From 5bd52e414d5245cbf473aa2afe78b5d2c4d36ab4 Mon Sep 17 00:00:00 2001
From: NotXia <35894453+NotXia@users.noreply.github.com>
Date: Thu, 30 Nov 2023 18:14:27 +0100
Subject: [PATCH] Add FAIKR1 Graphplan
---
.../module1/img/_graphplan.drawio | 496 ++++++++++++++++++
.../module1/img/_graphplan.pdf | Bin 0 -> 35739 bytes
.../module1/sections/_planning.tex | 252 ++++++++-
3 files changed, 746 insertions(+), 2 deletions(-)
create mode 100644 src/fundamentals-of-ai-and-kr/module1/img/_graphplan.drawio
create mode 100644 src/fundamentals-of-ai-and-kr/module1/img/_graphplan.pdf
diff --git a/src/fundamentals-of-ai-and-kr/module1/img/_graphplan.drawio b/src/fundamentals-of-ai-and-kr/module1/img/_graphplan.drawio
new file mode 100644
index 0000000..743e7f2
--- /dev/null
+++ b/src/fundamentals-of-ai-and-kr/module1/img/_graphplan.drawio
@@ -0,0 +1,496 @@
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diff --git a/src/fundamentals-of-ai-and-kr/module1/img/_graphplan.pdf b/src/fundamentals-of-ai-and-kr/module1/img/_graphplan.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..6c25f8af3cd3eabba19a67ed0ac32876705df8a5
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diff --git a/src/fundamentals-of-ai-and-kr/module1/sections/_planning.tex b/src/fundamentals-of-ai-and-kr/module1/sections/_planning.tex
index 751b937..9902615 100644
--- a/src/fundamentals-of-ai-and-kr/module1/sections/_planning.tex
+++ b/src/fundamentals-of-ai-and-kr/module1/sections/_planning.tex
@@ -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}
\ No newline at end of file
+\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}
\ No newline at end of file