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albertz committed Jul 19, 2012
1 parent 1dab4af commit 0ab82135648c1ce99c60ef0e5563b27e059c9d57
Showing with 21 additions and 8 deletions.
  1. +10 −1 bibliography.bib
  2. +3 −2 generic-star-omega-results.tex
  3. BIN index.pdf
  4. +5 −3 intro.tex
  5. +3 −2 star-languages.tex
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@@ -2,13 +2,22 @@
%% http://bibdesk.sourceforge.net/
-%% Created for Albert Zeyer at 2012-07-18 21:39:17 +0200
+%% Created for Albert Zeyer at 2012-07-19 19:04:54 +0200
%% Saved with string encoding Unicode (UTF-8)
+@article{McNaughton66,
+ Author = {Robert McNaughton},
+ Date-Added = {2012-07-19 17:04:39 +0000},
+ Date-Modified = {2012-07-19 17:04:52 +0000},
+ Journal = {Information and Control},
+ Pages = {521-530},
+ Title = {Testing and Generating Infinite Sequences by a Finite Automaton},
+ Year = {1966}}
+
@article{FinAutRabin59,
Author = {Rabin, M. O. and Scott, D.},
Date-Added = {2012-07-18 19:39:04 +0000},
@@ -518,7 +518,7 @@ \subsection{Classification for arbitrary language classes}
%\[ \BC \ext \Lang \subseteq \lim \cap \dlim \Lang . \]
%via \cref{gen:extInLim}
-We summarize some of the results from this chapter to represent all of the strict inclusions from the diagram in \cref{regomega-diagram}. The diagram is about $\Langreg$ but the following theorem is generic.
+We summarize some of the results from this section to represent all of the strict inclusions from the diagram in \cref{regomega-diagram}. The diagram is about $\Langreg$ but the following theorem is generic.
In any case for any $\Lang$, we obviously have
\[ \ext \cap \dext \Lang \subseteq
@@ -554,7 +554,8 @@ \subsection{Classification for arbitrary language classes}
\end{proof}
\end{theorem}
-Closure under suffix-independence is a very strong property, as can be seen in \cref{gen:R-suffix-indep->single-state-loops}. Note that we only needed this property in \cref{gen:main-theorem-inclusions} for the $\BC \ext \Lang = \lim \cap \dlim \Lang$ equality.
+Closure under suffix-independence is a very strong property, as can be seen in \cref{gen:R-suffix-indep->single-state-loops}. Note that we only needed this property in \cref{gen:main-theorem-inclusions} for the $\BC \ext \Lang = \lim \cap \dlim \Lang$ equality. However, without this equality, we don't have an easy connection between $\ext \Lang$ and $\lim \Lang$. We can have $\ext \Lang \supseteq \lim \Lang$ as can be seen in \cref{lang:finite} for $\Lang(\mathtext{finite})$. Or ...
+\cref{gen:example:ext<->lim}
Without the separating languages $L_a$, $L'_a$ from \cref{gen:main-theorem-inclusions}, we formulate a bit less restrictive version:
\begin{lemma}
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BIN index.pdf
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@@ -4,11 +4,13 @@ \section{Introduction}
The study of formal languages and finite-state automata theory is very old and fundamental in theoretical computer science. Regular expressions were introduced by Kleene in 1956 (\cite{Kleene56}). Research on the connection between formal languages, automata theory and mathematical logic began in the early 1960's by Büchi (\cite{Buchi60}). Good introductions into the theory are \cite{FinAutLogR109} and \cite{LangAutLogicR102}.
-We call languages over finite words the $*$-languages. Likewise, $\omega$-languages are over infinite words.
+We call languages over finite words the $*$-languages. Likewise, $\omega$-languages are over infinite words. Words are just sequences of input symbols.
-The class of regular $*$-languages is probably the most well studied language class. Its expressiveness is exactly equivalent to the class of finite-state automata as well as regular expressions. For many applications, less powerful subsets of the regular $*$-languages are interesting, like starfree $*$-languages, locally testable $*$-languages, etc., as well as more powerful supersets, like context-free $*$-languages.
+The class of regular $*$-languages is probably the most well studied language class. Its expressiveness is exactly equivalent to the class of finite-state automata as well as regular expressions. The connection between finite-state automata and regular languages was established by S. C. Kleene in \cite{Kleene56}. A more generic concept of finite-state automata, including the non-deterministic case, was introduced by M. O. Rabin and D. Scott in \cite{FinAutRabin59}. It was inspired as a strict subset of the Turing machine which has infinite many states given by its memory. Such finite-state automata get a sequence of input symbols and change their state on each input symbol. Depending on their state, we say that such automaton accepts a given word. Thus, an automaton is representing a language. And the class of languages given by such finite-state automata is equal to the class of regular languages.
-The research on $\omega$-languages and their connection to finite-state automata began a bit later by Büchi \cite{DecisionSOR111} and \cite{Muller63}. As for the $*$-languages, the most well studied $\omega$-language class are the regular $\omega$-languages. Good introductions into these theories are \cite{AutInfObjsR103}, \cite{InfCompR101}, \cite{OmLangR108} and \cite{InfWordsR110}.
+For many applications, less powerful subsets of the regular $*$-languages are interesting, like starfree $*$-languages, locally testable $*$-languages, etc., as well as more powerful supersets, like context-free $*$-languages.
+
+The research on $\omega$-languages and their connection to finite-state automata began a bit later by Büchi \cite{DecisionSOR111} and \cite{Muller63}. As for the $*$-languages, the most well studied $\omega$-language class are the regular $\omega$-languages. Good introductions into these theories are \cite{AutInfObjsR103}, \cite{InfCompR101}, \cite{OmLangR108} and \cite{InfWordsR110}. In contrast to the finite-word acceptance of automata, one can think of several different infinite-word acceptance conditions which lead to different automata, most importantly the Büchi and Muller automata. A central result by Robert McNaughton in \cite{McNaughton66} is the equivalence of non-deterministic Büchi automata and deterministic Muller automata.
The acceptance-condition in automata for $*$-languages is straight-forward. If we look at $\omega$-languages, several different types of automata and their acceptance have been thought of, like Büchi-acceptance or Muller-acceptance, or E-acceptance and A-acceptance.
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@@ -572,6 +572,7 @@ \subsection{Finite / Co-finite languages}
\item $\BC \lim \Lang(\mathtext{finite}) = \Set{\emptyset,\Sigma^\omega}$
\item $\ext \Lang(\mathtext{co-finite}) = \Set{\Sigma^\omega}$
\item $\neg\ext \Lang(\mathtext{co-finite}) = \Set{\emptyset}$
+\item $\BC \ext \Lang(\mathtext{co-finite}) = \Set{\emptyset,\Sigma^\omega}$
\item $\lim \Lang(\mathtext{co-finite}) = \Set{\Sigma^\omega}$
\item $\dlim \Lang(\mathtext{co-finite}) = \Set{\Sigma^\omega}$
\item $\neg\lim \Lang(\mathtext{co-finite}) = \dlim \Lang(\mathtext{finite}) = \Set{\emptyset}$
@@ -580,9 +581,9 @@ \subsection{Finite / Co-finite languages}
Thus, the usual inclusions don't hold at all here. Esp. we again have
\[ \BC \ext \Lang(\mathtext{finite}) \supsetneqq \BC \lim \Lang(\mathtext{finite}) , \ \ \
-\BC \ext \Lang(\mathtext{co-finite}) \supsetneqq \BC \lim \Lang(\mathtext{co-finite}) . \]
+\BC \ext \Lang(\mathtext{co-finite}) = \BC \lim \Lang(\mathtext{co-finite}) . \]
We also have
-\[ \BC \ext \Lang(\mathtext{finite}) = \BC \ext \Lang(\mathtext{co-finite}) , \ \ \
+\[ \BC \ext \Lang(\mathtext{co-finite}) =
\BC \lim \Lang(\mathtext{finite}) = \BC \lim \Lang(\mathtext{co-finite}) . \]
\subsection{Local languages}

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