# -*- coding: utf-8 -*-
#+TITLE: =randaut=
#+SETUPFILE: setup.org
#+HTML_LINK_UP: tools.html

The =randaut= tool generates random (connected) automata.

By default, it will generate a random TGBA with 10 states, no
acceptance sets, and using a set of atomic proposition you have to
supply.

#+NAME: randaut1
#+BEGIN_SRC sh :results verbatim :exports code
randaut a b
#+END_SRC
#+RESULTS: randaut1
#+begin_example
digraph G {
  rankdir=LR
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 8 [label=<!a &amp; !b>]
  0 -> 4 [label=<!a &amp; !b>]
  1 [label="1"]
  1 -> 2 [label=<a &amp; !b>]
  1 -> 7 [label=<!a &amp; !b>]
  1 -> 4 [label=<!a &amp; b>]
  2 [label="2"]
  2 -> 2 [label=<!a &amp; !b>]
  2 -> 0 [label=<a &amp; !b>]
  2 -> 5 [label=<a &amp; !b>]
  3 [label="3"]
  3 -> 6 [label=<!a &amp; b>]
  4 [label="4"]
  4 -> 8 [label=<a &amp; !b>]
  4 -> 2 [label=<!a &amp; b>]
  4 -> 3 [label=<a &amp; b>]
  4 -> 7 [label=<!a &amp; b>]
  5 [label="5"]
  5 -> 9 [label=<!a &amp; !b>]
  5 -> 3 [label=<a &amp; !b>]
  5 -> 7 [label=<!a &amp; !b>]
  6 [label="6"]
  6 -> 7 [label=<!a &amp; !b>]
  6 -> 1 [label=<!a &amp; b>]
  7 [label="7"]
  7 -> 3 [label=<!a &amp; !b>]
  8 [label="8"]
  8 -> 8 [label=<!a &amp; b>]
  9 [label="9"]
  9 -> 0 [label=<!a &amp; b>]
  9 -> 6 [label=<!a &amp; b>]
}
#+end_example

#+BEGIN_SRC dot :file randaut1.png :cmdline -Tpng :var txt=randaut1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:randaut1.png]]

As for [[file:randltl.org][=randltl=]], you can supply a number of atomic propositions
instead of giving a list of atomic propositions.

* States and density

The numbers of states can be controlled using the =-S= option.  This
option will accept a range as argument, so for instance =-S3..6= will
generate an automaton with 3 to 6 states.

The number of edges can be controlled using the =-d= (or
=--density=) option.  The argument should be a number between 0 and 1.
In an automaton with $S$ states and density $d$, the degree of each
state will follow a normal distribution with mean $1+(S-1)d$ and
variance $(S-1)d(1-d)$.

In particular =-d0= will cause all states to have 1 successors, and
=-d1= will cause all states to be interconnected.

#+NAME: randaut2
#+BEGIN_SRC sh :results verbatim :exports code
randaut -S3 -d0 2
#+END_SRC

#+RESULTS: randaut2
#+begin_example
digraph G {
  rankdir=LR
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 2 [label=<!p0 &amp; !p1>]
  1 [label="1"]
  1 -> 1 [label=<!p0 &amp; !p1>]
  2 [label="2"]
  2 -> 1 [label=<!p0 &amp; !p1>]
}
#+end_example

#+BEGIN_SRC dot :file randaut2.png :cmdline -Tpng :var txt=randaut2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:randaut2.png]]

#+NAME: randaut3
#+BEGIN_SRC sh :results verbatim :exports code
randaut -S3 -d1 2
#+END_SRC

#+RESULTS: randaut3
#+begin_example
digraph G {
  rankdir=LR
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 2 [label=<!p0 &amp; !p1>]
  0 -> 0 [label=<!p0 &amp; !p1>]
  0 -> 1 [label=<!p0 &amp; !p1>]
  1 [label="1"]
  1 -> 1 [label=<p0 &amp; p1>]
  1 -> 2 [label=<p0 &amp; !p1>]
  1 -> 0 [label=<p0 &amp; !p1>]
  2 [label="2"]
  2 -> 1 [label=<!p0 &amp; !p1>]
  2 -> 0 [label=<p0 &amp; !p1>]
  2 -> 2 [label=<p0 &amp; !p1>]
}
#+end_example

#+BEGIN_SRC dot :file randaut3.png :cmdline -Tpng :var txt=randaut3 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:randaut3.png]]


* Acceptance sets

The generation of acceptance sets is controlled with the following three parameters:

 - =-A= (or =--acc-sets=) controls the number of acceptance sets.
   This can be given as a range (e.g. =-A1..3=) in case you want this
   number to be picked at random in the range.
 - =-a= (or =--acc-probability=) controls the probability that any
   transition belong to a given acceptance set.
 - =--state-acc= requests that the automaton use state-based
   acceptance.  In this case, =-a= is the probability that a /state/
   belong to the acceptance set.  (Because Spot only deals with
   transition-based acceptance internally, this options force all
   transitions leaving a state to belong to the same acceptance sets.
   But if the output format allows state-acceptance, it will be used.)

In addition, =-B= (or =--ba=) is a shorthand for =-A1 --state-acc=.


#+NAME: randaut4
#+BEGIN_SRC sh :results verbatim :exports code
randaut -S3 -d0.5 -A3 -a0.5 2
#+END_SRC

#+RESULTS: randaut4
#+begin_example
digraph G {
  rankdir=LR
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 1 [label=<!p0 &amp; !p1>]
  0 -> 0 [label=<!p0 &amp; !p1<br/><font color="#FAA43A">❷</font>>]
  1 [label="1"]
  1 -> 2 [label=<!p0 &amp; p1<br/><font color="#F17CB0">❶</font><font color="#FAA43A">❷</font>>]
  2 [label="2"]
  2 -> 2 [label=<!p0 &amp; !p1<br/><font color="#5DA5DA">⓿</font>>]
  2 -> 1 [label=<p0 &amp; !p1<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
}
#+end_example

#+BEGIN_SRC dot :file randaut4.png :cmdline -Tpng :var txt=randaut4 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:randaut4.png]]


#+NAME: randaut5
#+BEGIN_SRC sh :results verbatim :exports code
randaut -S3 -d0.4 -B -a0.7 2
#+END_SRC

#+RESULTS: randaut5
#+begin_example
digraph G {
  rankdir=LR
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 2 [label=<!p0 &amp; !p1>]
  1 [label="1"]
  1 -> 2 [label=<!p0 &amp; p1>]
  2 [label="2", peripheries=2]
  2 -> 1 [label=<!p0 &amp; !p1>]
  2 -> 0 [label=<p0 &amp; !p1>]
}
#+end_example

#+BEGIN_SRC dot :file randaut5.png :cmdline -Tpng :var txt=randaut5 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:randaut5.png]]

* Determinism

The output can only contain a single edge between two given states.
By default, the label of this edge is a random assignment of all
atomic propositions.  Two edges leaving the same state may therefore
have the same label.

If the =-D= (or =--deterministic=) option is supplied, the labels
are generated differently: once the degree $m$ of a state has been
decided, the algorithm will compute a set of $m$ disjoint
Boolean formulas over the given atomic propositions, such that the
sum of all these formulas is $\top$.  The resulting automaton is
therefore deterministic and complete.

#+NAME: randaut6
#+BEGIN_SRC sh :results verbatim :exports code
randaut -D -S3 -d0.6 -A2 -a0.5 2
#+END_SRC

#+RESULTS: randaut6
#+begin_example
digraph G {
  rankdir=LR
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 1 [label=<p0>]
  0 -> 2 [label=<!p0>]
  1 [label="1"]
  1 -> 2 [label=<p0<br/><font color="#F17CB0">❶</font>>]
  1 -> 0 [label=<!p0<br/><font color="#5DA5DA">⓿</font>>]
  2 [label="2"]
  2 -> 2 [label=<!p0 &amp; p1<br/><font color="#5DA5DA">⓿</font>>]
  2 -> 0 [label=<!p0 &amp; !p1<br/><font color="#F17CB0">❶</font>>]
  2 -> 1 [label=<p0<br/><font color="#5DA5DA">⓿</font>>]
}
#+end_example

#+BEGIN_SRC dot :file randaut6.png :cmdline -Tpng :var txt=randaut6 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:randaut6.png]]


Note that in a deterministic automaton with $a$ atomic propositions,
it is not possible to have states with more than $2^a$ successors.  If
the combination of =-d= and =-S= allows the situation where a state
can have more than $2^a$ successors, the degree will be clipped to
$2^a$.  When working with random deterministic automata over $a$
atomic propositions, we suggest you always request more than $2^a$
states.

* Output formats

The output format can be controlled using [[file:oaut.org][the common output options]]
like =--hoaf=, =--dot==, =--lbtt=, and =--spin=.  Note that =--spin=
automatically implies =--ba=.

* Generating a stream of automata

Use option =-n= to specify a number of automata to build.  A negative
value will cause an infinite number of automata to be produced.  This
generation of multiple automata is probably useful only with =--hoaf=,
when piped to another tool that can read this format and process
automata in batches.