report2/implementation.tex

   1 \section{Implementation}
   2 \subsection{Screen encoding}
   3 When parsed the sokoban screen is stripped of all walls and unreachable empty
   4 spaces are removed. Moreover, a screen is rejected if either there are no
   5 boxes, there is no agent or the number of boxes is not equal to the number of
   6 targets.
   7
   8 Let $T=\{free,box,target,agent,targetagent,targetbox\}$ be the set of possible
   9 states of a tile. Tiles are numbered and thus a sokoban screen is the set $F$
  10 containing a $x_i\in T$ for every tile. We introduce a function $ord(x, y)$
  11 that returns the tile number for a given $x$ and $y$ coordinate and a function
  12 $iord(i)$ that does the reverse. To encode the
  13 state we introduce an encoding function that encodes a state in three boolean
  14 variables:
  15 $$encode(t)=\begin{cases}
  16         001 & \text{if }t=free\\
  17         010 & \text{if }t=box\\
  18         011 & \text{if }t=target\\
  19         100 & \text{if }t=targetbox\\
  20         101 & \text{if }t=agent\\
  21         110 & \text{if }t=agentbox
  22 \end{cases}$$
  23
  24 This means that the encoding of a screen takes $3*|F|$ variables.
  25
  26 \subsection{Transition encoding}
  27 We introduce a variable denoting the intended direction of movement $m \in
  28 \{\text{up}, \text{down}, \text{left}, \text{right}\}$. Per move we define a
  29 $\delta$ and $\gamma$ variable which represent the change in coordinate value
  30 respectively for the next position and the position next to the next postition.
  31 $$\delta_{(x,y)}(m)=\begin{cases}
  32         (x-1, y) & \text{if } m = left\\
  33         (x+1, y) & \text{if } m = right\\
  34         (x, y+1) & \text{if } m = down\\
  35         (x, y-1) & \text{if } m = up\\
  36 \end{cases}\quad
  37 \gamma{(x,y)}(m)=\begin{cases}
  38         (x-2, y) & \text{if } m = left\\
  39         (x+2, y) & \text{if } m = right\\
  40         (x, y+2) & \text{if } m = down\\
  41         (x, y-2) & \text{if } m = up\\
  42 \end{cases}$$
  43
  44 We define the tile update function $next(i_1, i_2, i_3)$ where $i_1$ contains
  45 the agent and $i_2$ and $i_3$ are adjacent to it in some direction.
  46 $$next(i_1, i_2, i_3)=\left\{\begin{array}{lp{2pt}l}
  47 % Three state transitions
  48         (free, agent, box) & \text{if } &
  49                 i_1=agent \wedge i_2=box \wedge i_3=free\\
  50         (target, agent, box) & \text{if } &
  51                 i_1=targetagent \wedge i_2=box \wedge i_3=free\\
  52         (free, targetagent, box) & \text{if } &
  53                 i_1=agent \wedge i_2=targetbox \wedge i_3=free\\
  54         (free, agent, targetbox) & \text{if } &
  55                 i_1=agent \wedge i_2=box \wedge i_3=targetbox\\
  56         (target, targetagent, box) & \text{if } &
  57                 i_1=targetagent \wedge i_2=targetbox \wedge i_3=free\\
  58         (target, agent, targetbox) & \text{if } &
  59                 i_1=targetagent \wedge i_2=box \wedge i_3=target\\
  60         (free, targetagent, targetbox) & \text{if } &
  61                 i_1=agent \wedge i_2=targetbox \wedge i_3=target\\
  62         (target, targetagent, targetbox) & \text{if } &
  63                 i_1=targetagent \wedge i_2=targetbox \wedge i_3=target\\
  64 % Two state transitions
  65         (free, agent, i_3) & \text{if } & i_1=agent \wedge i_2=free\\
  66         (free, targetagent, i_3) & \text{if } & i_1=agent \wedge i_2=target\\
  67         (target, agent, i_3) & \text{if } & i_1=targetagent \wedge i_2=free\\
  68         (target, targetagent, i_3) & \text{if } & i_1=targetagent \wedge i_2=target\\
  69 % One state transitions
  70         (agent, i_2, i_3) & \text{if } & i_1=agent\\
  71         (targetagent, i_2, i_3) & \text{if } & i_1=targetagent\\
  72 \end{array}\right.$$
  73
  74 \subsection{Goal encoding}
  75 Encoding the goal state is very trivial. Just make sure all $target$ tiles are
  76 $targetbox$ tiles.
  77
  78 \subsection{Example}
  79 For example, take the toy screen shown as the first representation in
  80 Listing~\ref{lst:toy}. When the screen is parsed the unreachable space is first
  81 removed by detecting reachable space with the flood fill algorithm. This
  82 results in the second representation in Listing~\ref{lst:toy}.
  83
  84 \lstinputlisting[label={lst:toy},caption={Example screen}]{toy.screen}
  85
  86 \subsubsection{Encoding}
  87 To encode the screen the tiles are numbered from left to right from top to
  88 bottom. Thus resulting in $i_1=agent, i_2=box, i_3=target$. When we translate
  89 these numbers to sylvan-sane variables we get the following bdd shown in
  90 Figure~\ref{fig:toy}. Variables $0, 2, 4$ represent $i_1$, $6, 8, 10$ represent
  91 $i_2$ and lastly $12, 14, 16$ represents $i_3$.
  92 \begin{figure}[p]
  93         \centering
  94         \includegraphics[width=\textwidth]{toy.png}
  95         \caption{Initial state encoding of the example~\label{fig:toy}}
  96 \end{figure}
  97
  98 Encoding the goal state happens in a similar way. The algorithm basically
  99 states that all $target$ positions should be $boxtarget$. This is results in
 100 the BDD shown in Figure~\ref{fig:toy2}.
 101
 102 \begin{figure}[p]
 103         \centering
 104         \includegraphics[width=.5\textwidth]{toy2.png}
 105         \caption{Goal state encoding of the example~\label{fig:toy2}}
 106 \end{figure}
 107
 108 Finally encoding the transitions already results in a long list of complicated
 109 BDDs that each encode a possible move in all possible directions. In this
 110 example there is only one move possible in the first state, thus all moves
 111 except for the one move result in a relative product that is empty. For example
 112 the sub-BDD that encodes the first position only looks is visualized in
 113 Figure~\ref{fig:toy3}. The only satisfying assignment is that $0, 2, 4$ hold
 114 the values $1, 0, 1$ thus encoding man and the variables $1, 3, 5$ holding the
 115 values $0, 0, 1$ thus encoding free.
 116
 117 \begin{figure}[p]
 118         \centering
 119         \includegraphics[width=.8\textwidth]{toy3.png}
 120         \caption{Sub-BDD of a move~\label{fig:toy3}}
 121 \end{figure}
 122
 123 \subsubsection{Results}