From 6df916b1bfe6e37cd1c087c493f297a34ccbc5f7 Mon Sep 17 00:00:00 2001
From: NotXia <35894453+NotXia@users.noreply.github.com>
Date: Wed, 8 Nov 2023 21:01:09 +0100
Subject: [PATCH] Add FAIKR1 STRIPS

---
 .../module1/img/_strips_example.pdf           | Bin 0 -> 77022 bytes
 .../module1/img/strips_example.drawio         | 506 ++++++++++++++++++
 .../module1/main.tex                          |   1 +
 .../module1/sections/_generative_planning.tex | 328 ++++++++++++
 .../module1/sections/_planning.tex            | 256 +--------
 5 files changed, 836 insertions(+), 255 deletions(-)
 create mode 100644 src/fundamentals-of-ai-and-kr/module1/img/_strips_example.pdf
 create mode 100644 src/fundamentals-of-ai-and-kr/module1/img/strips_example.drawio
 create mode 100644 src/fundamentals-of-ai-and-kr/module1/sections/_generative_planning.tex

diff --git a/src/fundamentals-of-ai-and-kr/module1/img/_strips_example.pdf b/src/fundamentals-of-ai-and-kr/module1/img/_strips_example.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..af462c2e23da579f596f61c70e83a37420cac340
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new file mode 100644
index 0000000..5f0fea7
--- /dev/null
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diff --git a/src/fundamentals-of-ai-and-kr/module1/main.tex b/src/fundamentals-of-ai-and-kr/module1/main.tex
index 2b61075..99160d2 100644
--- a/src/fundamentals-of-ai-and-kr/module1/main.tex
+++ b/src/fundamentals-of-ai-and-kr/module1/main.tex
@@ -13,5 +13,6 @@
     \input{sections/_swarm_intelligence.tex}
     \input{sections/_games.tex}
     \input{sections/_planning.tex}
+    \input{sections/_generative_planning.tex}
 
 \end{document}
\ No newline at end of file
diff --git a/src/fundamentals-of-ai-and-kr/module1/sections/_generative_planning.tex b/src/fundamentals-of-ai-and-kr/module1/sections/_generative_planning.tex
new file mode 100644
index 0000000..1997716
--- /dev/null
+++ b/src/fundamentals-of-ai-and-kr/module1/sections/_generative_planning.tex
@@ -0,0 +1,328 @@
+\chapter{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}
+
+
+\section{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}
+
+
+\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.
\ No newline at end of file
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 e50c2c8..4e1a57b 100644
--- a/src/fundamentals-of-ai-and-kr/module1/sections/_planning.tex
+++ b/src/fundamentals-of-ai-and-kr/module1/sections/_planning.tex
@@ -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}
\ No newline at end of file