xsemigroup.xml

<?xml version="1.0" encoding="ISO-8859-1"?>

  <!-- $Id: schutzenberger.xml,v 0.998  Exp $ -->
  <Chapter>
    <Heading>User friendly ways to give semigroups and automata</Heading>
This chapter describes two Tcl/Tk graphical interfaces that can be used to
define and edit semigroups and automata. 


<Section>
      <Heading> 
Finite automata 
      </Heading>
      <ManSection>
        <Func Arg="[A]" Name="XAutomaton" />
        <Description>
The function <C> Xautomaton </C> without arguments opens a new window where
an automaton may be specified. A finite automaton (which may then be edited)
may be given as argument.  
        </Description>
      </ManSection>
      <Example><![CDATA[
gap> XAutomaton();
      ]]></Example>
It opens a window like the following:
    <P />
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xaut.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xaut.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
<C> Var </C> is the &GAP; name of the automaton, <C> States </C> is the
number of states, <C> Alphabet </C> represents the alphabet and may be given
through a positive integer (in this case the alphabet is understood to be <C>
a,b,c,... </C>) or through a string whose symbols, in order, being the letters
of the alphabet. The numbers corresponding to the initial 
and accepting states are placed in the respective boxes. The automaton may be
specified to be deterministic, non deterministic or with epsilon
transitions. After pressing the <B> transition matrix </B> button the window
gets larger and the
transition matrix of the automaton may be given.
The <E>i</E>th row of the matrix describes the action of the
<E>i</E>th letter on the states.
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xautoma.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xautoma.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
A non deterministic automaton may be given as follows:
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xndAUT.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xndAUT.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
<P/>
By pressing the button <B> Ok </B> the &GAP; shell aquires the aspect shown
in the following picture and the automaton <C> ndAUT </C> may be used to do
computations. Some computations such as getting a rational expression
representing the language of the automaton, the (complete) minimal automaton
representing the same language or the transition semigroup of the automaton,
may be done directly after pressing the <B> Functions</B> button.
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xndAUTok.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xndAUTok.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
<P/>
By pressing the button <B> View </B> an image representing the automaton is
displayed in a new window.
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/ndAutImage.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/ndAutImage.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
An automaton with epsilon transitions may be given as follows shown in the
following picture. The last letter of the alphabet is always considered to be
the <M> \epsilon</M>. In the images it is represented by @.
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/EpsTrAut.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/EpsTrAut.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
<P/>
A new window with an image representing the automaton may be obtained by pressing the button <B> View </B>.
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/EpsTrAutImage.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/EpsTrAutImage.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>

<P/>
In the next example it is given an argument to the function <C>XAutomaton</C>.
      <Example><![CDATA[
gap> A := RandomAutomaton("det",2,2);
< deterministic automaton on 2 letters with 2 states >
gap> XAutomaton(A);
      ]]></Example>
It opens a window like the following:
    <P />
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xautgiven.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xautgiven.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>

     </Section>
    <Section>
    <Heading> Finite semigroups </Heading>
The most common ways to give a semigroup to are through generators and
relations, a set of (partial) transformations as generating set and as
syntactic semigroups of automata or rational languages. 
      <ManSection>
        <Func Arg="[S]" Name="XSemigroup" />
        <Description>
The function <C> XSemigroup </C> without arguments opens a new window where
a semigroup (or monoid) may be specified. A finite semigroup (which may then be edited) may be given as argument.  
        </Description>
      </ManSection>
      <Example><![CDATA[
gap> XSemigroup();
      ]]></Example>
It opens a window like the following:
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xsgp.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xsgp.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
where one may choose how to give the semigroup. 

    <Subsection>
      <Heading> 
Semigroups given through generators and relations 
      </Heading>
In the window opened by <C>XSemigroup</C>, by pressing the button
<B>Proceed</B> the window should enlarge and have the following aspect. 
(If the window does not enlarge automatically, use the mouse to do it.) 

    <P />
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xb211.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xb211.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
<C> GAP variable </C> is the &GAP; name of the semigroup. One has then to specify the
number of generators, the number of relations (which does not to be exact)
and whether one wants to produce a monoid or a semigroup.
Pressing the <B>Proceed</B> button one gets:
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xb212.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xb212.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>

     <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/xb21.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/xb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;

When giving the relations, the usual abbreviations "0" and "1" may be used.
<P/>
Pressing the <B>Done</B> button would output the following to the shell where <Package>GAP</Package>
is running:
    </Alt>
    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/gapb21.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/gapb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
    <Alt Only="LaTeX">
<P/>
The menu button <B>Functions</B> has the following commands:
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/funcsind.png}
        \includegraphics[width=0.4\textwidth]{images/funcsmenu.png}
        \end{center}

The interface allows to add and remove  &GAP; functions to the menu. When
adding a function, the name of the function should be provided.
(In its current version, it works only with functions that have
as only argument a semigroup.)

<P/>

By pressing the menu button <B>Functions</B> and selecting "Draw Schutzenberger Graphs" would
pop up the following window:
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/ShuzenbergerGraphb21.png}
        \end{center}
    </Alt>

    <Alt Only="HTML">
<P/>
The menu button <B>Functions</B> has the following commands:
&#60;br&#62;&#60;center&#62;
&#60;img src=&#34;images/funcsind.png&#34;&#62;
&#60;img src=&#34;images/funcsmenu.png&#34;&#62;
&#60;/center&#62;&#60;br&#62;

The interface allows to add and remove  &GAP; functions to the menu. When
adding a function, the name of the function should be provided.
(In its current version, it works only with functions that have
as only argument a semigroup.)

<P/>

By pressing the menu button <B>Functions</B> and selecting "Draw Schutzenberger Graphs" would
pop up the following window:
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/ShuzenbergerGraphb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>

    <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/CayleyGraphb21.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
By pressing the menu button <B>Functions</B> and selecting "Draw Cayley Graph" would
pop up the following window:

      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/CayleyGraphb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>

     <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/DClassesb21.png}
        \end{center}
    </Alt>
    <Alt Only="HTML">
By pressing the menu button <B>Functions</B> and selecting "Draw D-Classes" would
pop up the following window:

      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/DClassesb21.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>
     </Subsection>

     <Subsection>
      <Heading> 
Semigroups given by partial transformations
      </Heading>

<Code>XSemigroup(poi3);</Code> would pop up the following window, where everything should be clear:

     <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/Xsemigroup1.png}
        \end{center}
    </Alt>
<Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/Xsemigroup1.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>


    </Subsection>


    <Subsection>
      <Heading> 
Syntatic semigroups
      </Heading>

<Code>XSemigroup();</Code> would pop up the following window, where we would select "Syntatic semigroup",
press the <B>Proceed</B> button and then choose either to give a "Rational expression" or an "Automaton"
by pressing one of those buttons:

     <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/Xsemigroup2.png}
        \end{center}
    </Alt>
<Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/Xsemigroup2.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>

If "Rational expression" is chosen, a new window pops up where the expression can be specified:
     <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/regexp.png}
        \end{center}
    </Alt>
<Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/regexp.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>

After pressing the <B>Ok</B> button, notice that the menu button <B>Functions</B>
appears on the main window (lower right corner) meaning that <Package>GAP</Package>
already recognizes the given semigroup:
     <Alt Only="LaTeX">
        \begin{center}
        \includegraphics[width=0.4\textwidth]{images/regexp2.png}
        \end{center}
    </Alt>
<Alt Only="HTML">
      &#60;br&#62;&#60;center&#62;&#60;img src=&#34;images/regexp2.png&#34;&#62;&#60;/center&#62;&#60;br&#62;
    </Alt>


    </Subsection>



    </Section>

  </Chapter>