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Created FAIKR modules
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src/fundamentals-of-ai-and-kr/module1/ainotes.cls
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src/fundamentals-of-ai-and-kr/module1/ainotes.cls
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13
src/fundamentals-of-ai-and-kr/module1/main.tex
Normal file
13
src/fundamentals-of-ai-and-kr/module1/main.tex
Normal file
@ -0,0 +1,13 @@
|
||||
\documentclass[11pt]{ainotes}
|
||||
|
||||
\title{Fundamentals of Artificial Intelligence and Knowledge Representation\\(Module 1)}
|
||||
\date{2023 -- 2024}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\makenotesfront
|
||||
|
||||
\input{sections/_intro.tex}
|
||||
\input{sections/_search.tex}
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\end{document}
|
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215
src/fundamentals-of-ai-and-kr/module1/sections/_intro.tex
Normal file
215
src/fundamentals-of-ai-and-kr/module1/sections/_intro.tex
Normal file
@ -0,0 +1,215 @@
|
||||
\lohead{\color{gray} Not required for the exam}
|
||||
\lehead{\color{gray} Not required for the exam}
|
||||
\chapter{Introduction}
|
||||
|
||||
|
||||
\section{AI systems classification}
|
||||
|
||||
\subsection{Intelligence classification}
|
||||
Intelligence is defined as the ability to perceive or infer information and to retain the knowledge for future use.
|
||||
|
||||
\begin{description}
|
||||
\item[Weak AI] \marginnote{Weak AI}
|
||||
aims to build a system that acts as an intelligent system.
|
||||
|
||||
\item[Strong AI] \marginnote{Strong AI}
|
||||
aims to build a system that is actually intelligent.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Capability classification}
|
||||
\begin{description}
|
||||
\item[General AI] \marginnote{General AI}
|
||||
systems able to solve any generalized task.
|
||||
|
||||
\item[Narrow AI] \marginnote{Narrow AI}
|
||||
systems able to solve a particular task.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{AI approaches}
|
||||
\begin{description}
|
||||
\item[Symbolic AI (top-down)] \marginnote{Symbolic AI}
|
||||
Symbolic representation of knowledge, understandable by humans.
|
||||
|
||||
\item[Connectionist approach (bottom up)] \marginnote{Connectionist approach}
|
||||
Neural networks. Knowledge is encoded and not understandable by humans.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Symbolic AI}
|
||||
\begin{description}
|
||||
\item[Deductive reasoning] \marginnote{Deductive reasoning}
|
||||
Conclude something given some premises (general to specific).
|
||||
It is unable to produce new knowledge.
|
||||
\begin{example}
|
||||
"All men are mortal" and "Socrates is a man" $\rightarrow$ "Socrates is mortal"
|
||||
\end{example}
|
||||
|
||||
\item[Inductive reasoning] \marginnote{Inductive reasoning}
|
||||
A conclusion is derived from an observation (specific to general).
|
||||
Produces new knowledge, but correctness is not guaranteed.
|
||||
\begin{example}
|
||||
"Several birds fly" $\rightarrow$ "All birds fly"
|
||||
\end{example}
|
||||
|
||||
\item[Abduction reasoning] \marginnote{Abduction reasoning}
|
||||
An explanation of the conclusion is found from known premises.
|
||||
Differently from inductive reasoning, it does not search for a general rule.
|
||||
Produces new knowledge, but correctness is not guaranteed.
|
||||
\begin{example}
|
||||
"Socrates is dead" (conclusion) and "All men are mortal" (knowledge) $\rightarrow$ "Socrates is a man"
|
||||
\end{example}
|
||||
|
||||
\item[Reasoning by analogy] \marginnote{Reasoning by analogy}
|
||||
Principle of similarity (e.g. k-nearest-neighbor algorithm).
|
||||
\begin{example}
|
||||
"Socrates loves philosophy" and Socrates resembles John $\rightarrow$ "John loves philosophy"
|
||||
\end{example}
|
||||
|
||||
\item[Constraint reasoning and optimization] \marginnote{Constraint reasoning}
|
||||
Constraints, probability, statistics.
|
||||
\end{description}
|
||||
|
||||
|
||||
\section{Machine learning}
|
||||
|
||||
\subsection{Training approach}
|
||||
\begin{description}
|
||||
\item[Supervised learning] \marginnote{Supervised learning}
|
||||
Trained on labeled data (ground truth is known).\\
|
||||
Suitable for classification and regression tasks.
|
||||
|
||||
\item[Unsupervised learning] \marginnote{Unsupervised learning}
|
||||
Trained on unlabeled data (the system makes its own discoveries).\\
|
||||
Suitable for clustering and data mining.
|
||||
|
||||
\item[Semi-supervised learning] \marginnote{Semi-supervised learning}
|
||||
The system is first trained to synthesize data in an unsupervised manner,
|
||||
followed by a supervised phase.
|
||||
|
||||
\item[Reinforcement learning] \marginnote{Reinforcement learning}
|
||||
An agent learns by simulating actions in an environment with rewards and punishments depending on its choices.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Tasks}
|
||||
\begin{description}
|
||||
\item[Classification] \marginnote{Classification}
|
||||
Supervised task that, given the input variables $X$ and the output (discrete) categories $Y$,
|
||||
aims to approximate a mapping function $f: X \rightarrow Y$.
|
||||
|
||||
\item[Regression] \marginnote{Regression}
|
||||
Supervised task that, given the input variables $X$ and the output (continuous) variables $Y$,
|
||||
aims to approximate a mapping function $f: X \rightarrow Y$.
|
||||
|
||||
\item[Clustering] \marginnote{Clustering}
|
||||
Unsupervised task that aims to organize objects into groups.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Neural networks}
|
||||
\marginnote{Perceptron}
|
||||
A neuron (\textbf{perceptron}) computes a weighted sum of its inputs and
|
||||
passes the result to an activation function to produce the output.
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.40\textwidth]{img/neuron.png}
|
||||
\caption{Representation of an artificial neuron}
|
||||
\end{figure}
|
||||
|
||||
\marginnote{Feed-forward neural network}
|
||||
A \textbf{feed-forward neural network} is composed of multiple layers of neurons, each connected to the next one.
|
||||
The first layer is the input layer, while the last is the output layer.
|
||||
Intermediate layers are hidden layers.
|
||||
|
||||
The expressivity of a neural networks increases when more neurons are used:
|
||||
\begin{descriptionlist}
|
||||
\item[Single perceptron]
|
||||
Able to compute a linear separation.
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.25\textwidth]{img/1perceptron.png}
|
||||
\caption{Separation performed by one perceptron}
|
||||
\end{figure}
|
||||
\item[Three-layer network]
|
||||
Able to separate a convex region ($n_\text{edges} \leq n_\text{hidden neurons}$)
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.90\textwidth]{img/3layer.png}
|
||||
\caption{Separation performed by a three-layer network}
|
||||
\end{figure}
|
||||
\item[Four-layer network]
|
||||
Able to separate regions of arbitrary shape.
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.40\textwidth]{img/4layer.png}
|
||||
\caption{Separation performed by a four-layer network}
|
||||
\end{figure}
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{theorem}[Universal approximation theorem] \marginnote{Universal approximation theorem}
|
||||
A feed-forward network with one hidden layer and a finite number of neurons is
|
||||
able to approximate any continuous function with desired accuracy.
|
||||
\end{theorem}
|
||||
|
||||
\begin{description}
|
||||
\item[Deep learning] \marginnote{Deep learning}
|
||||
Neural network with a large number of layers and neurons.
|
||||
The learning process is hierarchical: the network exploits simple features in the first layers and
|
||||
synthesis more complex concepts while advancing through the layers.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Automated planning}
|
||||
Given an initial state, a set of actions and a goal,
|
||||
\textbf{automated planning} aims to find a partially or totally ordered sequence of actions to achieve a goal. \marginnote{Automated planning}
|
||||
|
||||
An \textbf{automated planner} is an agent that operates in a given domain described by:
|
||||
\begin{itemize}
|
||||
\item Representation of the initial state
|
||||
\item Representation of a goal
|
||||
\item Formal description of the possible actions (preconditions and effects)
|
||||
\end{itemize}
|
||||
|
||||
|
||||
|
||||
\section{Swarm intelligence}
|
||||
\marginnote{Swarm intelligence}
|
||||
Decentralized and self-organized systems that result in emergent behaviors.
|
||||
|
||||
|
||||
|
||||
\section{Decision support systems}
|
||||
|
||||
\begin{description}
|
||||
\item[Knowledge based system] \marginnote{Knowledge based system}
|
||||
Use knowledge (and data) to support human decisions.
|
||||
Bottlenecked by knowledge acquisition.
|
||||
\end{description}
|
||||
|
||||
Different levels of decision support exist:
|
||||
\begin{descriptionlist}
|
||||
\item[Descriptive analytics] \marginnote{Descriptive analytics}
|
||||
Data are used to describe the system (e.g. dashboards, reports, \dots).
|
||||
Human intervention is required.
|
||||
|
||||
\item[Diagnostic analytics] \marginnote{Diagnostic analytics}
|
||||
Data are used to understand causes (e.g. fault diagnosis)
|
||||
Decisions are made by humans.
|
||||
|
||||
\item[Predictive analytics] \marginnote{Predictive analytics}
|
||||
Data are used to predict future evolutions of the system.
|
||||
Uses machine learning models or simulators (digital twins)
|
||||
|
||||
\item[Prescriptive analytics] \marginnote{Prescriptive analytics}
|
||||
Make decisions by finding the preferred scenario.
|
||||
Uses optimization systems, combinatorial solvers or logical solvers.
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
\newpage
|
||||
\lohead{}
|
||||
\lehead{}
|
||||
409
src/fundamentals-of-ai-and-kr/module1/sections/_search.tex
Normal file
409
src/fundamentals-of-ai-and-kr/module1/sections/_search.tex
Normal file
@ -0,0 +1,409 @@
|
||||
\chapter{Search problems}
|
||||
|
||||
|
||||
|
||||
\section{Search strategies}
|
||||
\begin{description}
|
||||
\item[Solution space] \marginnote{Solution space}
|
||||
Set of all the possible sequences of actions an agent may apply.
|
||||
Some of these lead to a solution.
|
||||
|
||||
\item[Search algorithm] \marginnote{Search algorithm}
|
||||
Takes a problem as input and returns a sequence of actions that solves the problem (if exists).
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Search tree}
|
||||
\begin{description}
|
||||
\item[Expansion] \marginnote{Expansion}
|
||||
Starting from a state, apply a successor function and generate a new state.
|
||||
|
||||
\item[Search strategy] \marginnote{Search strategy}
|
||||
Choose which state to expand.
|
||||
Usually is implemented using a fringe that decides which is the next node to expand.
|
||||
|
||||
\item[Search tree] \marginnote{Search tree}
|
||||
Tree structure to represent the expansion of all states starting from a root
|
||||
(i.e. the representation of the solution space).
|
||||
|
||||
Nodes are states and branches are actions.
|
||||
A leaf can be a state to expand, a solution or a dead-end.
|
||||
\Cref{alg:search_tree_search} describes a generic tree search algorithm.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.25\textwidth]{img/_search_tree.pdf}
|
||||
\caption{Search tree}
|
||||
\end{figure}
|
||||
|
||||
Each node contains:
|
||||
\begin{itemize}
|
||||
\item The state
|
||||
\item The parent node
|
||||
\item The action that led to this node
|
||||
\item The depth of the node
|
||||
\item The cost of the path from the root to this node
|
||||
\end{itemize}
|
||||
\end{description}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Tree search} \label{alg:search_tree_search}
|
||||
\begin{lstlisting}
|
||||
def treeSearch(problem, fringe):
|
||||
fringe.push(problem.initial_state)
|
||||
# Get a node in the fringe and expand it if it is not a solution
|
||||
while fringe.notEmpty():
|
||||
node = fringe.pop()
|
||||
if problem.isGoal(node.state):
|
||||
return node.solution
|
||||
fringe.pushAll(expand(node, problem))
|
||||
return FAILURE
|
||||
|
||||
def expand(node, problem):
|
||||
successors = set()
|
||||
# List all neighboring nodes
|
||||
for action, result in problem.successor(node.state):
|
||||
s = new Node(
|
||||
parent=node, action=action, state=result, depth=node.dept+1,
|
||||
cost=node.cost + problem.pathCost(node, s, action)
|
||||
)
|
||||
successors.add(s)
|
||||
return successors
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\subsection{Strategies}
|
||||
\begin{description}
|
||||
\item[Non-informed strategy] \marginnote{Non-informed strategy}
|
||||
Domain knowledge not available. Usually does an exhaustive search.
|
||||
|
||||
\item[Informed strategy] \marginnote{Informed strategy}
|
||||
Use domain knowledge by using heuristics.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Evaluation}
|
||||
\begin{description}
|
||||
\item[Completeness] \marginnote{Completeness}
|
||||
if the strategy is guaranteed to find a solution (when exists).
|
||||
|
||||
\item[Time complexity] \marginnote{Time complexity}
|
||||
time needed to complete the search.
|
||||
|
||||
\item[Space complexity] \marginnote{Space complexity}
|
||||
memory needed to complete the search.
|
||||
|
||||
\item[Optimality] \marginnote{Optimality}
|
||||
if the strategy finds the best solution (when more solutions are possible).
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Non-informed search}
|
||||
|
||||
\subsection{Breadth-first search (BFS)}
|
||||
\marginnote{Breadth-first search}
|
||||
Always expands the less deep node. The fringe is implemented as a queue (FIFO).
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Only with uniform cost (i.e. all edges have same cost) \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, where the solution depth is $d$ and the branching factor is $b$ (i.e. each non-leaf node has $b$ children) \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
The exponential space complexity makes BFS impractical for large problems.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.30\textwidth]{img/_bfs.pdf}
|
||||
\caption{BFS visit order}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Uniform-cost search}
|
||||
\marginnote{Uniform-cost search}
|
||||
Same as BFS, but always expands the node with the lowest cumulative cost.
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Yes \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.50\textwidth]{img/_ucs.pdf}
|
||||
\caption{Uniform-cost search visit order. $(n)$ is the cumulative cost}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Depth-first search (DFS)}
|
||||
\marginnote{Depth-first search}
|
||||
Always expands the deepest node. The fringe is implemented as a stack (LIFO).
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & No (loops) \\
|
||||
\hline
|
||||
\textbf{Optimality} & No \\
|
||||
\hline
|
||||
\textbf{Time complexity}
|
||||
& $O(b^m)$, with maximum depth $m$ and branching factor $b$ \\
|
||||
\hline
|
||||
\textbf{Space complexity}
|
||||
& $O(b \cdot m)$, with maximum depth $m$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.30\textwidth]{img/_dfs.pdf}
|
||||
\caption{DFS visit order}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Depth-limited search}
|
||||
\marginnote{Depth-limited search}
|
||||
Same as DFS, but introduces a maximum depth.
|
||||
A node at the maximum depth will not be explored further.
|
||||
|
||||
This allows to avoid infinite branches (i.e. loops).
|
||||
|
||||
|
||||
|
||||
\subsection{Iterative deepening}
|
||||
\marginnote{Iterative deepening}
|
||||
Rus a depth-limited search by trying all possible depth limits.
|
||||
It is important to note that each iteration is executed from scratch (i.e. a new execution of depth-limited search).
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Iterative deepening}
|
||||
\begin{lstlisting}
|
||||
def iterativeDeepening(G):
|
||||
for c in range(G.max_depth):
|
||||
sol = depthLimitedSearch(G, c)
|
||||
if sol is not FAILURE:
|
||||
return sol
|
||||
return FAILURE
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
Both advantages of DFS and BFS are combined.
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Only with uniform cost \\
|
||||
\hline
|
||||
\textbf{Time complexity}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\textbf{Space complexity}
|
||||
& $O(b \cdot d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
|
||||
|
||||
\section{Informed search}
|
||||
\marginnote{Informed search}
|
||||
Informed search uses evaluation functions (heuristics) to reduce the search space and
|
||||
estimate the effort needed to reach the final goal.
|
||||
|
||||
|
||||
\subsection{Best-first search}
|
||||
\marginnote{Best-first seacrh}
|
||||
Uses heuristics to compute the desirability of the nodes (i.e. how close they are to the goal).
|
||||
The fringe is ordered according the estimated scores.
|
||||
|
||||
|
||||
\begin{description}
|
||||
\item[Greedy search / Hill climbing]
|
||||
\marginnote{Greedy search / Hill climbing}
|
||||
The heuristic only evaluates nodes individually and does not consider the path to the root
|
||||
(i.e. expands the node that currently seems closer to the goal).
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{9cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & No (loops) \\
|
||||
\hline
|
||||
\textbf{Optimality} & No \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
% The complexity can be reduced depending on the heuristic.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.65\textwidth]{img/_greedy_best_first_example.pdf}
|
||||
\caption{Hill climbing visit order}
|
||||
\end{figure}
|
||||
|
||||
\item[A$^\textbf{*}$]
|
||||
\marginnote{A$^*$}
|
||||
The heuristic also considers the cumulative cost needed to reach a node from the root.
|
||||
The score associated to a node $n$ is:
|
||||
\[ f(n) = g(n) + h'(n) \]
|
||||
where $g$ is the depth of the node and $h'$ is the heuristic that computes the distance to the goal.
|
||||
|
||||
\begin{description}
|
||||
\item[Optimistic/Feasible heuristic]
|
||||
\marginnote{Optimistic/Feasible heuristic}
|
||||
Given $t(n)$ that computes the true distance of a node $n$ to the goal.
|
||||
An heuristic $h'(n)$ is optimistic (i.e. feasible) if:
|
||||
\[ h'(n) \leq t(n) \]
|
||||
In other words, $h'$ is optimistic if it always underestimates the distance to the goal.
|
||||
\end{description}
|
||||
|
||||
\begin{theorem}
|
||||
If the heuristic used by A${^*}$ is optimistic $\Rightarrow$ A${^*}$ is optimal
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Consider a scenario where the queue contains:
|
||||
\begin{itemize}
|
||||
\item A node $n$ whose child is the optimal solution
|
||||
\item A sub-optimal solution $G_2$
|
||||
\end{itemize}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.5\textwidth]{img/_a_start_optimality.pdf}
|
||||
\end{center}
|
||||
We want to prove that A$^*$ will always expand $n$.
|
||||
|
||||
Given an optimistic heuristic $f(n) = g(n) + h'(n)$ and
|
||||
the true distance of a node $n$ to the goal $t(n)$,
|
||||
we have that:
|
||||
\[
|
||||
\begin{split}
|
||||
f(G_2) &= g(G_2) + h'(G_2) = g(G_2) \text{, as } G_2 \text{ is a solution: } h'(G_2)=0 \\
|
||||
f(G) &= g(G) + h'(G) = g(G) \text{, as } G \text{ is a solution: } h'(G)=0
|
||||
\end{split}
|
||||
\]
|
||||
Moreover, $g(G_2) > g(G)$ as $G_2$ is suboptimal.
|
||||
Therefore, $\bm{f(G_2) > f(G)}$.
|
||||
|
||||
Furthermore, as $h'$ is feasible, we have that:
|
||||
\[
|
||||
\begin{split}
|
||||
h'(n) \leq t(n) &\iff g(n) + h'(n) \leq g(n) + t(n) = g(G)=f(G) \\
|
||||
&\iff \bm{f(n) \leq f(G)}
|
||||
\end{split}
|
||||
\]
|
||||
In the end, we have that $f(G_2) > f(G) \geq f(n)$.
|
||||
So we can conclude that A$^*$ will never expand $G_2$ as:
|
||||
\[ f(G_2) > f(n) \]
|
||||
\end{proof}
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{9cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Only if the heuristic is optimistic \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
In generally, it is better to use heuristics with large values (i.e. heuristics that don't underestimate too much).
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.65\textwidth]{img/_a_start_example.pdf}
|
||||
\caption{A$^*$ visit order}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
\section{Graph search}
|
||||
\marginnote{Graph search}
|
||||
Differently from a tree search, searching in a graph requires to keep track of the explored nodes.
|
||||
\begin{algorithm}
|
||||
\caption{Graph search} \label{alg:search_graph_search}
|
||||
\begin{lstlisting}
|
||||
def graphSearch(problem, fringe):
|
||||
closed = set()
|
||||
fringe.push(problem.initial_state)
|
||||
# Get a node in the fringe and
|
||||
# expand it if it is not a solution and is not closed
|
||||
while fringe.notEmpty():
|
||||
node = fringe.pop()
|
||||
if problem.isGoal(node.state):
|
||||
return node.solution
|
||||
if node.state not in closed:
|
||||
closed.add(node.state)
|
||||
fringe.pushAll(expand(node, problem))
|
||||
return FAILURE
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\subsection{A$^\textbf{*}$ with graphs}
|
||||
\marginnote{A$^*$ with graphs}
|
||||
The algorithm keeps track of closed and open nodes.
|
||||
The heuristic $g(n)$ evaluates the minimum distance from the root to the node $n$.
|
||||
|
||||
\begin{description}
|
||||
\item[Consistent heuristic (monotone)] \marginnote{Consistent heuristic (monotone)}
|
||||
An heuristic is consistent if for each $n$, for any successor $n'$ of $n$ (i.e. nodes reachable from $n$ by making an action)
|
||||
holds that:
|
||||
\[
|
||||
\begin{cases}
|
||||
h(n) = 0 & \text{if the corresponding status is the goal} \\
|
||||
h(n) \leq c(n, a, n') + h(n') & \text{otherwise}
|
||||
\end{cases}
|
||||
\]
|
||||
where $c(n, a, n')$ is the cost to reach $n'$ from $n$ by taking the action $a$.
|
||||
|
||||
In other words, $f$ never decreases along a path.
|
||||
In fact:\\
|
||||
\begin{minipage}{.48\linewidth}
|
||||
\[
|
||||
\begin{split}
|
||||
f(n') &= g(n') + h(n') \\
|
||||
&= g(n) + c(n, a, n') + h(n') \\
|
||||
&\geq g(n) + h(n) \\
|
||||
&= f(n)
|
||||
\end{split}
|
||||
\]
|
||||
\end{minipage}
|
||||
\begin{minipage}{.48\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.3\textwidth]{img/monotone_heuristic.png}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\begin{theorem}
|
||||
If $h$ is a consistent heuristic, A$^*$ on graphs is optimal.
|
||||
\end{theorem}
|
||||
\end{description}
|
||||
Reference in New Issue
Block a user