Suppose that we wish to achieve some goal $A$. Suppose that, if we achieve $B$ and $C$ and $D$, then we will also achieve $A$. One can also think of $A$ as some problem to solve, where $A$ has the property that it will be solved if each of $B, C$, and $D$ is solved. When this relation holds, it is said that the goal or problem $A$ can be reduced to the subgoals or subproblems $B, C$, and $D$. In sentential calculus, this situation can be represented by:
$$
(B \wedge(C \wedge D)) \rightarrow A .
$$
We can also represent this situation by the tree in Figure 1-2. In this figure, $B, C, D$ are the child nodes of $A$. The curved arc indicates that $A$ is reducible to the conjunction of $B, C, D$.
Figure 1-2.
Now suppose that we achieve $B$ if we achieve $E$, so:
$$
E \rightarrow B .
$$
Suppose that we will also achieve $B$ if we achieve $F$, so:
$$
F \rightarrow B .
$$
We need $B$ and $C$ and $D$ to get $A$, but either child node $E$ or $F$ will get us $B$. This situation is represented in Figure 1-3, where there is no arc between the lines from $B$ to $E$ and from $B$ to $F$.Also suppose that $G$ is a "good" node, i.e., something that we can simply do (or a problem that we can directly solve), without depending on any further action (or further solutions). In sentential calculus, we represent this by:
$G$.
Suppose that $H$ is a "nogood" node, i.e., something that we cannot achieve or realize or solve; it is a dead end. In sentential calculus, we represent this by:
$\neg H$.
In the tree diagram, terminal nodes that are good are represented by * and terminal nodes that are nogood are represented by \#. The situation is now represented by Figure 1-4.
Figure 1-4.
Suppose we have the following additional information:
$$
\begin{gathered}
(I \wedge S) \rightarrow F \\
I \\
(J \wedge K) \rightarrow S \\
J \\
K \\
L \rightarrow C \\
M \rightarrow C \\
T \rightarrow C \\
\neg L \\
M
\end{gathered}
$$
\begin{gathered}
T \\
(N \wedge(O \wedge P)) \rightarrow D \\
N \\
O \\
Q \rightarrow P \\
R \rightarrow P \\
U \rightarrow P \\
\neg Q \\
R \\
U
\end{gathered}
FIGURE CANT COPY