Ferryman$-$Cabbage$-$Goat$-$Wolf problem using formal method semantics:

$($Q$1)$ Model Representation:

We can represent the problem using a labeled transition system $($LTS$).$ We define states as tuples indicating the locations of the ferryman, cabbage, goat, and wolf $(0$ for one side of the river, $1$ for the other side$).$ The transitions represent valid movements of the ferryman with one item across the river, considering the constraints provided.

States:

S $=\{($F$,$ C$,$ G$,$ W$)\}$

Where:

F: Ferryman's location $(0$ or $1)$

C: Cabbage's location $(0$ or $1)$

G: Goat's location $(0$ or $1)$

W: Wolf's location $(0$ or $1)$

Transitions:

T $=\{(0,0,0,0)->(1,0,0,0),(0,0,0,0)->(1,0,0,1),...\}$

Each transition represents a valid movement of the ferryman with one item, considering the constraints to avoid conflicts.

$($Q$2)$ Temporal Logic Formula:

We can express the property "There exists a safe path following which the ferryman will be able to transport all the goods to the other side of the river" using Linear Temporal Logic $($LTL$).$ The formula would be:

$\backslash$phi $=$p$.\backslash$geoquad $($p $->\backslash$geoquad $($F & $($C $|$ G $|$ W$)$ & $...))$

This formula states that there exists a path p such that, in all states along p$,$ the ferryman is on the same side as at least one item $($C$,$ G$,$ W$)$ and no conflicts occur.

$($Q$3)$ Model Checking:

Model checking solves the puzzle by exhaustively exploring all possible paths in the LTS to verify if the given property holds. It systematically checks if there exists a path where the property holds, i$.$e$.,$ there exists a safe path for the ferryman to transport all goods without any conflicts. By applying model checking algorithms to the LTS$,$ we can determine if the problem can be solved and find the solution path if it exists. Model checking ensures that all possible states and transitions are considered, providing a rigorous and systematic approach to solving the problem.