Once upon a time a farmer went to a market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and rented a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases-the fox, the goose, or the bag of beans. If left together, the fox would eat the goose, or the goose would eat the beans. The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. a. Find a solution to the puzzle. b. Is the solution unique? c. Assume you had a model checker (along with a state transition diagram for the puzzle). Define a property (in English, not temporal logic) that you can give to the model checker such that the tool solves the puzzle for you. d. What 'feature' of a model checker provides the solution to you
The Correct Answer and Explanation is:
a. Solution to the Puzzle
- The farmer takes the goose across the river and leaves it.
- He returns alone and takes the fox across the river.
- He leaves the fox on the far bank but takes the goose back with him.
- He leaves the goose on the starting bank and takes the beans across the river.
- He leaves the beans with the fox and returns alone.
- Finally, he takes the goose across the river again.
Now all three items—the fox, the goose, and the beans—are on the far side of the river, and none have been eaten.
b. Is the Solution Unique?
No, the exact sequence of steps may vary slightly in terms of wording or intermediate decisions, but functionally all valid solutions follow the same key idea: never leave the fox with the goose or the goose with the beans unsupervised. While there are alternative orders in which the farmer might pick up or return items, the structure of the solution is effectively unique due to the constraints.
c. Property for a Model Checker
The property you would give to a model checker could be stated as:
“Find a sequence of actions such that all three items—the fox, the goose, and the bag of beans—end up on the opposite bank of the river from the starting point, and at no time are the goose and the beans or the goose and the fox left together without the farmer present.”
This property ensures the model checker searches only for valid, safe paths through the state transition system.
d. Model Checker Feature that Provides the Solution
The feature of a model checker that provides the solution is called state space exploration or model traversal. The model checker systematically explores all possible states and transitions (based on the rules of the puzzle), checking against the defined safety and goal properties. When it finds a sequence that satisfies the constraints, it presents it as a valid path—effectively solving the puzzle.
Explanation
The fox, goose, and bag of beans puzzle is a classic example of a constraint satisfaction problem, often used to illustrate logical reasoning and state transition modeling. The constraints are clear: the goose cannot be left alone with the fox (or it will be eaten), and the goose cannot be left with the beans (or it will eat them). The challenge is further constrained by the boat’s capacity—it can only carry the farmer and one item at a time.
To solve the puzzle, the farmer must carefully manage the crossings to ensure that no dangerous combinations are left unsupervised. The key strategy involves temporarily “undoing” a crossing by bringing the goose back midway to prevent a conflict. Although variations in sequence wording are possible, the logic underlying any valid solution remains the same: prevent any pairing that leads to loss, while moving all items across.
Model checking offers a computational approach to solving such puzzles. By representing the system as a finite state transition diagram—where each state corresponds to a specific arrangement of the farmer and his items—one can use a model checker to explore every possible transition. A goal state is defined (all items across safely), and constraints are encoded to prevent invalid states (e.g., goose and fox alone). The model checker’s state space exploration engine examines the possible transitions and verifies which path meets the criteria.
Thus, model checking not only confirms that a solution exists but can also output the sequence of steps needed to reach the goal. This demonstrates the practical application of formal verification tools in solving logic puzzles and real-world decision problems.
