# -*- coding: utf-8 -*-
#+TITLE: SAT-based Minimization of Deterministic ω-Automata
#+DESCRIPTION: Spot command-line tools for minimizing ω-automata
#+SETUPFILE: setup.org
#+HTML_LINK_UP: tools.html

#+NAME: version
#+BEGIN_SRC sh :exports none
head -n 1 ../../picosat/VERSION | tr -d '\n'
#+END_SRC

This page explains how to use [[file:ltl2tgba.org][=ltl2tgba=]], [[file:dstar2tgba.org][=dstar2tgba=]], or [[file:autfilt.org][=autfilt=]]
to minimize deterministic automata using a SAT solver.

Let us first state a few facts about this minimization procedure.

1) The procedure works only on *deterministic* Büchi automata: any
   recurrence property can be converted into a deterministic Büchi
   automaton, and sometimes there are several ways of doing so.
2) Spot actually implements two SAT-based minimization procedures: one
   that builds a deterministic transition-based Büchi automaton
   (DTBA), and one that builds a deterministic transition-based
   ω-automaton with arbitrary acceptance condition (DTωA).  In
   [[file:ltl2tgba.org][=ltl2tgba=]] and [[file:dstar2tgba.org][=dstar2tgba=]], the latter procedure is restricted to
   TGBA.  In [[file:autfilt.org][=autfilt=]] it can use different and acceptance conditions
   for input and output, so you could for instance input a Rabin
   automaton, and produce a Streett automaton.
3) These two procedures can optionally constrain their output to
   use state-based acceptance. (They simply restrict all the outgoing
   transitions of a state to belong to the same acceptance sets.)
4) Spot is built using PicoSAT call_version()[:results raw].
   This solver was chosen for its performances, simplicity of
   integration and license compatibility.  However, it is
   still possible to use an external SAT solver (as described below).
5) [[file:ltl2tgba.org][=ltl2tgba=]] and [[file:dstar2tgba.org][=dstar2tgba=]] will always try to output an automaton.
   If they fail to determinize the property, they will simply output a
   nondeterministic automaton, if they managed to obtain a
   deterministic automaton but failed to minimize it (e.g., the
   requested number of states in the final automaton is too low), they
   will return that "unminimized" deterministic automaton.  There are
   only two cases where these tool will abort without returning an
   automaton: when the number of clauses output by Spot (and to be fed
   to the SAT solver) exceeds $2^{31}$, or when the SAT-solver was
   killed by a signal. [[file:autfilt.org][=autfilt --sat-minimize=]] will only output an
   automaton if the SAT-based minimization was successful.
6) Our [[https://www.lrde.epita.fr/~adl/dl/adl/baarir.14.forte.pdf][FORTE'14 paper]] describes the SAT encoding for the minimization
   of deterministic BA and TGBA.  Our [[https://www.lrde.epita.fr/~adl/dl/adl/baarir.15.lpar.pdf][LPAR'15 paper]] describes the
   generalization of the SAT encoding to deal with deterministic TωA
   with any acceptance condition.

* How to change the SAT solver used

By default Spot uses PicoSAT call_version()[:results raw]), this SAT-solver
is built into the Spot library, so that no temporary files are used to
store the problem.

The environment variable =SPOT_SATSOLVER= can be used to change the
SAT solver used by Spot.  This variable should describe a shell command
to run the SAT-solver on an input file called =%I= so that a model satisfying
the formula will be written in =%O=.  For instance to use [[http://www.labri.fr/perso/lsimon/glucose/][Glucose 3.0]], instead
of the builtin version of PicoSAT, define
#+BEGIN_SRC sh
export SPOT_SATSOLVER='glucose -verb=0 -model %I >%O'
#+END_SRC
We assume the SAT solver follows the input/output conventions of the
[[http://www.satcompetition.org/][SAT competition]]

* Enabling SAT-based minimization in =ltl2tgba= or =dstar2tgba=

Both tools follow the same interface, because they use the same
post-processing steps internally (i.e., the =spot::postprocessor=
class).

First, option =-D= should be used to declare that you are looking for
more determinism.  This will tweak the translation algorithm used by
=ltl2tgba= to improve determinism, and will also instruct the
post-processing routine used by both tools to prefer a
deterministic automaton over a smaller equivalent nondeterministic
automaton.

However =-D= is not a guarantee to obtain a deterministic automaton,
even if one exists.  For instance, =-D= fails to produce a
deterministic automaton for =GF(a <-> XXb)=.  Instead we get a 9-state
non-deterministic automaton.

#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -D 'GF(a <-> XXb)' --stats='states=%s, det=%d'
#+END_SRC
#+RESULTS:
: states=9, det=0

Option =-x tba-det= enables an additional
determinization procedure, that would otherwise not be used by =-D=
alone.  This procedure will work on any automaton that can be
represented by a DTBA; if the automaton to process use multiple
acceptance conditions, it will be degeneralized first.

On our example, =-x tba-det= successfully produces a deterministic
TBA, but a non-minimal one:

#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -D -x tba-det 'GF(a <-> XXb)' --stats='states=%s, det=%d'
#+END_SRC
#+RESULTS:
: states=7, det=1

Option =-x sat-minimize= will turn-on SAT-based minimization.  It also
implies =-x tba-det=, so there is no need to supply both options.

#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -D -x sat-minimize 'GF(a <-> XXb)' --stats='states=%s, det=%d'
#+END_SRC
#+RESULTS:
: states=4, det=1

We can draw it:

#+NAME: gfaexxb3
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -D -x sat-minimize 'GF(a <-> XXb)' -d
#+END_SRC
#+RESULTS: gfaexxb3
#+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<br/><font color="#5DA5DA">⓿</font>>]
  0 -> 2 [label=<!a &amp; b>]
  0 -> 3 [label=<a &amp; b>]
  0 -> 3 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  1 [label="1"]
  1 -> 0 [label=<a &amp; b>]
  1 -> 0 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  1 -> 1 [label=<!a &amp; b>]
  1 -> 1 [label=<!a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
  2 [label="2"]
  2 -> 0 [label=<a &amp; !b>]
  2 -> 1 [label=<!a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  2 -> 1 [label=<!a &amp; !b>]
  2 -> 3 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  3 [label="3"]
  3 -> 2 [label=<!a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  3 -> 2 [label=<!a &amp; !b>]
  3 -> 3 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  3 -> 3 [label=<a &amp; !b>]
}
#+end_example

#+BEGIN_SRC dot :file gfaexxb3.svg :var txt=gfaexxb3 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:gfaexxb3.svg]]

Clearly this automaton benefits from the transition-based
acceptance.  If we want a traditional Büchi automaton, with
state-based acceptance, we only need to add the =-B= option.  The
result will of course be slightly bigger.

#+NAME: gfaexxb4
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -BD -x sat-minimize 'GF(a <-> XXb)' -d
#+END_SRC
#+RESULTS: gfaexxb4
#+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 -> 0 [label=<!a &amp; b>]
  0 -> 1 [label=<!b>]
  0 -> 2 [label=<a &amp; b>]
  1 [label="1", peripheries=2]
  1 -> 4 [label=<!a>]
  1 -> 5 [label=<a>]
  2 [label="2"]
  2 -> 1 [label=<!b>]
  2 -> 4 [label=<!a &amp; b>]
  2 -> 5 [label=<a &amp; b>]
  3 [label="3", peripheries=2]
  3 -> 0 [label=<!a>]
  3 -> 1 [label=<a &amp; !b>]
  3 -> 2 [label=<a &amp; b>]
  4 [label="4"]
  4 -> 0 [label=<!a &amp; !b>]
  4 -> 1 [label=<a &amp; b>]
  4 -> 2 [label=<a &amp; !b>]
  4 -> 3 [label=<!a &amp; b>]
  5 [label="5"]
  5 -> 1 [label=<a &amp; b>]
  5 -> 3 [label=<!a &amp; b>]
  5 -> 4 [label=<!a &amp; !b>]
  5 -> 5 [label=<a &amp; !b>]
}
#+end_example

#+BEGIN_SRC dot :file gfaexxb4.svg :var txt=gfaexxb4 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:gfaexxb4.svg]]


There are cases where =ltl2tgba='s =tba-det= algorithm fails to produce a deterministic automaton.
In that case, SAT-based minimization is simply skipped.  For instance:

#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -D -x sat-minimize 'G(F(!b & (X!a M (F!a & F!b))) U !b)' --stats='states=%s, det=%d'
#+END_SRC
#+RESULTS:
: states=5, det=0

The question, of course, is whether there exist a deterministic
automaton for this formula, in other words: is this a recurrence
property?  There are two ways to answer this question using Spot and
some help from [[http://www.ltl2dstar.de/][=ltl2dstar=]].

The first is purely syntactic.  If a formula belongs to the class of
"syntactic recurrence formulas", it expresses a syntactic property.
(Of course there are formulas that expresses a syntactic properties
without being syntactic recurrences.)  [[file:ltlfilt.org][=ltlfilt=]] can be instructed to
print only formulas that are syntactic recurrences:

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --syntactic-recurrence -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)'
#+END_SRC
#+RESULTS:
: G(F(!b & (X!a M (F!a & F!b))) U !b)

Since our input formula was output, it expresses a recurrence property.

The second way to check whether a formula is a recurrence is by
converting a deterministic Rabin automaton using [[file:dstar2tgba.org][=dstar2tgba=]].  The
output is guaranteed to be deterministic if and only if the input DRA
expresses a recurrence property.

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --remove-wm -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)' -l |
ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
dstar2tgba -D --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'
#+END_SRC
#+RESULTS:
: input(states=11) output(states=9, acc-sets=1, det=1)

#+NAME: size
#+BEGIN_SRC sh :exports none
ltlfilt --remove-wm -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)' -l |
ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
dstar2tgba -D --stats=$arg
#+END_SRC

In the above command, =ltldo= is used to convert the LTL formula into
=ltl2dstar='s syntax.  Then =ltl2dstar= creates a deterministic Rabin
automaton (using =ltl2tgba= as an LTL to BA translator), and the
resulting call_size(arg="%S")[:results raw]-state DRA is converted
into a call_size(arg="%s")[:results raw]-state DTBA by =dstar2tgba=.
Since that result is deterministic, we can conclude that the formula
was a recurrence.

As far as SAT-based minimization goes, =dstar2tgba= will take the same
options as =ltl2tgba=.  For instance we can see that the smallest DTBA
has call_size(arg="%s -x sat-minimize")[:results raw] states:

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --remove-wm -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)' -l |
ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
dstar2tgba -D -x sat-minimize --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'
#+END_SRC
#+RESULTS:
: input(states=11) output(states=4, acc-sets=1, det=1)

* More acceptance sets

The formula "=G(F(!b & (X!a M (F!a & F!b))) U !b)=" can in fact be minimized into an
even smaller automaton if we use multiple acceptance sets.

Unfortunately because =dstar2tgba= does not know the formula being
translated, and it always convert a DRA into a DBA (with a single
acceptance set) before further processing, it does not know if using
more acceptance sets could be useful to further minimize it.   This
number of acceptance sets can however be specified on the command-line
with option =-x sat-acc=M=.  For instance:

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --remove-wm -f 'G(F(!b & (X!a M (F!a & F!b))) U !b)' -l |
ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
dstar2tgba -D -x sat-minimize,sat-acc=2 --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'
#+END_SRC
#+RESULTS:
: input(states=11) output(states=3, acc-sets=2, det=1)

Beware that the size of the SAT problem is exponential in the number
of acceptance sets (adding one acceptance set, in the input automaton
or in the output automaton, will double the size of the problem).

The case of =ltl2tgba= is slightly different because it can remember
the number of acceptance sets used by the translation algorithm, and
reuse that for SAT-minimization even if the automaton had to be
degeneralized in the meantime for the purpose of determinization.

* Low-level details

The following figure (from our [[https://www.lrde.epita.fr/~adl/dl/adl/baarir.14.forte.pdf][FORTE'14 paper]]) gives an overview of
the processing chains that can be used to turn an LTL formula into a
minimal DBA/DTBA/DTGBA.  The blue area at the top describes =ltl2tgba
-D -x sat-minimize=, while the purple area at the bottom corresponds
to =dstar2tgba -D -x stat-minimize=.

[[file:satmin.svg]]

The picture is slightly inaccurate in the sense that both =ltl2tgba=
and =dstar2tgba= are actually using the same post-processing chain:
only the initial translation to TGBA or conversion to DBA differs, the
rest is the same.  However in the case of =dstar2tgba=, no
degeneration or determinization are needed.

Also the picture does not show what happens when =-B= is used: any
DTBA is degeneralized into a DBA, before being sent to "DTBA SAT
minimization", with a special option to request state-based output.

The WDBA-minimization boxes are able to produce minimal Weak DBA from
any TGBA representing an obligation property.  In that case using
transition-based or generalized acceptance will not allow further
reduction.  This minimal WDBA is always used when =-D= is given
(otherwise, for the default =--small= option, the minimal WDBA is only
used if it is smaller than the nondeterministic automaton it has been
built from).

The "simplify" boxes are actually simulation-based reductions, and
SCC-based simplifications.

The red boxes "not in TCONG" or "not a recurrence" correspond to
situations where the tools will produce non-deterministic automata.

The following options can be used to fine-tune this procedure:

- =-x tba-det= :: attempt a powerset construction and check if
                  there exists an acceptance set such that the
                  resulting DTBA is equivalent to the input.
- =-x sat-minimize= :: enable SAT-based minimization. It is the same as
     =-x sat-minimize=1= (which is the default value). It performs a dichotomy
     to find the correct automaton size.This option implies =-x tba-det=.
- =-x sat-minimize=[2|3]= :: enable SAT-based
     minimization. Let us consider each intermediate automaton as a =step=
     towards the minimal automaton and assume =N= as the size of the starting
     automaton. =2= and =3= have been implemented with the aim of not
     restarting the encoding from scratch at each step. To do so, both restart
     the encoding after =N-1-(sat-incr-steps)= states have been won. Now,
     where is the difference? They both start by encoding the research of the
     =N-1= step and then:
       - =2= uses PicoSAT assumptions. It adds =sat-incr-steps= assumptions
         (each of them removing one more state) and then checks direclty the
         =N-1-(sat-incr-steps)= step. If such automaton is found, the process is
         restarted. Otherwise, a binary search begins between =N-1= and
         =N-1-sat-incr-steps=. If not provided, =sat-incr-steps= default value
         is 6.
       - =3= checks incrementally each =N-(2+i)= step, =i= ranging from =0= to
         =sat-incr-steps=. This process is fully repeated until the minimal
         automaton is found. The last SAT problem solved correspond to the
         minimal automaton. =sat-incr-steps= defaults to 2.
     Both implies =-x tba-det=.
- =-x sat-minimize=4= :: enable SAT-based minimization. It tries to reduce the
     size of the automaton one state at a time. This option implies
     =-x tba-det=.
- =-x sat-incr-steps=N= :: set the value of =sat-incr-steps= to N. It doest not
     make sense to use it without =-x sat-minimize=2= or =-x sat-minimize=3=.
- =-x sat-acc=$m$= :: attempt to build a minimal DTGBA with $m$ acceptance sets.
     This options implies =-x sat-minimize=.
- =-x sat-states=$n$= :: attempt to build an equivalent DTGBA with $n$
     states.  This also implies =-x sat-minimize= but won't perform
     any loop to lower the number of states.  Note that $n$ should be
     the number of states in a complete automaton, while =ltl2tgba=
     and =dstar2tgba= both remove sink states in their output by
     default (use option =--complete= to output a complete automaton).
     Also note that even with the =--complete= option, the output
     automaton may have appear to have less states because the other
     are unreachable.
- =-x state-based= :: for all outgoing transition of each state
     to belong to the same acceptance sets.
- =-x !wdba-minimize= :: disable WDBA minimization.

When options =-B= and =-x sat-minimize= are both used, =-x
state-based= and =-x sat-acc=1= are implied.  Similarly, when option
=-S= and =-x sat-minimize= are both used, then option =-x state-based=
is implied.

* Using =autfilt --sat-minimize= to minimize any deterministic ω-automaton

This interface is new in Spot 1.99 and allows to minimize any
deterministic ω-automaton, regardless of the acceptance condition
used.  By default, the procedure will try to use the same acceptance
condition (or any inferior one) and produce transition-based
acceptance.

For our example, let us first generate a deterministic Rabin
automaton with [[http://www.ltl2dstar.de/][=ltl2dstar=]].

#+BEGIN_SRC sh :results silent :exports both
ltlfilt -f 'FGa | FGb' -l |
ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds --output-format=hoa - - > output.hoa
#+END_SRC

Let's draw it:
#+NAME: autfiltsm1
#+BEGIN_SRC sh :results verbatim :exports code
autfilt output.hoa --dot
#+END_SRC
#+RESULTS: autfiltsm1
#+begin_example
digraph G {
  rankdir=LR
  label=<(Fin(<font color="#5DA5DA">â¿</font>) &amp; Inf(<font color="#F17CB0">â¶</font>)) | (Fin(<font color="#FAA43A">â·</font>) &amp; Inf(<font color="#B276B2">â¸</font>))>
  labelloc="t"
  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">â¿</font><font color="#FAA43A">â·</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<br/><font color="#F17CB0">â¶</font><font color="#FAA43A">â·</font>>]
  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="#5DA5DA">â¿</font><font color="#B276B2">â¸</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="#F17CB0">â¶</font><font color="#B276B2">â¸</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 autfiltsm1.svg :var txt=autfiltsm1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:autfiltsm1.svg]]

So this is a state-based Rabin automaton with two pairs.  If we call
=autfilt= with the =--sat-minimize= option, we can get the following
transition-based version (the output may change depending on the SAT
solver used):

#+NAME: autfiltsm2
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --sat-minimize output.hoa --dot
#+END_SRC

#+BEGIN_SRC dot :file autfiltsm2.svg :var txt=autfiltsm2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:autfiltsm2.svg]]

We can also attempt to build a state-based version with

#+NAME: autfiltsm3
#+BEGIN_SRC sh :results verbatim :exports code
autfilt -S --sat-minimize output.hoa --dot
#+END_SRC

#+BEGIN_SRC dot :file autfiltsm3.svg :var txt=autfiltsm3 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:autfiltsm3.svg]]

This is clearly smaller than the input automaton.  In this example the
acceptance condition did not change.  The SAT-based minimization only
tries to minimize the number of states, but sometime the
simplifications algorithms that are run before we attempt SAT-solving
will simplify the acceptance, because even removing a single
acceptance set can halve the run time.

You can however force a specific acceptance to be used as output.
Let's try with generalized co-Büchi for instance:

#+NAME: autfiltsm4
#+BEGIN_SRC sh :results verbatim :exports code
autfilt -S --sat-minimize='acc="generalized-co-Buchi 2"' output.hoa --dot
#+END_SRC

#+RESULTS: autfiltsm4
#+begin_example
digraph G {
  rankdir=LR
  label=<Fin(<font color="#5DA5DA">⓿</font>)|Fin(<font color="#F17CB0">❶</font>)>
  labelloc="t"
  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>>]
  0 -> 0 [label=<a>]
  0 -> 1 [label=<!a>]
  1 [label=<1<br/><font color="#F17CB0">❶</font>>]
  1 -> 0 [label=<!b>]
  1 -> 1 [label=<b>]
}
#+end_example

#+BEGIN_SRC dot :file autfiltsm4.svg :var txt=autfiltsm4 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfiltsm4.svg]]


Note that instead of naming the acceptance condition, you can actually
give an acceptance formula in the [[http://adl.github.io/hoaf/#acceptance][HOA syntax]].  For example we can
attempt to create a co-Büchi automaton with

#+NAME: autfiltsm5
#+BEGIN_SRC sh :results verbatim :exports code
autfilt -S --sat-minimize='acc="Fin(0)"' output.hoa --dot
#+END_SRC
#+RESULTS: autfiltsm5

#+BEGIN_SRC dot :file autfiltsm5.svg :var txt=autfiltsm5 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfiltsm5.svg]]


When forcing an acceptance condition, you should keep in mind that the
SAT-based minimization algorithm will look for automata that have
fewer states than the original automaton (after preliminary
simplifications).  This is not always reasonable.  For instance
constructing a Streett automaton from a Rabin automaton might require
more states.  An upper bound on the number of state can be passed
using a =max-states=123= argument to =--sat-minimize=.

If the input automaton is transition-based, but option =-S= is used to
produce a state-based automaton, then the original automaton is
temporarily converted into an automaton with state-based acceptance to
obtain an upper bound on the number of states if you haven't specified
=max-state=.  This upper bound might be larger than the one you would
specify by hand.

Here is an example demonstrating the case where the input automaton is
smaller than the output.   Let's take this small TGBA as input:

#+NAME: autfiltsm6
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba 'GFa & GFb' >output2.hoa
autfilt output2.hoa --dot
#+END_SRC

#+RESULTS: autfiltsm6
#+begin_example
digraph G {
  rankdir=LR
  label=<Inf(<font color="#5DA5DA">⓿</font>)&amp;Inf(<font color="#F17CB0">❶</font>)>
  labelloc="t"
  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 -> 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>>]
}
#+end_example

#+BEGIN_SRC dot :file autfiltsm6.svg :var txt=autfiltsm6 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfiltsm6.svg]]


If we attempt to minimize it into a transition-based Büchi automaton,
with fewer states, it will fail, output no result, and return with a
non-zero exit code (because no automata where output).

#+NAME: autfiltsm7
#+BEGIN_SRC sh :results verbatim :exports both
autfilt --sat-minimize='acc="Buchi"' output2.hoa
echo $?
#+END_SRC
#+RESULTS: autfiltsm7
: 1

However if we allow more states, it will work:

#+NAME: autfiltsm8
#+BEGIN_SRC sh :results verbatim :exports code
autfilt --sat-minimize='acc="Buchi",max-states=3' output2.hoa --dot
#+END_SRC

#+BEGIN_SRC dot :file autfiltsm8.svg :var txt=autfiltsm8 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfiltsm8.svg]]


By default, the SAT-based minimization tries to find a smaller automaton by
performing a binary search starting from =N/2= (N being the size of the
starting automaton). After various benchmarks, this algorithm proves to be the
best. However, in some cases, other rather similar methods might be better. The
algorithm to execute and some other parameters can be set thanks to the
=--sat-minimize= option.

The =--sat-minimize= option takes a comma separated list of arguments
that can be any of the following:

- =acc=DOUBLEQUOTEDSTRING= :: where the =DOUBLEQUOTEDSTRING= is an
     acceptance formula in the [[http://adl.github.io/hoaf/#acceptance][HOA syntax]], or a parametrized acceptance
     name (the different [[http://adl.github.io/hoaf/#acceptance-specifications][=acc-name:= options from HOA]]).
- =max-states=N= :: where =N= is an upper-bound on the maximum
     number of states of the constructed automaton.
- =states=M= :: where =M= is a fixed number of states to use in the
     result (all the states needs not be accessible in the result,
     so the output might be smaller nonetheless).  If this option is
     used the SAT-based procedure is just used once to synthesize
     one automaton, and no further minimization is attempted.
- =sat-incr=[1|2]= :: =1= and =2= correspond respectively to
     =-x sat-minimize=2= and =-x sat-minimize=3= options.
     They have been implemented with the aim of not restarting the
     encoding from scratch at each step - a step is a minimized intermediate
     automaton. To do so, both restart the encoding after
     =N-1-(sat-incr-steps)= states have been won - =N= being the size of the
     starting automaton. Now, where is the difference? They both start by
     encoding the research of the =N-1= step and then:
       - =1= uses PicoSAT assumptions. It adds =steps= assumptions (each of
         them removing one more state) and then checks direclty the
         =N-1-(sat-incr-steps)= step. If such automaton is found, the process is
         restarted. Otherwise, a binary search begins between =N-1= and
         =N-1-sat-incr-steps=. If not provided, =sat-incr-steps= defaults to 6.
       - =2= checks incrementally each =N-(2+i)= step, =i= ranging from =0= to
         =sat-incr-steps=. This process is fully repeated until the minimal
         automaton is found. The last SAT problem solved correspond to the
         minimal automaton. =sat-incr-steps= defaults to 2.
     Both implies =-x tba-det=.
- =sat-incr-steps=N= :: set the value of =sat-incr-steps= to =N=.
     This is used by =sat-incr= option.
- =sat-naive= :: use the =naive= algorithm to find a smaller automaton. It starts
     from =N= and then checks =N-1=, =N-2=, etc. until the last successful
     check.
- =sat-langmap= :: Find  the lower bound of default sat-minimize procedure. This
     relies on the fact that the size of the minimal automaton is at least equal
     to the  total  number  of different languages recognized by the automaton's
     states.
- =colored= :: force all transitions (or all states if =-S= is used)
     to belong to exactly one acceptance condition.


The =colored= option is useful when used in conjunction with a parity
acceptance condition.  Indeed, the parity acceptance condition by
itself does not require that the acceptance sets form a partition of
the automaton (which is expected from parity automata).

Compare the following, where parity acceptance is used, but the
automaton is not colored:

#+NAME: autfiltsm9
#+BEGIN_SRC sh :results verbatim :exports code
autfilt -S --sat-minimize='acc="parity max even 3"' output2.hoa --dot
#+END_SRC

#+BEGIN_SRC dot :file autfiltsm9.svg :var txt=autfiltsm9 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfiltsm9.svg]]

... to the following, where the automaton is colored, i.e., each state
belong to exactly one acceptance set:

#+NAME: autfiltsm10
#+BEGIN_SRC sh :results verbatim :exports code
autfilt -S --sat-minimize='acc="parity max even 3",colored' output2.hoa --dot
#+END_SRC

#+BEGIN_SRC dot :file autfiltsm10.svg :var txt=autfiltsm10 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:autfiltsm10.svg]]

* Logging statistics

If the environment variable =SPOT_SATLOG= is set to the name of a
file, the minimization function will append statistics about each of
its iterations in this file.

#+BEGIN_SRC sh :results verbatim :exports both
rm -f stats.csv
export SPOT_SATLOG=stats.csv
ltlfilt -f 'Ga R (F!b & (c U b))' -l |
ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - |
dstar2tgba -D -x sat-minimize,sat-acc=2 --stats='input(states=%S) output(states=%s, acc-sets=%a, det=%d)'
cat stats.csv
#+END_SRC
#+RESULTS:
: input(states=11) output(states=5, acc-sets=2, det=1)
: 9,9,36,72,44064,9043076,616,18,258,24
: 8,7,29,56,19712,3493822,236,9,135,6
: 6,6,25,48,10512,1362749,97,4,42,2
: 5,,,,7200,784142,65,2,40,2

The generated CSV file use the following columns:
- the n passed to the SAT-based minimization algorithm
  (it means the input automaton had n+1 states)
- number of reachable states in the output of
  the minimization.
- number of edges in the output
- number of transitions
- number of variables in the SAT problem
- number of clauses in the SAT problem
- user time for encoding the SAT problem
- system time for encoding the SAT problem
- user time for solving the SAT problem
- system time for solving the SAT problem
- automaton produced

Times are measured with the times() function, and expressed
in ticks (usually: 1/100 of seconds).

In the above example, the input DRA had 11
states.  In the first line of the =stats.csv= file, you can see the
minimization function searching for a 9 state DTBA and obtaining a
8-state solution.  (Since the minimization function searched for a
9-state DTBA, it means it received a 10-state complete DTBA, so the
processings performed before the minimization procedure managed to
convert the 11-state DRA into a 10-state DTBA.)  Starting from the
8-state solution, it looked for (and found) a 7-state solution, and
then a 6-state solution.  The search for a 5-state complete DTBA
failed.  The final output is reported with 5 states, because by
default we output trim automata. If the =--complete= option had been
given, the useless sink state would have been kept and the output
automaton would have 6 states.


#+BEGIN_SRC sh :results silent :exports results
rm -f output.hoa output2.hoa
#+END_SRC
