transitions formalized

diff --git a/report2/implementation.tex b/report2/implementation.tex
index 32a5bd7..9cfcbac 100644 (file)
--- a/report2/implementation.tex
+++ b/report2/implementation.tex
@@ -39,14 +39,32 @@ $$\delta_{(x,y)}(m)=\begin{cases}
        (x, y-2) & \text{if } m = up\\
 \end{cases}$$
 
-We define the tile update function $next(i)$ for $i \in F$.
-$$next(i)=\left\{\begin{array}{lll}
-       free & \text{if } & i=agent \wedge \delta(i)=free\\
-               & \vee & i = agent \wedge \delta(i)=target\\
-       target & \text{if } & i=targetagent \wedge \delta(i)=free\\
-               & \vee & i = targetagent \wedge \delta(i)=target\\
-       agent & \text{if } & i=free \wedge \delta(i)=agent\\
-               & \vee & i = free \wedge \delta(i)=targetagent\\
-       targetagent & \text{if } & i=target \wedge \delta(i)=agent\\
-               & \vee & i = target \wedge \delta(i)=targetagent\\
+We define the tile update function $next(i_1, i_2, i_3)$ where $i_1$ contains
+the agent and $i_2$ and $i_3$ are adjacent to it in some direction.
+$$next(i_1, i_2, i_3)=\left\{\begin{array}{lll}
+% Three state transitions
+       (free, agent, box) & \text{if } & 
+               i_1=agent \wedge i_2=box \wedge i_3=free\\
+       (target, agent, box) & \text{if } & 
+               i_1=targetagent \wedge i_2=box \wedge i_3=free\\
+       (free, targetagent, box) & \text{if } &
+               i_1=agent \wedge i_2=targetbox \wedge i_3=free\\
+       (free, agent, targetbox) & \text{if } &
+               i_1=agent \wedge i_2=box \wedge i_3=targetbox\\
+       (target, targetagent, box) & \text{if } &
+               i_1=targetagent \wedge i_2=targetbox \wedge i_3=free\\
+       (target, agent, targetbox) & \text{if } &
+               i_1=targetagent \wedge i_2=box \wedge i_3=target\\
+       (free, targetagent, targetbox) & \text{if } &
+               i_1=agent \wedge i_2=targetbox \wedge i_3=target\\
+       (target, targetagent, targetbox) & \text{if } &
+               i_1=targetagent \wedge i_2=targetbox \wedge i_3=target\\
+% Two state transitions
+       (free, agent, i_3) & \text{if } & i_1=agent \wedge i_2=free\\
+       (free, targetagent, i_3) & \text{if } & i_1=agent \wedge i_2=target\\
+       (target, agent, i_3) & \text{if } & i_1=targetagent \wedge i_2=free\\
+       (target, targetagent, i_3) & \text{if } & i_1=targetagent \wedge i_2=target\\
+% One state transitions
+       (agent, i_2, i_3) & \text{if } & i_1=agent\\
+       (targetagent, i_2, i_3) & \text{if } & i_1=targetagent\\
 \end{array}\right.$$