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Add FAIKR1 Graphplan

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Tian Cheng (Riccardo) Xia
2023-11-30 18:14:27 +01:00
parent 71cd9e14f0
commit 5bd52e414d
3 changed files with 746 additions and 2 deletions
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src/fundamentals-of-ai-and-kr/module1/img/_graphplan.drawio Normal file
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@ -40,7 +40,7 @@
\item[Correctness] \marginnote{Correct planner}
The planner always finds a solution that leads from the initial state to the goal.
\item[Completeness] \marginnote{Complete planner}
The planner always finds a plan when it exits (planning is semi-decidable).
The planner always finds a plan when it exists (planning is semi-decidable).
\end{descriptionlist}
\item[Execution] \marginnote{Execution}
@ -777,4 +777,252 @@ The steps the algorithm does are:
\item Checks the pre-conditions and post-conditions of the action it is going to execute.
\item Backtracks the effects of an action in case of a failure.
\item Corrects the plan if external events occur.
\end{itemize}
\end{itemize}
\section{Graphplan}
\marginnote{Graphplan}
Graphplan is an off-line, least-commitment planner (closed-world assumption) that
constructs a partially ordered set of actions through a planning graph based on time steps.
The planner is correct, complete, optimal and computationally efficient.
\begin{description}
\item[Action] \marginnote{Actions}
An action, as in STRIPS, has:
\begin{itemize}
\item Preconditions.
\item Add list.
\item Delete list.
\end{itemize}
\begin{description}
\item[\texttt{NO-OP}]
Action that does not change the state (to solve the frame problem).
Can be seen as an action with the same proposition as precondition and add list.
\end{description}
\item[State] \marginnote{State}
A state is represented by a set of propositions that are true in that time step.
\item[Planning graph] \marginnote{Planning graph}
Directed leveled graph where edges connect nodes of adjacent levels.
There are two possible levels that are alternated during construction:
\begin{descriptionlist}
\item[Proposition level]
Contains propositions that describe the state.
Note that interfering propositions can appear.
\item[Action level]
Contains all the possible actions that have as preconditions the propositions in the previous level.
Note that interfering actions can appear.
\end{descriptionlist}
The first level of the graph is a proposition level containing the initial state.
Edges can be:
\begin{descriptionlist}
\item[Precondition arcs] proposition $\rightarrow$ action.
\item[Add arcs] action $\rightarrow$ proposition.
\item[Delete arcs] action $\rightarrow$ proposition.
\end{descriptionlist}
\item[Inconsistency] \marginnote{Inconsistency}
Actions and propositions can be inconsistent in the same time step.
Possible causes are:
\begin{descriptionlist}
\item[Inconsistent effects] \marginnote{Inconsistent effects}
An action negates the effects of another one.
\item[Interference] \marginnote{Interference}
An action deletes the preconditions of another one.
\item[Competing needs] \marginnote{Competing needs}
Propositions that cannot appear together either
because one negates the other or
because they can be reached only through mutually exclusive paths.
(i.e. two actions have mutually exclusive preconditions).
\item[Domain dependent]
\end{descriptionlist}
\item[Plan extraction] \marginnote{Plan extraction}
Once a proposition level containing the goal as non-mutually exclusive propositions has been reached,
the algorithm can attempt to extract a plan.
A valid plan has the following properties:
\begin{itemize}
\item Actions in the same time step do not interfere and can be executed in any order.
\item Propositions at the same time step are non-mutually exclusive.
\item The last step contains the goal as non-mutually exclusive propositions.
\end{itemize}
Even if the last step is a superset of the goal, planning may still fail.
In this case, the algorithm has to continue generating levels.
% Loop detection can be used to stop when a plan cannot be found.
\item[Memoization] \marginnote{Memoization}
At each step, if the goal is not satisfiable, the result is saved and
when the same state is encountered in the future it will automatically fail.
\end{description}
\begin{theorem}
The following statements hold:
\begin{itemize}
\item If a valid plan exists, it can be found as a subgraph of the planning graph.
\item In a planning graph, two actions in a time step are mutually exclusive
if a valid plan containing both does not exist.
\item In a planning graph, two propositions are mutually exclusive if they are inconsistent.
\end{itemize}
\end{theorem}
\begin{corollary}
Inconsistencies found during the planning graph construction
prune paths in the search tree.
\end{corollary}
\begin{algorithm}[H]
\caption{Graphplan}
\begin{lstlisting}[mathescape=true]
def graphplan(initial_state, actions, goal):
graph = PlanningGraph()
graph.addPropositionLevel(initial_state)
while True:
if (goal in graph.lastPropositionLevel and
not mutexPropositions(goal, graph.lastPropositionLevel)):
plan = extractSolution(graph, goal)
if plan is not FAIL: return plan
graph.addActionLevel( selectActions(graph) )
graph.addPropositionLevel( selectPreconditions(graph, actions) )
def selectActions(graph, actions):
new_action_level = Level()
for action in actions.unify(graph.lastPropositionLevel):
preconds = action.preconditions
if not mutexPropositions(preconds, graph.lastPropositionLevel):
new_action_level.add(preconds, action)
for proposition in graph.lastPropositionLevel:
new_action_level.add(NO_OP(proposition))
new_action_level.findInconsistencies()
return new_action_level
def selectPreconditions(graph):
new_props_level = Level()
for action in graph.lastActionLevel:
for prop in action.add_list:
new_props_level.add(action, prop, "add")
for prop in action.delete_list:
new_props_level.add(action, prop, "delete")
new_props_level.findInconsistencies()
return new_props_level
def extractSolution(graph, goal):
plan = Plan()
actions = graph.lastActionLevel.getActionsWithEffect(goal)
if mutexActions(actions, graph.lastActionLevel): return FAIL
plan.addLevel(actions)
graph = graph.popLastActionLevel()
return plan.merge(extractSolution(graph, actions.preconditions))
\end{lstlisting}
\end{algorithm}
\begin{example}[Moving objects with a cart]
Given the actions:
\begin{descriptionlist}
\item[\texttt{MOVE(R, PosA, PosB)}] \phantom{}
\begin{description}
\item[Preconditions] $\texttt{at(R, PosA)}$, $\texttt{hasFuel(R)}$
\item[Add list] $\texttt{at(R, PosB)}$
\item[Delete list] $\texttt{at(R, PosA)}$, $\texttt{hasFuel(R)}$
\end{description}
\end{descriptionlist}
\begin{minipage}{0.5\textwidth}
\begin{descriptionlist}
\item[\texttt{LOAD(Obj, Pos)}] \phantom{}
\begin{description}
\item[Preconds] $\texttt{at(R, Pos)}$, $\texttt{at(Obj, Pos)}$
\item[Add list] $\texttt{in(R, Obj)}$
\item[Delete list] $\texttt{at(Obj, Pos)}$
\end{description}
\end{descriptionlist}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{descriptionlist}
\item[\texttt{UNLOAD(Obj, Pos)}] \phantom{}
\begin{description}
\item[Preconds] $\texttt{in(R, Obj)}$, $\texttt{at(R, Pos)}$
\item[Add list] $\texttt{at(Obj, Pos)}$
\item[Delete list] $\texttt{in(R, Obj)}$
\end{description}
\end{descriptionlist}
\end{minipage}
Given a scenario where:
\begin{itemize}
\item \texttt{r} is a cart.
\item \texttt{a} and \texttt{b} are objects.
\item \texttt{l} and \texttt{p} are locations.
\end{itemize}
and the initial state is:
\begin{center}
\texttt{at(a, l)} $\cdot$ \texttt{at(b, l)} $\cdot$ \texttt{at(r, l)} $\cdot$ \texttt{hasFuel(r)}
\end{center}
The first four time steps of the planning graph are:
\begin{center}
\includegraphics[width=\textwidth]{img/_graphplan.pdf}
\end{center}
The inconsistencies at $t=1$ are:
\begin{descriptionlist}
\item[Inconsistent effects] \phantom{}\\[0.5em]
$\begin{cases}\texttt{NO-OP} \\ \texttt{LOAD(a, r)}\end{cases} \text{for } \texttt{at(a, l)}$,
$\begin{cases}\texttt{NO-OP} \\ \texttt{LOAD(b, r)}\end{cases} \text{for } \texttt{at(b, l)}$,\\[0.3em]
$\begin{cases}\texttt{NO-OP} \\ \texttt{MOVE(r, l, p)}\end{cases} \text{for } \texttt{at(r, l)}$,
$\begin{cases}\texttt{NO-OP} \\ \texttt{MOVE(r, l, p)}\end{cases} \text{for } \texttt{hasFuel(r)}$
\item[Interference] \phantom{}\\[0.5em]
$\begin{cases}\texttt{MOVE(r, l, p)} \\ \texttt{LOAD(a, r)}\end{cases} \text{for } \texttt{at(r, l)}$,
$\begin{cases}\texttt{MOVE(r, l, p)} \\ \texttt{LOAD(b, r)}\end{cases} \text{for } \texttt{at(r, l)}$
\end{descriptionlist}
The inconsistencies of at $t=2$ are:
\begin{descriptionlist}
\item[Competing needs] \phantom{}\\[0.5em]
Consequence of the add and delete list of each action:\\[0.3em]
$\begin{cases}\texttt{at(a, l)} \\ \texttt{in(r, a)}\end{cases}$,
$\begin{cases}\texttt{at(b, l)} \\ \texttt{in(r, b)}\end{cases}$,
$\begin{cases}\texttt{at(r, l)} \\ \texttt{at(r, p)}\end{cases}$,
$\begin{cases}\texttt{at(r, p)} \\ \texttt{hasFuel(r)}\end{cases}$\\[0.5em]
Consequence of the add list of interfering actions (mutual exclusion):\\[0.3em]
$\begin{cases}\texttt{in(r, a)} \\ \texttt{at(r, p)}\end{cases}$,
$\begin{cases}\texttt{in(r, b)} \\ \texttt{at(r, p)}\end{cases}$
\end{descriptionlist}
Note that because of the mutually exclusive propositions at $t=2$,
at $t=3$ the actions \texttt{UNLOAD($\cdot$, p)} cannot be performed.
\end{example}
\subsection{Fast forward}
\marginnote{Fast forward}
Heuristic planner based on Graphplan, hill climbing and A$^*$.
\begin{description}
\item[Heuristic]
Given a problem $P$, the algorithm considers a relaxation $P^+$
where delete effects are ignored.
$P^+$ is solved using Graphplan and the number of actions required to solve it is used as lower bound heuristic for $P$.
\item[Algorithm] \phantom{}
\begin{enumerate}
\item From a state $S$, examine the successors.
\item If there is a successor $S'$ better than $S$, move into it and return to point 1.
\item Otherwise, run a complete A$^*$ search.
\end{enumerate}
\end{description}