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Add FAIKR1 non-linear planning
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src/fundamentals-of-ai-and-kr/module1/img/_pop_example1.pdf
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src/fundamentals-of-ai-and-kr/module1/img/_threat_example.pdf
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\section{Linear planning}
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\marginnote{Linear planning}
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Formulates the planning problem as a search problem where:
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@ -326,3 +327,240 @@ Conjunction of goals are solved separately, but this could lead to the \marginno
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where a sub-goal destroys what another sub-goal has done.
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and is left as a final check.
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\section{Non-linear planning}
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\marginnote{Non-linear planning}
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Non-linear planning finds a plan as a search problem in the space of plans (instead of states as in linear planning).
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Each node of the search tree is a partial plan. Edges represent plan refinement operations.
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A non-linear plan is represented by:
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\begin{descriptionlist}
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\item[Actions{\normalfont.}] \marginnote{Actions set}
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\item[Orderings] \marginnote{Orderings set}
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between actions.
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\item[Causal links] \marginnote{Causal links}
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triplet $\langle S_i, S_j, c \rangle$ where $S_i$ and $S_j$ are actions and $c$ is a sub-goal.
|
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$c$ should be the effect of $S_i$ and precondition of $S_j$.
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Causal links represent causal relations between actions (i.e. interaction between sub-goals):
|
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to execute $S_j$, the effect $c$ of $S_i$ is required first.
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\end{descriptionlist}
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The initial plan is an empty plan with two fake actions \texttt{start} and \texttt{stop}
|
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with ordering $\texttt{start} < \texttt{stop}$:
|
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\begin{descriptionlist}
|
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\item[\texttt{start}] has no preconditions and the effects match the initial state.
|
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\item[\texttt{stop}] has no effects and the preconditions match the goal.
|
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\end{descriptionlist}
|
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At each step, one of the following refinement operations can be applied until the goal is reached:
|
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\begin{itemize}
|
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\item Add an action to the set of actions.
|
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\item Add an ordering to the set of orderings.
|
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\item Add a causal link to the set of causal links.
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\end{itemize}
|
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|
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\begin{figure}[h]
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\centering
|
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\includegraphics[width=0.4\textwidth]{img/_nonlinear_plan_example.pdf}
|
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\caption{Example of search tree in non-linear planning}
|
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\end{figure}
|
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|
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|
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\begin{description}
|
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\item[Least commitment planning] \marginnote{Least commitment planning}
|
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Only strictly necessary restrictions (e.g. ordering) are imposed.
|
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Non-linear planning is a least commitment planning.
|
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|
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\item[Linearization] \marginnote{Linearization}
|
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At the end, the partially ordered actions should be linearized,
|
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respecting the ordering constraints, to obtain the final plan.
|
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\end{description}
|
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|
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|
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\begin{description}
|
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\item[Threat] \marginnote{Threat}
|
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An action $S_k$ is a threat to a causal link $\langle S_i, S_j, c \rangle$
|
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if its effects cancel $c$.
|
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$S_k$ should not be executed in between $S_i$ and $S_j$.
|
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|
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\begin{figure}[H]
|
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\centering
|
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\includegraphics[width=0.3\textwidth]{img/_threat_example.pdf}
|
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\caption{Example of threat. Causal links are represented using thick arrows.}
|
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\end{figure}
|
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|
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Possible solutions to a threat $S_k$ to $\langle S_i, S_j, c \rangle$ are:
|
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\begin{descriptionlist}
|
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\item[Demotion] \marginnote{Demotion}
|
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Add the ordering constraint $S_k < S_i$ (i.e. threat executed before).
|
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\item[Promotion] \marginnote{Promotion}
|
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Add the ordering constraint $S_k > S_i$ (i.e. threat executed after).
|
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\end{descriptionlist}
|
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\end{description}
|
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|
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\begin{algorithm}
|
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\caption{Partial order planning (POP)}
|
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\begin{lstlisting}[mathescape=true]
|
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def pop(initial_state, goal, actions):
|
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plan = init_empty_plan(initial_state, goal)
|
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while not plan.isSolution():
|
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try:
|
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sn, c = selectSubgoal(plan)
|
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chooseOperator(plan, actions, sn, c)
|
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resolveThreats(plan)
|
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except PlanFailError:
|
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plan.backtrack()
|
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return plan
|
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|
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def selectSubgoal(plan):
|
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sn, c = random([sn, c in plan.steps if c in sn.unsolved_preconditions])
|
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return sn, c
|
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|
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def chooseOperator(plan, actions, sn, c):
|
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s = random([s in (actions + plan.steps) if c in s.effects])
|
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if s is None: raise(PlanFailError)
|
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plan.addCausalLink($\langle$s, sn, c$\rangle$)
|
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plan.addOrdering(s < sn)
|
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if s not in plan.steps:
|
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plan.addAction(s)
|
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plan.addOrdering(start < s < stop)
|
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|
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def resolveThreats(plan):
|
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for s_k, s_i, s_j in plan.threats():
|
||||
resolution = random([ DEMOTION, PROMOTION ])
|
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if resolution == DEMOTION:
|
||||
plan.addOrdering(s_k < s_i)
|
||||
elif resolution == PROMOTION:
|
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plan.addOrdering(s_k > s_j)
|
||||
if plan.isNotConsistent(): raise(PlanFailError)
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
\begin{example}[Purchasing schedule]
|
||||
The initial state is:
|
||||
\[ \texttt{at(home)}, \texttt{sells(hws, drill)}, \texttt{sells(sm, milk)}, \texttt{sells(sm, banana)} \]
|
||||
where $\texttt{hws}$ means "hardware store" and $\texttt{sm}$ means "supermarket".
|
||||
|
||||
The goal is:
|
||||
\[ \texttt{at(home)}, \texttt{have(drill)}, \texttt{have(milk)}, \texttt{have(banana)} \]
|
||||
|
||||
The possible actions are:\\[0.5em]
|
||||
\begin{minipage}{.5\linewidth}
|
||||
\begin{descriptionlist}
|
||||
\item[\texttt{GO(X, Y)}] \phantom{}
|
||||
\begin{description}
|
||||
\item[Preconditions] $\texttt{at(X)}$
|
||||
\item[Effects] $\texttt{at(Y)}$, $\lnot \texttt{at(X)}$
|
||||
\end{description}
|
||||
\end{descriptionlist}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.5\linewidth}
|
||||
\begin{descriptionlist}
|
||||
\item[\texttt{BUY(S, Y)}] \phantom{}
|
||||
\begin{description}
|
||||
\item[Preconditions] $\texttt{at(S)}$, $\texttt{sells(S, Y)}$
|
||||
\item[Effects] $\texttt{have(Y)}$
|
||||
\end{description}
|
||||
\end{descriptionlist}
|
||||
\end{minipage}\\[0.5em]
|
||||
|
||||
Partial order planning steps are:
|
||||
\begin{enumerate}
|
||||
\item Define the initial plan:
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{img/_pop_example1.pdf}
|
||||
\end{center}
|
||||
|
||||
\item The loop of POP is:
|
||||
\begin{itemize}
|
||||
\item Choose an action $a_i$ and one of its unsolved preconditions $c$.
|
||||
\item Select an action $a_j$ with the precondition $c$ in its effects.
|
||||
\item Add the ordering constraint $\texttt{start} < a_j < \texttt{stop}$.
|
||||
\item Add the causal link $\langle a_j, a_i, c \rangle$ (and ordering $a_j < a_i$).
|
||||
\item Solve threats.
|
||||
\end{itemize}
|
||||
|
||||
We choose the action $a_i = \texttt{stop}$ and the precondition $c = \texttt{have(drill)}$.
|
||||
We choose as action with $c$ in its effects $a_j = \texttt{BUY(X, drill)}$.
|
||||
We therefore add to the plan the ordering $\texttt{start} < \texttt{BUY(X, drill)} < \texttt{stop}$ and
|
||||
the causal link $\langle \texttt{BUY(X, drill)}, \texttt{stop}, \texttt{have(drill)} \rangle$:
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{img/_pop_example2.pdf}
|
||||
\end{center}
|
||||
|
||||
\item Repeat the previous point for the preconditions $\texttt{have(milk)}$ and $\texttt{have(banana)}$:
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{img/_pop_example3.pdf}
|
||||
\end{center}
|
||||
|
||||
\item Now, we choose as action $\texttt{BUY(X, drill)}$ and as unsolved precondition $\texttt{sells(X, drill)}$.
|
||||
This can be solved from the action $\texttt{start}$ with effect $\texttt{sells(hws, drill)}$.
|
||||
We make the substitution $\texttt{X}/\texttt{drill}$ and
|
||||
add $\langle \texttt{start}, \texttt{BUY(hws, drill)}, \texttt{sells(hws, drill)} \rangle$ to the causal links.
|
||||
The same process can be repeated for $\texttt{BUY(X, milk)}$ and $\texttt{BUY(X, banana)}$:
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{img/_pop_example4.pdf}
|
||||
\end{center}
|
||||
|
||||
\item Now, we choose as action $\texttt{BUY(hws, drill)}$ and as unsolved precondition $\texttt{at(hws)}$.
|
||||
This can be solved using the action $\texttt{GO(X, hws)}$.
|
||||
We add $\langle \texttt{GO(X, hws)}, \texttt{BUY(hws, drill)}, \texttt{at(hws)} \rangle$ to the causal links.
|
||||
|
||||
We continue by choosing as action $\texttt{GO(X, hws)}$ and as unsolved precondition $\texttt{at(X)}$.
|
||||
This can be solved from $\texttt{start}$ with effect $\texttt{at(home)}$.
|
||||
We therefore make the substitution $\texttt{X}/\texttt{home}$ and
|
||||
add $\langle \texttt{start}, \texttt{GO(home, hws)}, \texttt{at(home)} \rangle$ to the causal links.
|
||||
|
||||
The same process can be repeated for the $\texttt{milk}$ and $\texttt{banana}$ branch:
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{img/_pop_example5.pdf}
|
||||
\end{center}
|
||||
|
||||
\item We have a threat between $\texttt{GO(home, hws)}$ and $\texttt{GO(home, sm)}$ as they both
|
||||
require the precondition $\texttt{at(home)}$ and both have as effect $\lnot\texttt{at(home)}$.
|
||||
It can be easily seen that neither promotion nor demotion solves the conflict.
|
||||
We are therefore forced to backtrack.
|
||||
|
||||
We backtrack at the previous point, where we chose as action $\texttt{GO(X, sm)}$ and as precondition $\texttt{at(X)}$
|
||||
(this step has been implicitly done in the previous point).
|
||||
\begin{itemize}
|
||||
\item Instead of choosing the action $\texttt{start}$, we choose $\texttt{GO(home, hws)}$ with the effect $\texttt{at(hws)}$.
|
||||
We therefore make the substitution $\texttt{X}/\texttt{hws}$ and update the causal links.
|
||||
\item We also resolve the threat $\texttt{GO(hws, sm)}$ to $\texttt{BUY(hws, drill)}$
|
||||
(it removes the precondition $\texttt{at(hws)}$)
|
||||
by promoting $\texttt{GO(hws, sm)}$
|
||||
and adding the ordering constraint $\texttt{BUY(hws, drill)} < \texttt{GO(hws, sm)}$:
|
||||
\end{itemize}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{img/_pop_example6.pdf}
|
||||
\end{center}
|
||||
|
||||
\item Now, we choose as action $\texttt{stop}$ and as precondition $\texttt{at(home)}$.
|
||||
We choose as action $\texttt{GO(sm, home)}$ and update the causal links.
|
||||
|
||||
Finally, we solve the threat $\texttt{GO(sm, home)}$ to
|
||||
both $\texttt{BUY(sm, milk)}$ and $\texttt{BUY(sm, banana)}$ (it removes the required precondition $\texttt{at(sm)}$)
|
||||
by promoting $\texttt{GO(sm, home)}$.
|
||||
The newly added ordering constraints are
|
||||
$\texttt{BUY(sm, milk)} < \texttt{GO(sm, home)}$ and
|
||||
$\texttt{BUY(sm, banana)} < \texttt{GO(sm, home)}$.
|
||||
|
||||
The final plan is:
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{img/_pop_example7.pdf}
|
||||
\end{center}
|
||||
By considering the ordering constraints, a linearization could be:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{GO(home, hws)} &\rightarrow
|
||||
\texttt{BUY(hws, drill)} \rightarrow
|
||||
\texttt{GO(hws, sm)} \rightarrow\\
|
||||
&\texttt{BUY(sm, milk)} \rightarrow
|
||||
\texttt{BUY(sm, banana)} \rightarrow
|
||||
\texttt{GO(sm, home)}
|
||||
\end{split}
|
||||
\]
|
||||
\end{enumerate}
|
||||
\end{example}
|
||||
Reference in New Issue
Block a user