1.1 --- a/paper/src/paper.tex	Wed Jun 01 11:18:05 2011 +0200
     1.2 +++ b/paper/src/paper.tex	Wed Jun 01 18:53:27 2011 +0200
     1.3 @@ -83,13 +83,9 @@
     1.4  
     1.5  \section{$\omega$-regular languages}
     1.6  \begin{def:finite words}
     1.7 -Let $\Sigma$ be a non-empty set of symbols, the alphabet. $\Sigma^*$ is the set of all \emph{finite} words over $\Sigma$. A \emph{finite} word $w \in \Sigma^*$ is a \emph{finite} sequence $v_0...v_{n-1}$ of symbols from alphabet $\Sigma$ with length $n = |w|$. $\varepsilon$ denotes the empty word with length $|\varepsilon| = 0$.
     1.8 +Let $\Sigma$ be a non-empty set of symbols, called the alphabet. $\Sigma^*$ is the set of all \emph{finite} words over $\Sigma$. A \emph{finite} word $w \in \Sigma^*$ is a \emph{finite} sequence $v_0...v_{n-1}$ of symbols from alphabet $\Sigma$ with length $n = |w|$. $\varepsilon$ denotes the empty word with length $|\varepsilon| = 0$.
     1.9  \end{def:finite words}
    1.10  
    1.11 -\begin{def:infinite words}
    1.12 -$\Sigma^\omega$ is the set of all \emph{infinite} words over $\Sigma$. An \emph{infinite} word $w \in \Sigma^\omega$ is an \emph{infinite} sequence $v_0...v_\infty$ with length $\infty$. To address the elements of the infinite sequence $w$, view the word as a function $w : \mathbb{N}_0 \to \Sigma$ with $w(i) = v_i$; thus $w(i)$ denotes the symbol at sequence position $i$ of word $w$, another notation used for $w(i)$ is $w_i$.
    1.13 -\end{def:infinite words}
    1.14 -
    1.15  \begin{def:regular languages}
    1.16  The class of regular languages is defined recursively over an alphabet $\Sigma$:
    1.17  \begin{itemize}
    1.18 @@ -108,6 +104,10 @@
    1.19  \end{itemize}
    1.20  \end{def:regular languages closure}
    1.21  
    1.22 +\begin{def:infinite words}
    1.23 +$\Sigma^\omega$ is the set of all \emph{infinite} words over $\Sigma$. An \emph{infinite} word $w \in \Sigma^\omega$ is an \emph{infinite} sequence $v_0...v_\infty$ with length $\infty$. To address the elements of the infinite sequence $w$, view the word as a function $w : \mathbb{N}_0 \to \Sigma$ with $w(i) = v_i$; thus $w(i)$ denotes the symbol at sequence position $i$ of word $w$; another notation used for $w(i)$ is $w_i$.
    1.24 +\end{def:infinite words}
    1.25 +
    1.26  \begin{def:omega regular languages}
    1.27  Set $L$ is an $\omega$-language over alphabet $\Sigma$ iff $L \subseteq \Sigma^\omega$. Let $L_R \subseteq \Sigma^*$ be a non-empty regular finite language and $\varepsilon \notin L_R$. A set $L$ is $\omega$-regular iff $L$ is an $\omega$-language and $L = L_R^\omega$.
    1.28  \end{def:omega regular languages}