least 100 painters in total: $x_1 + x_2 \geq 100$.
2. At least 30 of the painters must be full-time employees: $x_2 \geq 30$.
3. The number of full-time employees should be at least half the number of students: $x_2 \geq \frac{1}{2}x_1$.

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of students'), ('x2', 'number of full-time employees')],
    'objective_function': '200*x1 + 500*x2',
    'constraints': ['x1 + x2 >= 100', 'x2 >= 30', 'x2 >= 0.5*x1']
}
```

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Painting_Company")

# Define the variables
x1 = m.addVar(vtype=GRB.INTEGER, name="students")
x2 = m.addVar(vtype=GRB.INTEGER, name="full_time_employees")

# Set the objective function
m.setObjective(200*x1 + 500*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1 + x2 >= 100, "total_painters")
m.addConstr(x2 >= 30, "full_time_min")
m.addConstr(x2 >= 0.5*x1, "experience_balance")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of students: {x1.x}")
    print(f"Number of full-time employees: {x2.x}")
    print(f"Total wage bill: ${200*x1.x + 500*x2.x}")
else:
    print("No optimal solution found")
```