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# -*- coding: utf-8 -*-
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#+TITLE: =ltl2tgba=
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#+SETUPFILE: setup.org
#+HTML_LINK_UP: tools.html
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This tool translates LTL or PSL formulas into two kinds of Büchi
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automata, or to monitors.  The default is to output Transition-based
Generalized Büchi Automata (hereinafter abbreviated TGBA), but more
traditional Büchi automata (BA) may be requested using the =-B=
option.
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* TGBA and BA

Formulas to translate may be specified using [[file:ioltl.org][common input options for
LTL/PSL formulas]].

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#+BEGIN_SRC sh :results verbatim :exports code
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ltl2tgba -f 'Fa & GFb'
#+END_SRC
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#+BEGIN_SRC sh :results verbatim :exports results
SPOT_DOTEXTRA= ltl2tgba -f 'Fa & GFb' --dot=
#+END_SRC
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#+RESULTS:
#+begin_example
digraph G {
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  rankdir=LR
  I [label="", style=invis, width=0]
  I -> 1
  0 [label="0"]
  0 -> 0 [label="b\n{0}"]
  0 -> 0 [label="!b"]
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  1 [label="1"]
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  1 -> 0 [label="a"]
  1 -> 1 [label="!a"]
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}
#+end_example

Actually, because =ltl2tgba= is often used with a single formula
passed on the command line, the =-f= option can be omitted and any
command-line parameter that is not the argument of some option will be
assumed to be a formula to translate (this differs from [[file:ltlfilt.org][=ltlfilt=]],
where such parameters are assumed to be filenames).

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=ltl2tgba= honors the [[file:oaut.org][common options for selecting the output format]].
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The default output format, as shown above, is [[http://www.graphviz.org/][GraphViz]]'s format.  This
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can converted into a picture, or into vectorial format using =dot= or
=dotty=.  Typically, you could get a =pdf= of this TGBA using
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba "Fa & GFb" | dot -Tpdf > tgba.pdf
#+END_SRC
#+RESULTS:

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The result would look like this (note that in this documentation
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we use some [[file:aout.org][environment variables]] to produce a more colorful
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output by default)
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#+NAME: dotex
#+BEGIN_SRC sh :results verbatim :exports none
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ltl2tgba "Fa & GFb"
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#+END_SRC
#+RESULTS: dotex
#+begin_example
digraph G {
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  rankdir=LR
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 1
  0 [label="0"]
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  0 -> 0 [label=<b<br/><font color="#5DA5DA">⓿</font>>]
  0 -> 0 [label=<!b>]
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  1 [label="1"]
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  1 -> 0 [label=<a>]
  1 -> 1 [label=<!a>]
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}
#+end_example

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

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

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Characters like ⓿, ❶, etc. denotes the acceptance sets a transition
belongs to.  In this case, there is only one acceptance set, called
=0=, containing a single transition.  You may have many transitions in
the same acceptance set, and a transition may also belong to multiple
acceptance sets.  An infinite path through this automaton is accepting
iff it visit each acceptance set infinitely often.  Therefore, in the
above example, any accepted path will /necessarily/ leave the initial
state after a finite amount of steps, and then it will verify the
property =b= infinitely often.  It is also possible that an automaton
do not use any acceptance set at all, in which any run is accepting.
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Here is a TGBA with multiple acceptance sets (we omit the call to
=dot= to render the output of =ltl2tgba= from now on):

#+NAME: dotex2
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#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba "GFa & GFb"
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#+END_SRC
#+RESULTS: dotex2
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#+begin_example
digraph G {
  rankdir=LR
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
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  0 -> 0 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
  0 -> 0 [label=<!a &amp; !b>]
  0 -> 0 [label=<!a &amp; b<br/><font color="#F17CB0">❶</font>>]
  0 -> 0 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
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}
#+end_example
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#+BEGIN_SRC dot :file dotex2.png :cmdline -Tpng :var txt=dotex2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2.png]]

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The above TGBA has two acceptance sets: ⓿ and ❶.  The position of
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these acceptance sets ensures that atomic propositions =a= and =b= must
be true infinitely often.
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A Büchi automaton for the previous formula can be obtained with the
=-B= option:

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#+NAME: dotex2ba
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#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B 'GFa & GFb'
#+END_SRC
#+RESULTS: dotex2ba
#+begin_example
digraph G {
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  rankdir=LR
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  node [shape="circle"]
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 0
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  0 [label="0", peripheries=2]
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  0 -> 0 [label=<a &amp; b>]
  0 -> 1 [label=<!b>]
  0 -> 2 [label=<!a &amp; b>]
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  1 [label="1"]
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  1 -> 0 [label=<a &amp; b>]
  1 -> 1 [label=<!b>]
  1 -> 2 [label=<!a &amp; b>]
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  2 [label="2"]
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  2 -> 0 [label=<a>]
  2 -> 2 [label=<!a>]
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}
#+end_example

#+BEGIN_SRC dot :file dotex2ba.png :cmdline -Tpng :var txt=dotex2ba :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2ba.png]]

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Although accepting states in the Büchi automaton are (traditionally)
pictured with double-lines, internally this automaton is still handled
as a TGBA with a single acceptance set such that the transitions
leaving the state are either all accepting, or all non-accepting.  You
can see this underlying TGBA if you pass the =--dot=t= option (the =t=
requests the use of transition-based acceptance as it is done
internally):
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#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba --dot=t -B 'GFa & GFb'
#+END_SRC

#+NAME: dotex2ba-t
#+BEGIN_SRC sh :results verbatim :exports none
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ltl2tgba --dot=.t -B 'GFa & GFb'
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#+END_SRC

#+RESULTS: dotex2ba-t
#+begin_example
digraph G {
  rankdir=LR
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
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  0 -> 0 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  0 -> 1 [label=<!b<br/><font color="#5DA5DA">⓿</font>>]
  0 -> 2 [label=<!a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
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  1 [label="1"]
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  1 -> 0 [label=<a &amp; b>]
  1 -> 1 [label=<!b>]
  1 -> 2 [label=<!a &amp; b>]
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  2 [label="2"]
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  2 -> 0 [label=<a>]
  2 -> 2 [label=<!a>]
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}
#+end_example

#+BEGIN_SRC dot :file dotex2ba-t.png :cmdline -Tpng :var txt=dotex2ba-t :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2ba-t.png]]

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Using option =-S= instead of option =-B= you can obtain generalized
Büchi automata with state-based acceptance.  Here is the same formula
as above, for comparison.

#+NAME: dotex2gba
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -S 'GFa & GFb'
#+END_SRC

#+RESULTS: dotex2gba
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label=<0<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
  0 -> 0 [label=<a &amp; b>]
  0 -> 1 [label=<!a &amp; !b>]
  0 -> 2 [label=<!a &amp; b>]
  0 -> 3 [label=<a &amp; !b>]
  1 [label=<1>]
  1 -> 0 [label=<a &amp; b>]
  1 -> 1 [label=<!a &amp; !b>]
  1 -> 2 [label=<!a &amp; b>]
  1 -> 3 [label=<a &amp; !b>]
  2 [label=<2<br/><font color="#F17CB0">❶</font>>]
  2 -> 0 [label=<a &amp; b>]
  2 -> 1 [label=<!a &amp; !b>]
  2 -> 2 [label=<!a &amp; b>]
  2 -> 3 [label=<a &amp; !b>]
  3 [label=<3<br/><font color="#5DA5DA">⓿</font>>]
  3 -> 0 [label=<a &amp; b>]
  3 -> 1 [label=<!a &amp; !b>]
  3 -> 2 [label=<!a &amp; b>]
  3 -> 3 [label=<a &amp; !b>]
}
#+end_example

#+BEGIN_SRC dot :file dotex2gba.png :cmdline -Tpng :var txt=dotex2gba :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2gba.png]]

Note that =ltl2tgba= is not very good at generating state-based
generalized Büchi automata (GBA): all it does is generating a
transition-based one internally, and then pushing acceptance sets onto
states.  On this example, the resulting GBA produced by =-S= is larger
than the BA produced by =-B=.

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As already discussed on the page about [[file:oaut.org][common output options]], various
options controls the output format of =ltl2tgba=:
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#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/Output format:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
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#+begin_example
  -8, --utf8                 enable UTF-8 characters in output (ignored with
                             --lbtt or --spin)
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      --check[=PROP]         test for the additional property PROP and output
                             the result in the HOA format (implies -H).  PROP
                             may be any prefix of 'all' (default),
                             'unambiguous', or 'stutter-invariant'.
      --dot[=1|a|b|B|c|e|f(FONT)|h|n|N|o|r|R|s|t|v]
                             GraphViz's format (default).  Add letters for (1)
                             force numbered states, (a) acceptance display, (b)
                             acceptance sets as bullets, (B) bullets except for
                             Büchi/co-Büchi automata, (c) force circular
                             nodes, (e) force elliptic nodes, (f(FONT)) use
                             FONT, (h) horizontal layout, (v) vertical layout,
                             (n) with name, (N) without name, (o) ordered
                             transitions, (r) rainbow colors for acceptance
                             sets, (R) color acceptance sets by Inf/Fin, (s)
                             with SCCs, (t) force transition-based acceptance.
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  -H, --hoaf[=i|s|t|m|l]     Output the automaton in HOA format.  Add letters
                             to select (i) use implicit labels for complete
                             deterministic automata, (s) prefer state-based
                             acceptance when possible [default], (t) force
                             transition-based acceptance, (m) mix state and
                             transition-based acceptance, (l) single-line
                             output
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      --lbtt[=t]             LBTT's format (add =t to force transition-based
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                             acceptance even on Büchi automata)
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      --name=FORMAT          set the name of the output automaton
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  -o, --output=FORMAT        send output to a file named FORMAT instead of
                             standard output.  The first automaton sent to a
                             file truncates it unless FORMAT starts with '>>'.
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  -q, --quiet                suppress all normal output
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  -s, --spin[=6|c]           Spin neverclaim (implies --ba).  Add letters to
                             select (6) Spin's 6.2.4 style, (c) comments on
                             states
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      --stats=FORMAT         output statistics about the automaton
#+end_example
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Option =-8= can be used to improve the readability of the output
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if your system can display UTF-8 correctly.

#+NAME: dotex2ba8
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#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B8 "GFa & GFb"
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#+END_SRC
#+RESULTS: dotex2ba8
#+begin_example
digraph G {
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  rankdir=LR
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 0
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  0 [label=<0<br/><font color="#5DA5DA">⓿</font>>]
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  0 -> 0 [label=<a∧b>]
  0 -> 1 [label=<b̅>]
  0 -> 2 [label=<a̅∧b>]
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  1 [label=<1>]
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  1 -> 0 [label=<a∧b>]
  1 -> 1 [label=<b̅>]
  1 -> 2 [label=<a̅∧b>]
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  2 [label=<2>]
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  2 -> 0 [label=<a>]
  2 -> 2 [label=<a̅>]
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}
#+end_example

#+BEGIN_SRC dot :file dotex2ba8.png :cmdline -Tpng :var txt=dotex2ba8 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2ba8.png]]

* Spin output

Using the =--spin= or =-s= option, =ltl2tgba= will produce a Büchi automaton
(the =-B= option is implied) as a never claim that can be fed to Spin.
=ltl2tgba -s= is therefore a drop-in replacement for =spin -f=.


#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -s 'GFa & GFb'
#+END_SRC
#+RESULTS:
#+begin_example
never { /* G(Fa & Fb) */
accept_init:
  if
  :: ((a) && (b)) -> goto accept_init
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  :: ((!(b))) -> goto T0_S2
  :: ((!(a)) && (b)) -> goto T0_S3
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  fi;
T0_S2:
  if
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  :: ((a) && (b)) -> goto accept_init
  :: ((!(b))) -> goto T0_S2
  :: ((!(a)) && (b)) -> goto T0_S3
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  fi;
T0_S3:
  if
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  :: ((a)) -> goto accept_init
  :: ((!(a))) -> goto T0_S3
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  fi;
}
#+end_example

Since Spin 6 extended its syntax to support arbitrary atomic
propositions, you may also need put the parser in =--lenient= mode to
support these:

#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -s --lenient '(a < b) U (process[2]@ok)'
#+END_SRC
#+RESULTS:
: never { /* "a < b" U "process[2]@ok" */
: T0_init:
:   if
:   :: ((process[2]@ok)) -> goto accept_all
:   :: ((a < b) && (!(process[2]@ok))) -> goto T0_init
:   fi;
: accept_all:
:   skip
: }

* Do you favor deterministic or small automata?

The translation procedure can be controled by a few switches.  A first
set of options specifies the intent of the translation: whenever
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possible, would you prefer a small automaton (=--small=) or a
deterministic (=--deterministic=) automaton?
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#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/Translation intent:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
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:   -a, --any                  no preference, do not bother making it small or
:                              deterministic
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:   -C, --complete             output a complete automaton (combine with other
:                              intents)
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:   -D, --deterministic        prefer deterministic automata
:       --small                prefer small automata (default)
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:   -U, --unambiguous          output unambiguous automata (combine with other
:                              intents)
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The =--any= option tells the translator that it should attempt to
reduce or produce a deterministic result result: any automaton
denoting the given formula is OK.  This effectively disables
post-processings and speeds up the translation.
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With the =-D= or =--deterministic= option, the translator will
/attempt/ to produce a deterministic automaton, even if this requires
a lot of states.  =ltl2tgba= knows how to produce the minimal
deterministic Büchi automaton for any obligation property (this
includes safety properties).
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With the =--small= option (the default), the translator will not
produce a deterministic automaton when it knows how to build smaller
automaton.

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Note that options =--deterministic= and =--small= express
/preferences/.  They certainly do /not/ guarantee that the output will
be deterministic, or will be the smallest automaton possible.

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An example formula where the difference between =-D= and =--small= is
flagrant is =Ga|Gb|Gc=:

#+NAME: gagbgc1
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#+BEGIN_SRC sh :results verbatim :exports code
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ltl2tgba 'Ga|Gb|Gc'
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#+END_SRC
#+RESULTS: gagbgc1
#+begin_example
digraph G {
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  rankdir=LR
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  node [shape="circle"]
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 0
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  0 [label=<0>]
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  0 -> 1 [label=<a>]
  0 -> 2 [label=<b>]
  0 -> 3 [label=<c>]
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  1 [label=<1>]
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  1 -> 1 [label=<a>]
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  2 [label=<2>]
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  2 -> 2 [label=<b>]
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  3 [label=<3>]
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  3 -> 3 [label=<c>]
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}
#+end_example

#+BEGIN_SRC dot :file gagbgc1.png :cmdline -Tpng :var txt=gagbgc1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:gagbgc1.png]]

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#+NAME: gagbgc2
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#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -D 'Ga|Gb|Gc'
#+END_SRC
#+RESULTS: gagbgc2
#+begin_example
digraph G {
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  rankdir=LR
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  node [shape="circle"]
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 6
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  0 [label="0", peripheries=2]
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  0 -> 0 [label=<c>]
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  1 [label="1", peripheries=2]
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  1 -> 0 [label=<!b &amp; c>]
  1 -> 1 [label=<b &amp; c>]
  1 -> 2 [label=<b &amp; !c>]
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  2 [label="2", peripheries=2]
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  2 -> 2 [label=<b>]
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  3 [label="3", peripheries=2]
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  3 -> 2 [label=<!a &amp; b>]
  3 -> 3 [label=<a &amp; b>]
  3 -> 5 [label=<a &amp; !b>]
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  4 [label="4", peripheries=2]
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  4 -> 0 [label=<!a &amp; c>]
  4 -> 4 [label=<a &amp; c>]
  4 -> 5 [label=<a &amp; !c>]
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  5 [label="5", peripheries=2]
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  5 -> 5 [label=<a>]
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  6 [label="6", peripheries=2]
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  6 -> 0 [label=<!a &amp; !b &amp; c>]
  6 -> 1 [label=<!a &amp; b &amp; c>]
  6 -> 2 [label=<!a &amp; b &amp; !c>]
  6 -> 3 [label=<a &amp; b &amp; !c>]
  6 -> 4 [label=<a &amp; !b &amp; c>]
  6 -> 5 [label=<a &amp; !b &amp; !c>]
  6 -> 6 [label=<a &amp; b &amp; c>]
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}
#+end_example

#+BEGIN_SRC dot :file gagbgc2.png :cmdline -Tpng :var txt=gagbgc2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:gagbgc2.png]]

You can augment the number of terms in the disjunction to magnify the
difference.  For N terms, the =--small= automaton has N+1 states,
while the =--deterministic= automaton needs 2^N-1 states.

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Add the =-C= or =--complete= option if you want to obtain a complete
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automaton, with a sink state capturing that rejected words that would
not otherwise have a run in the output automaton.

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Add the =-U= or =--unambiguous= option if you want unambiguous
automata to be produced.  An automaton is unambiguous if any word is
recognized by at most one accepting run of the automaton (however a
word can be rejected by multiple runs, so unambiguous automata can be
non-deterministic).

The following example is an ambiguous Büchi automaton, because the are
two ways to accept a run that repeats continuously the configuration
$\bar ab$.

#+NAME: ambig1
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B 'GFa -> GFb'
#+END_SRC
#+RESULTS: ambig1
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 1
  0 [label="0", peripheries=2]
  0 -> 0 [label=<!a>]
  1 [label="1"]
  1 -> 0 [label=<!a>]
  1 -> 1 [label=<1>]
  1 -> 2 [label=<b>]
  2 [label="2", peripheries=2]
  2 -> 2 [label=<b>]
  2 -> 3 [label=<!b>]
  3 [label="3"]
  3 -> 2 [label=<b>]
  3 -> 3 [label=<!b>]
}
#+end_example

#+BEGIN_SRC dot :file ambig1.png :cmdline -Tpng :var txt=ambig1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:ambig1.png]]

Here is an unambiguous automaton for the same formula, in which there
is only one run that recognizes this example word:

#+NAME: ambig2
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B -U 'GFa -> GFb'
#+END_SRC

#+RESULTS: ambig2
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 1 [label=<!a &amp; !b>]
  0 -> 2 [label=<1>]
  0 -> 3 [label=<a | b>]
  0 -> 4 [label=<!a &amp; !b>]
  1 [label="1", peripheries=2]
  1 -> 1 [label=<!a &amp; !b>]
  2 [label="2", peripheries=2]
  2 -> 2 [label=<b>]
  2 -> 5 [label=<!b>]
  3 [label="3"]
  3 -> 1 [label=<!a &amp; !b>]
  3 -> 3 [label=<a | b>]
  3 -> 4 [label=<!a &amp; !b>]
  4 [label="4"]
  4 -> 3 [label=<a | b>]
  4 -> 4 [label=<!a &amp; !b>]
  5 [label="5"]
  5 -> 2 [label=<b>]
  5 -> 5 [label=<!b>]
}
#+end_example

#+BEGIN_SRC dot :file ambig2.png :cmdline -Tpng :var txt=ambig2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:ambig2.png]]


Unlike =--small= and =--deterministic= that express preferences,
options =--complete= and =--unambiguous= do guarantee that the output
will be complete and unambiguous.

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A last parameter that can be used to tune the translation is the amount
of pre- and post-processing performed.  These two steps can be adjusted
via a common set of switches:
#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/Optimization level:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
:       --high                 all available optimizations (slow, default)
:       --low                  minimal optimizations (fast)
:       --medium               moderate optimizations

Pre-processings are rewritings done on the LTL formulas, usually to
reduce its size, but mainly to put it in a form that will help the
translator (for instance =F(a|b)= is easier to translate than
=F(a)|F(b)=).  At =--low= level, only simple syntactic rewritings are
performed.  At =--medium= level, additional simplifications based on
syntactic implications are performed.  At =--high= level, language
containment is used instead of syntactic implications.

Post-processings are cleanups and simplifications of the automaton
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produced by the core translator.  The algorithms used during
post-processing are
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- SCC filtering: removing useless strongly connected components,
  and useless acceptance sets.
- direct simulation: merge states based on suffix inclusion.
- iterated simulations: merge states based on suffix inclusion,
  or prefix inclusion, in a loop.
- WDBA minimization: determinize and minimize automata representing
  obligation properties.
- degeneralization: convert a TGBA into a BA
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- BA simulation (again direct or iterated)
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The chaining of these various algorithms depends on the selected
combination of optimization level (=--low=, =--medium=, =--high=),
translation intent (=--small=, =--deterministic=) and type of
automaton desired (=--tgba=, =--ba=).

A notable configuration is =--any --low=, which will produce a TGBA as
fast as possible.  In this case, post-processing is disabled, and only
syntactic rewritings are performed.  This can be used for
satisfiability checking, although in this context even building an
automaton is overkill (you only need an accepted run).

Finally, it should be noted that the default optimization options
(=--small --high=) are usually overkill.  =--low= will produce good
automata most of the time.  Most of pattern formulas of [[file:genltl.org][=genltl=]] will
be efficiently translated in this configuration (meaning that =--small
--high= will not produce a better automaton).  If you are planning to
generate automata for large family of pattern formulas, it makes sense
to experiment with the different settings on a small version of the
pattern, and select the lowest setting that satisfies your
expectations.

* Translating multiple formulas for statistics

If multiple formulas are given to =ltl2tgba=, the corresponding
automata will be output one after the other.  This is not very
convenient, since most of these output formats are not designed to
represent multiple automata, and tools like =dot= will only display
the first one.

One situation where passing many formulas to =ltl2tgba= is useful is
in combination with the =--stats=FORMAT= option.  This option will
output statistics about the translated automata instead of the
automata themselves.  The =FORMAT= string should indicate which
statistics should be output, and how they should be output using the
following sequence of characters (other characters are output as-is):

#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/^ *%/p'
#+END_SRC
#+RESULTS:
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#+begin_example
  %%                         a single %
  %a                         number of acceptance sets
  %c                         number of SCCs
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  %d                         1 if the output is deterministic, 0 otherwise
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  %e                         number of edges
  %f                         the formula, in Spot's syntax
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  %F                         name of the input file
  %L                         location in the input file
  %m                         name of the automaton
  %n                         number of nondeterministic states in output
  %p                         1 if the output is complete, 0 otherwise
  %r                         processing time (excluding parsing) in seconds
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  %s                         number of states
  %t                         number of transitions
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  %w                         one word accepted by the output automaton
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#+end_example
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For instance we can study the size of the automata generated for the
right-nested =U= formulas as follows:

#+BEGIN_SRC sh :results verbatim :exports both
genltl --u-right=1..8 | ltl2tgba -F - --stats '%s states and %e edges for "%f"'
#+END_SRC
#+RESULTS:
: 2 states and 2 edges for "p1"
: 2 states and 3 edges for "p1 U p2"
: 3 states and 6 edges for "p1 U (p2 U p3)"
: 4 states and 10 edges for "p1 U (p2 U (p3 U p4))"
: 5 states and 15 edges for "p1 U (p2 U (p3 U (p4 U p5)))"
: 6 states and 21 edges for "p1 U (p2 U (p3 U (p4 U (p5 U p6))))"
: 7 states and 28 edges for "p1 U (p2 U (p3 U (p4 U (p5 U (p6 U p7)))))"
: 8 states and 36 edges for "p1 U (p2 U (p3 U (p4 U (p5 U (p6 U (p7 U p8))))))"

Here =-F -= means that formulas should be read from the standard input.

When computing the size of an automaton, we distinguish /transitions/
and /edges/.  An edge between two states is labeled by a Boolean
formula and may in fact represent several transitions labeled by
compatible Boolean assignment.

For instance if the atomic propositions are =x= and =y=, an edge labeled
by the formula =!x= actually represents two transitions labeled respectively
with =!x&y= and =!x&!y=.

Two automata with the same structures (states and edges) but differing
labels, may have a different count of transitions, e.g., if one has
more restricted labels.

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[[file:csv.org][More examples of how to use =--stats= to create CSV
files are on a separate page]].

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* Building Monitors

In addition to TGBA and BA, =ltl2tgba= can output /monitor/ using the
=-M= option.  These are finite automata that accept all prefixes of a
formula.  The idea is that you can use these automata to monitor a
system as it is running, and report a violation as soon as no
compatible outgoing transition exist.

=ltl2tgba -M= may output non-deterministic monitors while =ltl2tgba
-MD= (short for =--monitor --deterministic=) will output the minimal
deterministic monitor for the given formula.

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#+NAME: monitor1
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#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -M '(Xa & Fb) | Gc'
#+END_SRC

#+RESULTS: monitor1
#+begin_example
digraph G {
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  rankdir=LR
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 0
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  0 [label=<0>]
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  0 -> 1 [label=<1>]
  0 -> 3 [label=<c>]
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  1 [label=<1>]
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  1 -> 2 [label=<a>]
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  2 [label=<2>]
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  2 -> 2 [label=<1>]
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  3 [label=<3>]
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  3 -> 3 [label=<c>]
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}
#+end_example

#+BEGIN_SRC dot :file monitor1.png :cmdline -Tpng :var txt=monitor1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:monitor1.png]]

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#+NAME: monitor2
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#+BEGIN_SRC sh :results verbatim :exports code
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ltl2tgba -MD '(Xa & Fb) | Gc'
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#+END_SRC

#+RESULTS: monitor2
#+begin_example
digraph G {
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  rankdir=LR
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  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
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  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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  I [label="", style=invis, width=0]
  I -> 3
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  0 [label=<0>]
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  0 -> 0 [label=<1>]
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  1 [label=<1>]
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  1 -> 0 [label=<a>]
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  2 [label=<2>]
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  2 -> 2 [label=<c>]
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  3 [label=<3>]
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  3 -> 1 [label=<!c>]
  3 -> 4 [label=<c>]
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  4 [label=<4>]
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  4 -> 0 [label=<a>]
  4 -> 2 [label=<!a &amp; c>]
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}
#+end_example

#+BEGIN_SRC dot :file monitor2.png :cmdline -Tpng :var txt=monitor2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:monitor2.png]]

Because they accept all finite executions that could be extended to
match the formula, monitor cannot be used to check for eventualities
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such as =F(a)=: indeed, any finite execution can be extended to match
=F(a)=.
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#  LocalWords:  ltl tgba num toc PSL Büchi automata SRC GFb invis Acc
#  LocalWords:  ltlfilt filenames GraphViz vectorial pdf Tpdf dotex
#  LocalWords:  sed png cmdline Tpng txt iff GFa ba utf UTF lbtt Fb
#  LocalWords:  GraphViz's LBTT's neverclaim SPOT's init goto fi Gb
#  LocalWords:  controled Gc gagbgc disjunction pre rewritings SCC Xa
#  LocalWords:  WDBA determinize degeneralization satisfiability SCCs
#  LocalWords:  genltl nondeterministic eval setenv concat getenv
#  LocalWords:  setq