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Add FAIKR1 STRIPS
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src/fundamentals-of-ai-and-kr/module1/img/_strips_example.pdf
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src/fundamentals-of-ai-and-kr/module1/img/_strips_example.pdf
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src/fundamentals-of-ai-and-kr/module1/img/strips_example.drawio
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src/fundamentals-of-ai-and-kr/module1/img/strips_example.drawio
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@ -13,5 +13,6 @@
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\input{sections/_swarm_intelligence.tex}
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\input{sections/_games.tex}
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\input{sections/_planning.tex}
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\input{sections/_generative_planning.tex}
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\end{document}
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@ -0,0 +1,328 @@
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\chapter{Generative planning}
|
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\begin{description}
|
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\item[Generative planning] \marginnote{Generative planning}
|
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Offline planning that creates the entire plan before execution based on
|
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a snapshot of the current state of the world.
|
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It relies on the following assumptions:
|
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\begin{descriptionlist}
|
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\item[Atomic time]
|
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Actions cannot be interrupted.
|
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\item[Determinism]
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Actions are deterministic.
|
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\item[Closed world]
|
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The initial state is fully known,
|
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what is not in the initial state is considered false (which is different from unknown).
|
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\item[No interference] Only the execution of the plan changes the state of the world.
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\end{descriptionlist}
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\end{description}
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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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\begin{itemize}
|
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\item Nodes contain the state of the world.
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\item Edges represent possible actions.
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\end{itemize}
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Produces a totally ordered list of actions.
|
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The direction of the search can be:
|
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\begin{descriptionlist}
|
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\item[Forward] \marginnote{Forward search}
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Starting from the initial state, the search terminates when a state containing a superset of the goal is reached.
|
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\item[Backward] \marginnote{Backward search}
|
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Starting from the goal, the search terminates when a state containing a subset of the initial state is reached.
|
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|
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Goal regression is used to reduce the goal into sub-goals.
|
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Given a (sub-)goal $G$ and a rule (action) $R$ with delete-list (states that are false after the action) \texttt{d\_list}
|
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and add-list (states that are true after the action) \texttt{a\_list}, regression of $G$ through $R$ is define as:
|
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\[
|
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\begin{split}
|
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\texttt{regr[$G$, $R$]} &= \texttt{true} \text{ if } G \in \texttt{a\_list} \text{ (i.e. regression possible)} \\
|
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\texttt{regr[$G$, $R$]} &= \texttt{false} \text{ if } G \in \texttt{d\_list} \text{ (i.e. regression not possible)} \\
|
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\texttt{regr[$G$, $R$]} &= G \text{ otherwise} \text{ (i.e. $R$ does not influence $G$)} \\
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\end{split}
|
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\]
|
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|
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\begin{example}[Moving blocks]
|
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Given the action \texttt{unstack(X, Y)} with:
|
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\[
|
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\begin{split}
|
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\texttt{d\_list} &= \{ \texttt{handempty}, \texttt{on(X, Y)}, \texttt{clear(X)} \} \\
|
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\texttt{a\_list} &= \{ \texttt{holding(X)}, \texttt{clear(Y)} \}
|
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\end{split}
|
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\]
|
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We have that:
|
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\[
|
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\begin{split}
|
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\texttt{regr[holding(b), unstack(b, Y)]} &= \texttt{true} \\
|
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\texttt{regr[handempty, unstack(X, Y)]} &= \texttt{false} \\
|
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\texttt{regr[ontable(c), unstack(X, Y)]} &= \texttt{ontable(c)} \\
|
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\texttt{regr[clear(c), unstack(X, Y)]} &= \begin{cases}
|
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\texttt{true} & \text{if \texttt{Y}=\texttt{c}} \\
|
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\texttt{clear(c)} & \text{otherwise}
|
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\end{cases}
|
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\end{split}
|
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\]
|
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\end{example}
|
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\end{descriptionlist}
|
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|
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|
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\subsection{Deductive planning}
|
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\marginnote{Deductive planning}
|
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Formulates the planning problem using first order logic to represent states, goals and actions.
|
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Plans are generated as theorem proofs.
|
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|
||||
\subsubsection{Green's formulation}
|
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\marginnote{Green's formulation}
|
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Green's formulation is based on \textbf{situation calculus}.
|
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To find a plan, the goal is negated and it is proven that it leads to an inconsistency.
|
||||
|
||||
The main concepts are:
|
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\begin{descriptionlist}
|
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\item[Situation]
|
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Properties (fluents) that hold in a given state \texttt{s}.
|
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\begin{example}[Moving blocks]
|
||||
To denote that \texttt{ontable(c)} holds in a state \texttt{s}, we use the axiom:
|
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\[ \texttt{ontable(c, s)} \]
|
||||
\end{example}
|
||||
The operator \texttt{do} allows to evolve the state such that:
|
||||
\[ \texttt{do(A, S)} = \texttt{S'} \]
|
||||
\texttt{S'} is the new state obtained by applying the action \texttt{A} in the state \texttt{S}.
|
||||
|
||||
\item[Actions]
|
||||
Define the pre-condition and post-condition fluents of an action in the form:
|
||||
\[ \texttt{pre-conditions} \rightarrow \texttt{post-conditions} \]
|
||||
Applying the equivalence $A \rightarrow B \equiv \lnot A \vee B$, actions can be described by means of disjunctions.
|
||||
\begin{example}[Moving blocks]
|
||||
The action \texttt{stack(X, Y)} has pre-conditions \texttt{holding(X)} and \texttt{clear(Y)}, and
|
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post-conditions \texttt{on(X, Y)}, \texttt{clear(X)} and \texttt{handfree}.
|
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Its representation in Green's formulation is:
|
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\[
|
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\begin{split}
|
||||
\texttt{holding(X, S)} \land \texttt{clear(Y, S)} &\rightarrow \\
|
||||
&\texttt{on(X, Y, do(stack(X, Y), s))} \land \\
|
||||
&\texttt{clear(X, do(stack(X, Y), s))} \land \\
|
||||
&\texttt{handfree(do(stack(X, Y), s))} \\
|
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\end{split}
|
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\]
|
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\end{example}
|
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|
||||
\item[Frame axioms]
|
||||
Besides the effects of actions, each state also have to define for all non-changing fluents their frame axioms.
|
||||
If the problem is complex, the number of frame axioms becomes unreasonable.
|
||||
\begin{example}[Moving blocks]
|
||||
\[ \texttt{on(U, V, S)}, \texttt{diff(U, X)} \rightarrow \texttt{on(U, V, do(move(X, Y, Z), S))} \]
|
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\end{example}
|
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\end{descriptionlist}
|
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|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
The initial state is described by the following axioms:\\[0.5em]
|
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\begin{minipage}{.3\linewidth}
|
||||
\centering
|
||||
\texttt{on(a, d, s0)} \\
|
||||
\texttt{on(b, e, s0)} \\
|
||||
\texttt{on(c, f, s0)} \\
|
||||
\texttt{clear(a, s0)} \\
|
||||
\texttt{clear(b, s0)} \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.3\linewidth}
|
||||
\centering
|
||||
\texttt{clear(c, s0)} \\
|
||||
\texttt{clear(g, s0)} \\
|
||||
\texttt{diff(a, b)} \\
|
||||
\texttt{diff(a, c)} \\
|
||||
\texttt{diff(a, d)} \dots \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.3\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{img/_moving_block_example_green.pdf}
|
||||
\end{minipage}\\[0.5em]
|
||||
|
||||
For simplicity, we only consider the action \texttt{move(X, Y, Z)} that moves \texttt{X} from \texttt{Y} to \texttt{Z}.
|
||||
It is defined as:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{clear(X, S)}&, \texttt{clear(Z, S)}, \texttt{on(X, Y, S)}, \texttt{diff(X, Z)} \rightarrow \\
|
||||
&\texttt{clear(Y, do(move(X, Y, Z), S))}, \texttt{on(X, Z, do(move(X, Y, Z), S))}
|
||||
\end{split}
|
||||
\]
|
||||
This action can be translated into the following effect axioms:
|
||||
\[
|
||||
\begin{split}
|
||||
\lnot\texttt{clear(X, S)} &\vee \lnot\texttt{clear(Z, S)} \vee \lnot\texttt{on(X, Y, S)} \vee \lnot\texttt{diff(X, Z)} \vee \\
|
||||
&\texttt{clear(Y, do(move(X, Y, Z), S))}
|
||||
\end{split}
|
||||
\]
|
||||
\[
|
||||
\begin{split}
|
||||
\lnot\texttt{clear(X, S)} &\vee \lnot\texttt{clear(Z, S)} \vee \lnot\texttt{on(X, Y, S)} \vee \lnot\texttt{diff(X, Z)} \vee \\
|
||||
&\texttt{on(X, Z, do(move(X, Y, Z), S))}
|
||||
\end{split}
|
||||
\]
|
||||
\end{example}
|
||||
|
||||
Given the goal \texttt{on(a, b, s1)}, we prove that $\lnot\texttt{on(a, b, s1)}$ leads to an inconsistency.
|
||||
We decide to make the following substitutions:
|
||||
\[ \{ \texttt{X}/\texttt{a}, \texttt{Z}/\texttt{b}, \texttt{s1}/\texttt{do(move(a, Y, b), S)} \} \]
|
||||
The premise of \texttt{move} leads to an inconsistency (when applying \texttt{move} its premise is false):
|
||||
\begin{center}
|
||||
\begin{tabular}{c|c|c|c}
|
||||
$\lnot\texttt{clear(a, S)}$ & $\lnot\texttt{clear(b, S)}$ & $\lnot\texttt{on(a, Y, S)}$ & $\lnot\texttt{diff(a, b)}$ \\
|
||||
False with $\{ \texttt{S}/\texttt{s0} \}$ & False with $\{ \texttt{S}/\texttt{s0} \}$
|
||||
& False with $\{ \texttt{S}/\texttt{s0}, \texttt{Y}/\texttt{d} \}$ & False
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Therefore, the substitution $\{ \texttt{s1}/\texttt{do(move(a, Y, b), S)} \}$ defines the plan to reach the goal \texttt{on(a, b, s1)}.
|
||||
|
||||
|
||||
\subsubsection{Kowalsky's formulation}
|
||||
\marginnote{Kowalsky's formulation}
|
||||
Kowalsky's formulation avoids the frame axioms problem by using a set of fixed predicates:
|
||||
\begin{descriptionlist}
|
||||
\item[\texttt{holds(rel, s/a)}]
|
||||
Describes the relations \texttt{rel} that are true in a state \texttt{s} or after the execution of an action \texttt{a}.
|
||||
\item[\texttt{poss(s)}]
|
||||
Indicates if a state \texttt{s} is possible.
|
||||
\item[\texttt{pact(a, s)}]
|
||||
Indicates if an action \texttt{a} can be executed in a state \texttt{s}.
|
||||
\end{descriptionlist}
|
||||
Actions can be described as:
|
||||
\[ \texttt{poss(S)} \land \texttt{pact(A, S)} \rightarrow \texttt{poss(do(A, S))} \]
|
||||
|
||||
In the Kowalsky's formulation, each action requires a frame assertion (in Green's formulation, each state requires frame axioms).
|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
An initial state can be described by the following axioms:\\[0.5em]
|
||||
\begin{minipage}{.35\linewidth}
|
||||
\centering
|
||||
\texttt{holds(on(a, b), s0)} \\
|
||||
\texttt{holds(ontable(b), s0)} \\
|
||||
\texttt{holds(ontable(c), s0)} \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.35\linewidth}
|
||||
\centering
|
||||
\texttt{holds(clear(a), s0)} \\
|
||||
\texttt{holds(clear(c), s0)} \\
|
||||
\texttt{holds(handempty, s0)} \\
|
||||
\texttt{poss(s0)} \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.2\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{img/_moving_block_example_kowalsky.pdf}
|
||||
\end{minipage}\\[0.5em]
|
||||
\end{example}
|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
The action \texttt{unstack(X, Y)} has:
|
||||
\begin{descriptionlist}
|
||||
\item[Pre-conditions] \texttt{on(X, Y)}, \texttt{clear(X)} and \texttt{handempty}
|
||||
\item[Effects] \phantom{}
|
||||
\begin{description}
|
||||
\item[Add-list] \texttt{holding(X)} and \texttt{clear(Y)}
|
||||
\item[Delete-list] \texttt{on(X, Y)}, \texttt{clear(X)} and \texttt{handempty}
|
||||
\end{description}
|
||||
\end{descriptionlist}
|
||||
|
||||
Its description in Kowalsky's formulation is:
|
||||
\begin{descriptionlist}
|
||||
\item[Pre-conditions]
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{holds(on(X, Y), S)}&, \texttt{holds(clear(X), S)}, \texttt{holds(handempty, S)} \rightarrow \\
|
||||
&\texttt{pact(unstack(X, Y), S)}
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\item[Effects] (use add-list)
|
||||
\[ \texttt{holds(holding(X), do(unstack(X, Y), S))} \]
|
||||
\[ \texttt{holds(clear(Y), do(unstack(X, Y), S))} \]
|
||||
|
||||
\item[Frame condition] (uses delete-list)
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{holds(V, S)}&, \texttt{V} \neq \texttt{on(X, Y)}, \texttt{V} \neq \texttt{clear(X)}, \texttt{V} \neq \texttt{handempty}
|
||||
\rightarrow \\
|
||||
& \texttt{holds(V, do(unstack(X, Y), S))}
|
||||
\end{split}
|
||||
\]
|
||||
\end{descriptionlist}
|
||||
\end{example}
|
||||
|
||||
|
||||
\subsection{STRIPS}
|
||||
\marginnote{STRIPS}
|
||||
STRIPS (Stanford Research Institute Problem Solver) is an ad-hoc algorithm
|
||||
for linear planning resolution.
|
||||
The elements of the problem are represented as:
|
||||
\begin{descriptionlist}
|
||||
\item[State] represented with its true fluents.
|
||||
\item[Goal] represented with its true fluents.
|
||||
\item[Action] represented using three lists:
|
||||
\begin{descriptionlist}
|
||||
\item[Preconditions] Fluents that are required to be true in order to apply the action.
|
||||
\item[Delete-list] Fluents that become false after the action.
|
||||
\item[Add-list] Fluents that become true after the action.
|
||||
\end{descriptionlist}
|
||||
Add-list and delete-list can be combined in an effect list with positive (add-list) and negative (delete-list) axioms.
|
||||
|
||||
\begin{description}
|
||||
\item[STRIPS assumption] Everything that is not in the add-list or delete-list is unchanged in the next state.
|
||||
\end{description}
|
||||
\end{descriptionlist}
|
||||
|
||||
STRIPS uses two data structures:
|
||||
\begin{descriptionlist}
|
||||
\item[Goal stack] Does a backward search to reach the initial state.
|
||||
\item[Current state] Represents the forward application of the actions found using the goal stack.
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{STRIPS}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def strips(problem):
|
||||
goal_stack = Stack()
|
||||
current_state = State(problem.initial_state)
|
||||
goal_stack.push(problem.goal)
|
||||
plan = []
|
||||
while not goal_stack.empty():
|
||||
if (goal_stack.top() is a single/conjunction of goals and
|
||||
there is a substitution $\theta$ that makes it $\subseteq$ current_state):
|
||||
A = goal_stack.pop()
|
||||
$\theta$ = find_substitution(A, current_state)
|
||||
goal_stack.apply_substitution($\theta$)
|
||||
elif goal_stack.top() is a single goal:
|
||||
R = rule with a $\in$ R.add_list
|
||||
_ = goal_stack.pop() # Pop goal
|
||||
goal_stack.push(R)
|
||||
goal_stack.push(R.preconditions)
|
||||
elif goal_stack.top() is a conjunction of goals:
|
||||
for g in permutation(goal_stack.top()):
|
||||
goal_stack.push(g)
|
||||
# Note that there is no pop
|
||||
elif goal_stack.top() is an action:
|
||||
action = goal_stack.pop()
|
||||
current_state.apply(action)
|
||||
plan.append(action)
|
||||
return plan
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
\begin{center}
|
||||
\includegraphics[trim={0 32.2cm 0 0}, clip, width=0.85\textwidth]{img/_strips_example.pdf}
|
||||
\end{center}
|
||||
\begin{center}
|
||||
\includegraphics[trim={0 0 0 17.5cm}, clip, width=0.85\textwidth]{img/_strips_example.pdf}
|
||||
\end{center}
|
||||
\end{example}
|
||||
|
||||
Since there are non-deterministic choices, the search space may become very large.
|
||||
Heuristics may be used to avoid this.
|
||||
|
||||
Conjunction of goals are solved separately, but this could lead to the \marginnote{Sussman anomaly} \textbf{Sussman anomaly}
|
||||
where a sub-goal destroys what another sub-goal has done.
|
||||
For this reason, when a conjunction is encountered, it is not immediately popped from the goal stack
|
||||
and is left as a final check.
|
||||
@ -1,4 +1,4 @@
|
||||
\chapter{Automated planning}
|
||||
\chapter{Automated planning definitions}
|
||||
|
||||
\begin{description}
|
||||
\item[Automated planning] \marginnote{Automated planning}
|
||||
@ -50,257 +50,3 @@
|
||||
An action applied to the real world may have unexpected effects due to uncertainty.
|
||||
\end{descriptionlist}
|
||||
\end{description}
|
||||
|
||||
|
||||
\section{Generative planning}
|
||||
|
||||
\begin{description}
|
||||
\item[Generative planning] \marginnote{Generative planning}
|
||||
Offline planning that creates the entire plan before execution based on
|
||||
a snapshot of the current state of the world.
|
||||
It relies on the following assumptions:
|
||||
\begin{descriptionlist}
|
||||
\item[Atomic time]
|
||||
Actions cannot be interrupted.
|
||||
\item[Determinism]
|
||||
Actions are deterministic.
|
||||
\item[Closed world]
|
||||
The initial state is fully known,
|
||||
what is not in the initial state is considered false (which is different from unknown).
|
||||
\item[No interference] Only the execution of the plan changes the state of the world.
|
||||
\end{descriptionlist}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Linear planning}
|
||||
\marginnote{Linear planning}
|
||||
Formulates the planning problem as a search problem where:
|
||||
\begin{itemize}
|
||||
\item Nodes contain the state of the world.
|
||||
\item Edges represent possible actions.
|
||||
\end{itemize}
|
||||
Produces a totally ordered list of actions.
|
||||
|
||||
The direction of the search can be:
|
||||
\begin{descriptionlist}
|
||||
\item[Forward] \marginnote{Forward search}
|
||||
Starting from the initial state, the search terminates when a state containing a superset of the goal is reached.
|
||||
\item[Backward] \marginnote{Backward search}
|
||||
Starting from the goal, the search terminates when a state containing a subset of the initial state is reached.
|
||||
|
||||
Goal regression is used to reduce the goal into sub-goals.
|
||||
Given a (sub-)goal $G$ and a rule (action) $R$ with delete-list (states that are false after the action) \texttt{d\_list}
|
||||
and add-list (states that are true after the action) \texttt{a\_list}, regression of $G$ through $R$ is define as:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{regr[$G$, $R$]} &= \texttt{true} \text{ if } G \in \texttt{a\_list} \text{ (i.e. regression possible)} \\
|
||||
\texttt{regr[$G$, $R$]} &= \texttt{false} \text{ if } G \in \texttt{d\_list} \text{ (i.e. regression not possible)} \\
|
||||
\texttt{regr[$G$, $R$]} &= G \text{ otherwise} \text{ (i.e. $R$ does not influence $G$)} \\
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
Given the action \texttt{unstack(X, Y)} with:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{d\_list} &= \{ \texttt{handempty}, \texttt{on(X, Y)}, \texttt{clear(X)} \} \\
|
||||
\texttt{a\_list} &= \{ \texttt{holding(X)}, \texttt{clear(Y)} \}
|
||||
\end{split}
|
||||
\]
|
||||
We have that:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{regr[holding(b), unstack(b, Y)]} &= \texttt{true} \\
|
||||
\texttt{regr[handempty, unstack(X, Y)]} &= \texttt{false} \\
|
||||
\texttt{regr[ontable(c), unstack(X, Y)]} &= \texttt{ontable(c)} \\
|
||||
\texttt{regr[clear(c), unstack(X, Y)]} &= \begin{cases}
|
||||
\texttt{true} & \text{if \texttt{Y}=\texttt{c}} \\
|
||||
\texttt{clear(c)} & \text{otherwise}
|
||||
\end{cases}
|
||||
\end{split}
|
||||
\]
|
||||
\end{example}
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
\subsection{Deductive planning}
|
||||
\marginnote{Deductive planning}
|
||||
Formulates the planning problem using first order logic to represent states, goals and actions.
|
||||
Plans are generated as theorem proofs.
|
||||
|
||||
\subsubsection{Green's formulation}
|
||||
\marginnote{Green's formulation}
|
||||
Green's formulation is based on \textbf{situation calculus}.
|
||||
To find a plan, the goal is negated and it is proven that it leads to an inconsistency.
|
||||
|
||||
The main concepts are:
|
||||
\begin{descriptionlist}
|
||||
\item[Situation]
|
||||
Properties (fluents) that hold in a given state \texttt{s}.
|
||||
\begin{example}[Moving blocks]
|
||||
To denote that \texttt{ontable(c)} holds in a state \texttt{s}, we use the axiom:
|
||||
\[ \texttt{ontable(c, s)} \]
|
||||
\end{example}
|
||||
The operator \texttt{do} allows to evolve the state such that:
|
||||
\[ \texttt{do(A, S)} = \texttt{S'} \]
|
||||
\texttt{S'} is the new state obtained by applying the action \texttt{A} in the state \texttt{S}.
|
||||
|
||||
\item[Actions]
|
||||
Define the pre-condition and post-condition fluents of an action in the form:
|
||||
\[ \texttt{pre-conditions} \rightarrow \texttt{post-conditions} \]
|
||||
Applying the equivalence $A \rightarrow B \equiv \lnot A \vee B$, actions can be described by means of disjunctions.
|
||||
\begin{example}[Moving blocks]
|
||||
The action \texttt{stack(X, Y)} has pre-conditions \texttt{holding(X)} and \texttt{clear(Y)}, and
|
||||
post-conditions \texttt{on(X, Y)}, \texttt{clear(X)} and \texttt{handfree}.
|
||||
Its representation in Green's formulation is:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{holding(X, S)} \land \texttt{clear(Y, S)} &\rightarrow \\
|
||||
&\texttt{on(X, Y, do(stack(X, Y), s))} \land \\
|
||||
&\texttt{clear(X, do(stack(X, Y), s))} \land \\
|
||||
&\texttt{handfree(do(stack(X, Y), s))} \\
|
||||
\end{split}
|
||||
\]
|
||||
\end{example}
|
||||
|
||||
\item[Frame axioms]
|
||||
Besides the effects of actions, each state also have to define for all non-changing fluents their frame axioms.
|
||||
If the problem is complex, the number of frame axioms becomes unreasonable.
|
||||
\begin{example}[Moving blocks]
|
||||
\[ \texttt{on(U, V, S)}, \texttt{diff(U, X)} \rightarrow \texttt{on(U, V, do(move(X, Y, Z), S))} \]
|
||||
\end{example}
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
The initial state is described by the following axioms:\\[0.5em]
|
||||
\begin{minipage}{.3\linewidth}
|
||||
\centering
|
||||
\texttt{on(a, d, s0)} \\
|
||||
\texttt{on(b, e, s0)} \\
|
||||
\texttt{on(c, f, s0)} \\
|
||||
\texttt{clear(a, s0)} \\
|
||||
\texttt{clear(b, s0)} \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.3\linewidth}
|
||||
\centering
|
||||
\texttt{clear(c, s0)} \\
|
||||
\texttt{clear(g, s0)} \\
|
||||
\texttt{diff(a, b)} \\
|
||||
\texttt{diff(a, c)} \\
|
||||
\texttt{diff(a, d)} \dots \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.3\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{img/_moving_block_example_green.pdf}
|
||||
\end{minipage}\\[0.5em]
|
||||
|
||||
For simplicity, we only consider the action \texttt{move(X, Y, Z)} that moves \texttt{X} from \texttt{Y} to \texttt{Z}.
|
||||
It is defined as:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{clear(X, S)}&, \texttt{clear(Z, S)}, \texttt{on(X, Y, S)}, \texttt{diff(X, Z)} \rightarrow \\
|
||||
&\texttt{clear(Y, do(move(X, Y, Z), S))}, \texttt{on(X, Z, do(move(X, Y, Z), S))}
|
||||
\end{split}
|
||||
\]
|
||||
This action can be translated into the following effect axioms:
|
||||
\[
|
||||
\begin{split}
|
||||
\lnot\texttt{clear(X, S)} &\vee \lnot\texttt{clear(Z, S)} \vee \lnot\texttt{on(X, Y, S)} \vee \lnot\texttt{diff(X, Z)} \vee \\
|
||||
&\texttt{clear(Y, do(move(X, Y, Z), S))}
|
||||
\end{split}
|
||||
\]
|
||||
\[
|
||||
\begin{split}
|
||||
\lnot\texttt{clear(X, S)} &\vee \lnot\texttt{clear(Z, S)} \vee \lnot\texttt{on(X, Y, S)} \vee \lnot\texttt{diff(X, Z)} \vee \\
|
||||
&\texttt{on(X, Z, do(move(X, Y, Z), S))}
|
||||
\end{split}
|
||||
\]
|
||||
\end{example}
|
||||
|
||||
Given the goal \texttt{on(a, b, s1)}, we prove that $\lnot\texttt{on(a, b, s1)}$ leads to an inconsistency.
|
||||
We decide to make the following substitutions:
|
||||
\[ \{ \texttt{X}/\texttt{a}, \texttt{Z}/\texttt{b}, \texttt{s1}/\texttt{do(move(a, Y, b), S)} \} \]
|
||||
The premise of \texttt{move} leads to an inconsistency (when applying \texttt{move} its premise is false):
|
||||
\begin{center}
|
||||
\begin{tabular}{c|c|c|c}
|
||||
$\lnot\texttt{clear(a, S)}$ & $\lnot\texttt{clear(b, S)}$ & $\lnot\texttt{on(a, Y, S)}$ & $\lnot\texttt{diff(a, b)}$ \\
|
||||
False with $\{ \texttt{S}/\texttt{s0} \}$ & False with $\{ \texttt{S}/\texttt{s0} \}$
|
||||
& False with $\{ \texttt{S}/\texttt{s0}, \texttt{Y}/\texttt{d} \}$ & False
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Therefore, the substitution $\{ \texttt{s1}/\texttt{do(move(a, Y, b), S)} \}$ defines the plan to reach the goal \texttt{on(a, b, s1)}.
|
||||
|
||||
|
||||
\subsubsection{Kowalsky's formulation}
|
||||
\marginnote{Kowalsky's formulation}
|
||||
Kowalsky's formulation avoids the frame axioms problem by using a set of fixed predicates:
|
||||
\begin{descriptionlist}
|
||||
\item[\texttt{holds(rel, s/a)}]
|
||||
Describes the relations \texttt{rel} that are true in a state \texttt{s} or after the execution of an action \texttt{a}.
|
||||
\item[\texttt{poss(s)}]
|
||||
Indicates if a state \texttt{s} is possible.
|
||||
\item[\texttt{pact(a, s)}]
|
||||
Indicates if an action \texttt{a} can be executed in a state \texttt{s}.
|
||||
\end{descriptionlist}
|
||||
Actions can be described as:
|
||||
\[ \texttt{poss(S)} \land \texttt{pact(A, S)} \rightarrow \texttt{poss(do(A, S))} \]
|
||||
|
||||
In the Kowalsky's formulation, each action requires a frame assertion (in Green's formulation, each state requires frame axioms).
|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
An initial state can be described by the following axioms:\\[0.5em]
|
||||
\begin{minipage}{.35\linewidth}
|
||||
\centering
|
||||
\texttt{holds(on(a, b), s0)} \\
|
||||
\texttt{holds(ontable(b), s0)} \\
|
||||
\texttt{holds(ontable(c), s0)} \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.35\linewidth}
|
||||
\centering
|
||||
\texttt{holds(clear(a), s0)} \\
|
||||
\texttt{holds(clear(c), s0)} \\
|
||||
\texttt{holds(handempty, s0)} \\
|
||||
\texttt{poss(s0)} \\
|
||||
\end{minipage}
|
||||
\begin{minipage}{.2\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{img/_moving_block_example_kowalsky.pdf}
|
||||
\end{minipage}\\[0.5em]
|
||||
\end{example}
|
||||
|
||||
\begin{example}[Moving blocks]
|
||||
The action \texttt{unstack(X, Y)} has:
|
||||
\begin{descriptionlist}
|
||||
\item[Pre-conditions] \texttt{on(X, Y)}, \texttt{clear(X)} and \texttt{handempty}
|
||||
\item[Effects] \phantom{}
|
||||
\begin{description}
|
||||
\item[Add-list] \texttt{holding(X)} and \texttt{clear(Y)}
|
||||
\item[Delete-list] \texttt{on(X, Y)}, \texttt{clear(X)} and \texttt{handempty}
|
||||
\end{description}
|
||||
\end{descriptionlist}
|
||||
|
||||
Its description in Kowalsky's formulation is:
|
||||
\begin{descriptionlist}
|
||||
\item[Pre-conditions]
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{holds(on(X, Y), S)}&, \texttt{holds(clear(X), S)}, \texttt{holds(handempty, S)} \rightarrow \\
|
||||
&\texttt{pact(unstack(X, Y), S)}
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\item[Effects] (use add-list)
|
||||
\[ \texttt{holds(holding(X), do(unstack(X, Y), S))} \]
|
||||
\[ \texttt{holds(clear(Y), do(unstack(X, Y), S))} \]
|
||||
|
||||
\item[Frame condition] (uses delete-list)
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{holds(V, S)}&, \texttt{V} \neq \texttt{on(X, Y)}, \texttt{V} \neq \texttt{clear(X)}, \texttt{V} \neq \texttt{handempty}
|
||||
\rightarrow \\
|
||||
& \texttt{holds(V, do(unstack(X, Y), S))}
|
||||
\end{split}
|
||||
\]
|
||||
\end{descriptionlist}
|
||||
\end{example}
|
||||
Reference in New Issue
Block a user