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Fix typos <noupdate>

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Tian Cheng (Riccardo) Xia
2024-01-07 17:26:21 +01:00
parent a7933cf3ba
commit 73fe58ed0b
6 changed files with 28 additions and 27 deletions
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2
src/fundamentals-of-ai-and-kr/module2/sections/_business_process.tex
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@ -212,7 +212,7 @@ The most common combination of approaches are:
\begin{description}
\item[Token play]
A transition is enabled when all the input places of a transition have a token and all the output places have not.
A transition is enabled when all its input places have a token and all its output places have not.
An enabled transition can be fired: tokens are removed from the inputs and moved to the outputs.
\end{description}

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src/fundamentals-of-ai-and-kr/module2/sections/_descriptive_logic.tex
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@ -54,7 +54,7 @@
\end{description}
\item[Assetion box (A-box)] \marginnote{Assetion box (A-box)}
\item[Assertion box (A-box)] \marginnote{Assertion box (A-box)}
List of facts about individuals.
\item[Terminological box (T-box)] \marginnote{Terminological box (T-box)}
@ -71,7 +71,7 @@ Let \texttt{r} be a role, \texttt{d} be a concept, \texttt{c} be a constant and
The semantics of concept-forming operators are:
\begin{descriptionlist}
\item[\texttt{[ALL r d]}]
Individuals \texttt{r}-related to the individuals of the category \texttt{d}.
Individuals \texttt{r}-related to only individuals of the category \texttt{d}.
\begin{example}
\texttt{[ALL :HasChild Male]} individuals that have zero children or only male children.
\end{example}
@ -140,13 +140,13 @@ The semantics of sentences are:
\[
\mathcal{I}[\texttt{[ALL r d]}] =
\{ \texttt{x} \in \mathcal{D} \mid \forall \texttt{y}:
\langle \texttt{x}, \texttt{y} \rangle \in \mathcal{I}[r] \text{ then } \texttt{y} \in \mathcal{I}[d] \}
\langle \texttt{x}, \texttt{y} \rangle \in \mathcal{I}[\texttt{r}] \text{ then } \texttt{y} \in \mathcal{I}[\texttt{d}] \}
\]
\item[\texttt{[EXISTS $n$ r]}]
\[
\mathcal{I}[\texttt{[EXISTS $n$ r]}] =
\{ \texttt{x} \in \mathcal{D} \mid \text{ exists at least $n$ distinct } \texttt{y}:
\langle \texttt{x}, \texttt{y} \rangle \in \mathcal{I}[r] \}
\langle \texttt{x}, \texttt{y} \rangle \in \mathcal{I}[\texttt{r}] \}
\]
\item[\texttt{[FILLS r c]}]
\[
@ -186,23 +186,23 @@ The semantics of sentences are:
\subsection{T-box reasoning}
Given a knowledge base of a set of sentences $S$, we would like to be able to determine the following:
\begin{description}
\begin{descriptionlist}
\item[Satisfiability] \marginnote{Satisfiability}
A concept \texttt{d} is satisfiable w.r.t. $S$ if:
\[ \exists \mathfrak{I}, (\mathfrak{I} \models S): \mathfrak{I}[\texttt{d}] \neq \varnothing \]
\[ \exists \mathfrak{I}, (\mathfrak{I} \models S): \mathcal{I}[\texttt{d}] \neq \varnothing \]
\item[Subsumption] \marginnote{Subsumption}
A concept $\texttt{d}_1$ is subsumed by $\texttt{d}_2$ w.r.t. $S$ if:
\[ \forall \mathfrak{I}, (\mathfrak{I} \models S): \mathfrak{I}[\texttt{d}_1] \subseteq \mathfrak{I}[\texttt{d}_2] \]
\[ \forall \mathfrak{I}, (\mathfrak{I} \models S): \mathcal{I}[\texttt{d}_1] \subseteq \mathcal{I}[\texttt{d}_2] \]
\item[Equivalence] \marginnote{Equivalence}
A concept $\texttt{d}_1$ is equivalent to $\texttt{d}_2$ w.r.t. $S$ if:
\[ \forall \mathfrak{I}, (\mathfrak{I} \models S): \mathfrak{I}[\texttt{d}_1] = \mathfrak{I}[\texttt{d}_2] \]
\[ \forall \mathfrak{I}, (\mathfrak{I} \models S): \mathcal{I}[\texttt{d}_1] = \mathcal{I}[\texttt{d}_2] \]
\item[Disjointness] \marginnote{Disjointness}
A concept $\texttt{d}_1$ is disjoint to $\texttt{d}_2$ w.r.t. $S$ if:
\[ \forall \mathfrak{I}, (\mathfrak{I} \models S): \mathfrak{I}[\texttt{d}_1] \neq \mathfrak{I}[\texttt{d}_2] \]
\end{description}
\[ \forall \mathfrak{I}, (\mathfrak{I} \models S): \mathcal{I}[\texttt{d}_1] \neq \mathcal{I}[\texttt{d}_2] \]
\end{descriptionlist}
\begin{theorem}[Reduction to subsumption]
\marginnote{Reduction to subsumption}
@ -216,13 +216,13 @@ Given a knowledge base of a set of sentences $S$, we would like to be able to de
\subsection{A-box reasoning}
Given a constant \texttt{c}, a concept \texttt{d} and a set of sentences $S$, we can determine the following:
\begin{description}
\begin{descriptionlist}
\item[Satisfiability] \marginnote{Satisfiability}
A constant \texttt{c} satisfies the concept \texttt{d} if:
\[ S \models (\texttt{c} \rightarrow \texttt{d}) \]
Note that it can be reduced to subsumption.
\end{description}
\end{descriptionlist}
\subsection{Computing subsumptions}
@ -237,18 +237,19 @@ The following algorithms can be employed:
\begin{enumerate}
\item Normalize \texttt{d} and \texttt{e} into a conjunctive form:
\[ \texttt{d} = \texttt{[AND d$_1$ \dots d$_n$]} \hspace*{1cm} \texttt{e} = \texttt{[AND e$_1$ \dots e$_m$]} \]
\item Check if each part of \texttt{e} is accounted by at least a component of \texttt{d}.
\item Check if each part of \texttt{d} is accounted by at least a component of \texttt{e}.
\end{enumerate}
\item[Tableaux-based algorithms] \marginnote{Tableaux-based algorithms}
Exploit the following theorem:
\[ (KB \models (C \sqsubseteq D)) \iff (KB \cup (x : C \sqcap \lnot D)) \text{ is inconsistent} \]
\[ \big( KB \models (\texttt{d} \sqsubseteq \texttt{e}) \big) \iff
\big( KB \cup (x : \texttt{[AND d $\lnot$e]}) \big) \text{ is inconsistent} \]
Note: similar to refutation.
\end{descriptionlist}
\subsection{Open world assumption}
\subsection{Open-world assumption}
\begin{description}
\item[Open-world assumption] \marginnote{Open-world assumption}
@ -256,7 +257,7 @@ The following algorithms can be employed:
\end{description}
Description logics are based on the open-world assumption.
To reason in open world assumption, all the possible models are split upon encountering unknown facts
To reason in open-world assumption, all the possible models are split upon encountering unknown facts
depending on the possible cases (Oedipus example).

2
src/fundamentals-of-ai-and-kr/module2/sections/_forward_reasoning.tex
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@ -162,7 +162,7 @@ RETE-based rule engine that uses Java.
Event detected outside an event processing system (e.g. a sensor). It does not provide any information alone.
\item[Complex event] \marginnote{Complex event}
Event generated by an event processing system and provides a higher informative payload.
Event generated by an event processing system and able to provides a higher informative payload.
\item[Complex event processing (CEP)] \marginnote{Complex event processing}
Paradigm for dealing with a large amount of information.

2
src/fundamentals-of-ai-and-kr/module2/sections/_ontologies.tex
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@ -75,7 +75,7 @@
\item[Exhaustive decomposition] \marginnote{Exhaustive decomposition}
Given a category $c$ and a set of categories $S$, $S$ is an exhaustive decomposition of $c$ iff
any element in $c$ belongs to at least a category in $S$:
\[ \texttt{exhaustiveDecomposition($S$, $c$)} \iff (\forall o \in c \iff \exists c_2 \in S: o \in c_2) \]
\[ \texttt{exhaustiveDecomposition($S$, $c$)} \iff \forall o: (o \in c \iff \exists c_2 \in S: o \in c_2) \]
\item[Partition] \marginnote{Partition}
Given a category $c$ and a set of categories $S$, $S$ is a partition of $c$ when:

4
src/fundamentals-of-ai-and-kr/module2/sections/_semantic_web.tex
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@ -130,7 +130,7 @@ xmlns:contact=http://www.w3.org/2000/10/swap/pim/contact#>
\item[Graph embedding] \marginnote{Graph embedding}
Project entities and relations into a vectorial space for ML applications.
\begin{description}
\item[Entity prediction] Given two entities \texttt{h} and \texttt{t}, determine the relation \texttt{r} between them.
\item[Link prediction] Given an entity \texttt{h} and a relation \texttt{t}, determine an entity \texttt{t} related to \texttt{h}.
\item[Link prediction] Given two entities \texttt{h} and \texttt{t}, determine the relation \texttt{r} between them.
\item[Entity prediction] Given an entity \texttt{h} and a relation \texttt{t}, determine an entity \texttt{t}-related to \texttt{h}.
\end{description}
\end{description}

12
src/fundamentals-of-ai-and-kr/module2/sections/_time_reasoning.tex
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@ -251,11 +251,11 @@ Logic-based on interacting agents with their own knowledge base.
Formally, semantics is defined on a set of primitive propositions $\phi$
using a Kripke structure $M = (S, \pi, K_\texttt{1}, \dots, K_\texttt{n})$ where:
\begin{itemize}
\item $S$ is a set of states of the world.
\item $S$ is a set of the states of the world.
\item $\pi: \phi \rightarrow 2^S$ specifies in which states each primitive proposition holds.
\item $K_\texttt{i} \subseteq S \times S$ is a binary relation where
$(s, t) \in K_\texttt{i}$ if an agent \texttt{i} considers the world $t$ possible (accessible) from $s$.
In other words, when the agent is in the world $s$, it considers $t$ to be a possibly valid world.
In other words, when the agent is in the world $s$, it considers $t$ to also be a possibly valid world.
Obviously, $(s, s) \in K_\texttt{i}$ for all states.
\end{itemize}
@ -272,7 +272,7 @@ Logic-based on interacting agents with their own knowledge base.
$\pi(\texttt{heads}) = \{ h_1, h_2 \}$\\
$\pi(\texttt{tails}) = \{ t_1, t_2 \}$
\item
$K_\texttt{a} = \{ (s, s) \mid s \in S \}$ as Alice observes everything in each world and does not have doubts.\\
$K_\texttt{a} = \{ (s, s) \mid s \in S \}$ as Alice observes everything in each world and does not have uncertainty.\\
$K_\texttt{b} = \{ (s, s) \mid s \in S \} \cup \{ (h_1, t_1), (t_1, h_1) \}$ as Bob is unsure of what happens in the second stage.\\
\end{itemize}
\vspace*{-1em}
@ -313,7 +313,7 @@ Logic-based on interacting agents with their own knowledge base.
\end{description}
\end{description}
Different modal logics can be defined based on the valid axioms.
Different modal logics can be defined based on the validity of these axioms.
\end{description}
@ -346,7 +346,7 @@ The accessibility relation maps into the temporal dimension with two possible ev
$\varphi$ is always true from now on.
\item[Future ($\diamond \varphi$)] \marginnote{Future}
$\varphi$ is true sometimes in the future.
$\varphi$ is sometimes true in the future.
It is equivalent to $\lnot\square(\lnot \varphi)$.
\item[Until ($\varphi \mathcal{U} \psi$)] \marginnote{Until}
@ -360,7 +360,7 @@ The accessibility relation maps into the temporal dimension with two possible ev
\item[Semantics]
Given a Kripke structure, $M = (S, \pi, K_\texttt{1}, \dots, K_\texttt{n})$ where states are represented using integers,
the semantic of the operators is the following:
the semantics of the operators is the following:
\begin{itemize}
\item $(M, i) \models P \iff i \in \pi(P)$.
\item $(M, i) \models \bigcirc\varphi \iff (M, i+1) \models \varphi$.