add move variables

diff --git a/report2/implementation.tex b/report2/implementation.tex
index bb00db8..9657891 100644 (file)
--- a/report2/implementation.tex
+++ b/report2/implementation.tex
@@ -23,59 +23,23 @@ This means that the encoding of a screen takes $3*|F|$ variables.
 
 \subsection{Transition encoding}
 We introduce a variable denoting the intended direction of movement $m \in
-\{\text{up}, \text{down}, \text{left}, \text{right}\}$. Tiles 
-%
-%Let\\
-%$\delta_{x}(x,m) =
-%  \begin{cases}
-%    (x+1) & \quad \text{if } m = left\\
-%    (x-1) & \quad \text{if } m = right\\
-%    x & \quad \text{otherwise}
-%  \end{cases}$\quad
-%$\delta'_{x}(x,m) =
-%  \begin{cases}
-%    (x-1) & \quad \text{if } m = left\\
-%    (x+1) & \quad \text{if } m = right\\
-%    x & \quad \text{otherwise}
-%  \end{cases}$\\
-%$\delta_{y}(y,m) =
-%  \begin{cases}
-%    (y+1) & \quad \text{if } m = up\\
-%    (y-1) & \quad \text{if } m = down\\
-%    y & \quad \text{otherwise}
-%  \end{cases}$\quad
-%$\delta'_{y}(y,m) =
-%  \begin{cases}
-%    (y-1) & \quad \text{if } m = up\\
-%    (y+1) & \quad \text{if } m = down\\
-%    y & \quad \text{otherwise}
-%  \end{cases}$\\
-%  $\gamma_{x}(x,m) =
-%    \begin{cases}
-%      (x+2) & \quad \text{if } m = left\\
-%      (x-2) & \quad \text{if } m = right\\
-%      x & \quad \text{otherwise}
-%    \end{cases}$\quad
-%  $\gamma_{y}(y,m) =
-%    \begin{cases}
-%      (y+2) & \quad \text{if } m = up\\
-%      (y-2) & \quad \text{if } m = down\\
-%      y & \quad \text{otherwise}
-%    \end{cases}$
-%
-%%    "    x" <+ x <+ "_" <+ y <+ " = BoxOnTarget & (x" <+ (checkX p x) <+ "_" <+ (checkY p x  (y+1)) <+ " = Agent | x"
-%%                            <+ (checkX p x) <+ "_" <+ (checkY p x  (y+1)) <+ " = AgentOnTarget) & (x" <+ (checkX p x) <+ "_" <+ (checkY p x (y-1)) <+ " = Free | x" <+ (checkX p x) <+ "_" <+ (checkY p x (y-1)) <+ " = Target) & move = Left: AgentOnTarget;",
-%%                    "    x" <+ x <+ "_" <+ y <+ " = BoxOnTarget & (x" <+ (checkX p (x+1)) <+ "_" <+ (checkY p (x+1) y) <+ " = Agent | x"
-%%                            <+ (checkX p (x+1)) <+ "_" <+ (checkY p (x+1) y) <+ " = AgentOnTarget) & (x" <+ (checkX p (x-1)) <+ "_" <+ (checkY p (x-1) y) <+ " = Free | x" <+ (checkX p (x-1)) <+ "_"
-%%                            <+ (checkY p (x-1) y) <+ " = Target) & move = Up: AgentOnTarget;",
-%%                    "    x" <+ x <+ "_" <+ y <+ " = BoxOnTarget & (x" <+ (checkX p x) <+ "_" <+ (checkY p x (y-1)) <+ " = Agent | x"
-%%                            <+ (checkX p x) <+ "_" <+ (checkY p x (y-1)) <+ " = AgentOnTarget) & (x" <+ (checkX p x) <+ "_" <+ (checkY p x  (y+1)) <+ " = Free | x" <+ x <+ "_" <+ (checkY p x  (y+1)) <+ " = Target) & move = Right : AgentOnTarget;",
-%%                    "    x" <+ x <+ "_" <+ y <+ " = BoxOnTarget & (x" <+ (checkX p (x-1)) <+ "_" <+ (checkY p (x-1) y) <+ " = Agent | x"
-%%                            <+ (checkX p (x-1)) <+ "_" <+ (checkY p (x-1) y) <+ " = AgentOnTarget) & (x" <+ (checkX p (x+1)) <+ "_" <+ (checkY p (x+1) y) <+ " = Free | x" <+ (checkX p (x+1)) <+ "_"
-%%                            <+ (checkY p (x+1) y) <+ " = Target) & move = Down : AgentOnTarget;",
-%
-%
-%
+\{\text{up}, \text{down}, \text{left}, \text{right}\}$. Per move we define a
+$\delta$ and $\gamma$ variable which represent the change in coordinate value
+respectively for the next position and the position next to the next postition.
+\\
+$\delta_{(x,y)}(m)=\begin{cases}
+       (x-1, y) & \text{if } m = left\\
+       (x+1, y) & \text{if } m = right\\
+       (x, y+1) & \text{if } m = down\\
+       (x, y-1) & \text{if } m = up\\
+\end{cases}\quad
+\gamma{(x,y)}(m)=\begin{cases}
+       (x-2, y) & \text{if } m = left\\
+       (x+2, y) & \text{if } m = right\\
+       (x, y+2) & \text{if } m = down\\
+       (x, y-2) & \text{if } m = up\\
+\end{cases} $
+
 %We define the tile update function $next(x_{i,j}), x_{i,j} \in F, \forall i,j \text{ s.t.} x_{i,j} \neq \perp$ as\\
 %$
 %next(x_{i,j}) =