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# -*- coding: utf-8 -*-
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#+TITLE: =ltlcross=
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#+DESCRIPTION: Spot command-line tool for cross-comparing the output of LTL translators.
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#+SETUPFILE: setup.org
#+HTML_LINK_UP: tools.html
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=ltlcross= is a tool for cross-comparing the output of LTL-to-automata
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translators.  It is actually a Spot-based clone of [[http://www.tcs.hut.fi/Software/lbtt/][LBTT]], the
/LTL-to-Büchi Translator Testbench/, that essentially performs the
same sanity checks.

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The main differences with LBTT are:
  - *support for PSL formulas in addition to LTL*
  - support for (non-alternating) automata with *any type of acceptance condition*,
  - support for *weak alternating automata*,
  - additional intersection *checks with the complement*, allowing
    to check equivalence of automata more precisely,
  - *more statistics*, especially:
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    - the number of logical transitions represented by each physical edge,
    - the number of deterministic states and automata
    - the number of SCCs with their various strengths (nonaccepting, terminal, weak, strong)
    - the number of terminal, weak, and strong automata
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  - an option to *reduce counterexample* by attempting to mutate and
    shorten troublesome formulas,
  - statistics output in *CSV* for easier post-processing,
  - *more precise time measurement* (LBTT was only precise to
    1/100 of a second, reporting most times as "0.00s").
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Although =ltlcross= performs the same sanity checks as LBTT, it does
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not implement any of the interactive features of LBTT.  In our almost
10-year usage of LBTT, we never had to use its interactive features to
understand bugs in our translation.  Therefore =ltlcross= will report
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problems, maybe with a conterexample, but you will be on your own to
investigate and fix them.
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The core of =ltlcross= is a loop that does the following steps:
  - Input a formula
  - Translate the formula and its negation using each configured translator.
    If there are 3 translators, the positive and negative translations
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    will be denoted =P0=, =N0=, =P1=, =N1=, =P2=, =N2=.  Optionally
    build complemented automata denoted =Comp(P0)=, =Comp(N0)=, etc.
  - Perform sanity checks between all these automata to detect any problem.
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  - Build the products of these automata with a random state-space (the same
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    state-space for all translations).  (If the =--products=N= option is given,
    =N= products are performed instead.)
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  - Gather statistics if requested.

* Formula selection

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Formulas to translate should be specified using the [[file:ioltl.org][common input
options]].  Standard input is read if it is not connected to a terminal,
and no =-f= or =-F= options are given.
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* Configuring translators

Each translator should be specified as a string that use some of the
following character sequences:

#+BEGIN_SRC sh :results verbatim :exports results
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  ltlcross --help | sed -n '/character sequences:/,/^$/p' | sed '1d;$d'
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#+END_SRC
#+RESULTS:
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:   %%                         a single %
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:   %f,%s,%l,%w                the formula as a (quoted) string in Spot, Spin,
:                              LBT, or Wring's syntax
:   %F,%S,%L,%W                the formula as a file in Spot, Spin, LBT, or
:                              Wring's syntax
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:   %O                         the automaton is output in HOA, never claim, LBTT,
:                              or ltl2dstar's format
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For instance here is how we could cross-compare the never claims
output by =spin= and =ltl2tgba= for the formulas =GFa= and =X(a U b)=.

#+BEGIN_SRC sh :results verbatim :exports code
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ltlcross -f 'GFa' -f 'X(a U b)' 'ltl2tgba -s %s >%O' 'spin -f %s >%O'
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#+END_SRC
#+RESULTS:

When =ltlcross= executes these commands, =%s= will be replaced
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by the formula in Spin's syntax, and =%O= will be replaced by a
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temporary file into which the output of the translator is redirected
before it is read back by =ltlcross=.

#+BEGIN_SRC sh :results verbatim :exports results
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ltlcross -f 'GFa' -f 'X(a U b)' 'ltl2tgba -s %s >%O' 'spin -f %s >%O' 2>&1
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#+END_SRC
#+RESULTS:
#+begin_example
([](<>(a)))
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Running [P0]: ltl2tgba -s '([](<>(a)))' >'lcr-o0-hvzgTC'
Running [P1]: spin -f '([](<>(a)))' >'lcr-o1-iYh45L'
Running [N0]: ltl2tgba -s '(!([](<>(a))))' >'lcr-o0-z6nzjV'
Running [N1]: spin -f '(!([](<>(a))))' >'lcr-o1-8JB5F4'
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Performing sanity checks and gathering statistics...

(X((a) U (b)))
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Running [P0]: ltl2tgba -s '(X((a) U (b)))' >'lcr-o0-rqfB6d'
Running [P1]: spin -f '(X((a) U (b)))' >'lcr-o1-OUNHEn'
Running [N0]: ltl2tgba -s '(!(X((a) U (b))))' >'lcr-o0-qzVvdx'
Running [N1]: spin -f '(!(X((a) U (b))))' >'lcr-o1-eUfHTG'
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Performing sanity checks and gathering statistics...

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No problem detected.
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#+end_example

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To handle tools that do not support some LTL operators, the character
sequences ~%f~, ~%s~, ~%l~, ~%w~, ~%F~, ~%S~, ~%L~, and ~%W~ can be
"infixed" by a bracketed list of operators to rewrite away.  For
instance if a tool reads LTL formulas from a file in LBT's syntax, but
does not support operators ~M~ (strong until) and ~W~ (weak until),
use ~%[WM]L~ instead of just ~%L~; this way operators ~W~ and ~M~ will
be rewritten using the other supported operators.

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=ltlcross= can only read four kinds of output:
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  - Never claims (only if they are restricted to representing an
    automaton using =if=, =goto=, and =skip= statements) such as those
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    output by [[http://spinroot.com/][=spin=]], [[http://www.lsv.ens-cachan.fr/~gastin/ltl2ba/][=ltl2ba=]], [[https://sourceforge.net/projects/ltl3ba/][=ltl3ba=]], or =ltl2tgba --spin=.  The
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    newer syntax introduced by Spin 6.24, using =do= instead of =if=,
    is also supported.
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  - [[http://www.tcs.hut.fi/Software/lbtt/doc/html/Format-for-automata.html][LBTT's format]], which supports generalized Büchi automata with
    either state-based acceptance or transition-based acceptance.
    This output is used for instance by [[http://www.tcs.hut.fi/Software/maria/tools/lbt/][=lbt=]], [[http://web.archive.org/web/20080607170403/http://www.science.unitn.it/~stonetta/modella.html][=modella=]], or =ltl2tgba
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    --lbtt=.
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  - Non-alternating automata in [[file:http://adl.github.io/hoaf/][the HOA format]] with any acceptance
    condition.
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  - [[file:concepts.org::#property-flags][Weak]] alternating automata in [[file:http://adl.github.io/hoaf/][the HOA format]].
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  - [[http://www.ltl2dstar.de/docs/ltl2dstar.html][=ltl2dstar='s format]], which supports deterministic Rabin or Streett
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    automata.

Files in any of these format should be indicated with =%O=.  (Past
versions of =ltlcross= used different letters for each format, but the
four parsers have been merged into a single one.)
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Of course all configured tools need not use the same =%= sequences.
The following list shows some typical configurations for some existing
tools:

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  - '=spin -f %s >%O='
  - '=ltl2ba -f %s >%O='
  - '=ltl3ba -M0 -f %s >%O=' (less deterministic output, can be smaller)
  - '=ltl3ba -M1 -f %s >%O=' (more deterministic output)
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  - '=modella -r12 -g -e %[MWei^]L %O='
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  - '=/path/to/script4lbtt.py %L %O=' (script supplied by [[http://web.archive.org/web/20070214050826/http://estragon.ti.informatik.uni-kiel.de/~fritz/][ltl2nba]] for
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    its interface with LBTT)
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  - '=ltl2tgba -s %f >%O=' (smaller output, Büchi automaton)
  - '=ltl2tgba -s -D %f >%O=' (more deterministic output, Büchi automaton)
  - '=ltl2tgba -H %f >%O=' (smaller output, TGBA)
  - '=ltl2tgba -H -D %f >%O=' (more deterministic output, TGBA)
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  - '=lbt <%L >%O='
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  - '~ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD
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    --output-format=hoa %[MW]L %O~' deterministic Rabin output in HOA, as
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    supported since version 0.5.2 of =ltl2dstar=.
  - '~ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD --automata=streett
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    --output-format=hoa %[MW]L %O~' deterministic Streett output in HOA,
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    as supported since version 0.5.2 of =ltl2dstar=.
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  - '=ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD %[MW]L %O=' (Rabin
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    output in DSTAR format, as supported in older versions of
    =ltl2dstar=.
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  - '=ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD %L - | dstar2tgba
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    -s >%O=' (external conversion from Rabin to Büchi done by
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    =dstar2tgba= for more reduction of the Büchi automaton than what
    =ltlcross= would provide)
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  - '=java -jar Rabinizer.jar -ltl2dstar %[MW]F %O; mv %O.dst %O=' (Rabinizer
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    uses the last =%O= argument as a prefix to which it always append =.dst=,
    so we have to rename =%O.dst= as =%O= so that =ltlcross= can find the file)
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  - '~java -jar rabinizer3.1.jar -in=formula -silent -out=std -format=hoa -auto=tr %[MWR]f >%O~'
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    (rabinizer 3.1 can output automata in the HOA format)
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  - '=ltl3dra -f %s >%O=' (The HOA format is the default for =ltl2dra=.)
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To simplify the use of some of the above tools, a set of predefined
shorthands are available.  Those can be listed with the
=--list-shorthands= option.

#+BEGIN_SRC sh :results verbatim :exports both
ltlcross --list-shorthands
#+END_SRC
#+RESULTS:
#+begin_example
If a COMMANDFMT does not use any %-sequence, and starts with one of
the following words, then the string on the right is appended.

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  lbt          <%L>%O
  ltl2ba       -f %s>%O
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  ltl2dstar    --output-format=hoa %[MW]L %O
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  ltl2tgba     -H %f>%O
  ltl3ba       -f %s>%O
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  ltl3dra      -f %s>%O
  modella      %[MWei^]L %O
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  spin         -f %s>%O
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Any {name} and directory component is skipped for the purpose of
matching those prefixes.  So for instance
  '{DRA} ~/mytools/ltl2dstar-0.5.2'
will changed into
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  '{DRA} ~/mytools/ltl2dstar-0.5.2 --output-format=hoa %[MR]L %O'
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#+end_example

What this implies is that running =ltlcross ltl2ba ltl3ba ...= is
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the same as running =ltlcross 'ltl2ba -f %s>%O' 'ltl3ba -f %s>%O' ...=
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Because only the prefix of the actual command is checked, you can
still specify some options.  For instance =ltlcross 'ltl2tgba -D' ...=
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is short for =ltlcross 'ltl2tgba -D -H %F>%O' ...=
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* Getting statistics

Detailed statistics about the result of each translation, and the
product of that resulting automaton with the random state-space, can
be obtained using the =--csv=FILE= or =--json=FILE= option.

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** CSV or JSON output (or both!)

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The following compare =ltl2tgba=, =spin=, and =lbt= on three random
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formulas (where =W= and =M= operators have been rewritten away because
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they are not supported by =spin= and =lbt=).
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#+BEGIN_SRC sh :results verbatim :exports code
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randltl -n 3 a b |
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ltlfilt --remove-wm |
ltlcross --csv=results.csv \
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         'ltl2tgba -s %f >%O' \
         'spin -f %s >%O' \
         'lbt < %L >%O'
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#+END_SRC
#+RESULTS:

#+BEGIN_SRC sh :results verbatim :exports results
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randltl -n 3 a b | ltlfilt --remove-wm |
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ltlcross --csv=results.csv --json=results.json \
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         'ltl2tgba -s %f >%O' \
         'spin -f %s >%O' \
         'lbt < %L >%O' --csv=results.csv 2>&1
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#+END_SRC
#+RESULTS:
#+begin_example
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-:1: 0
Running [P0]: ltl2tgba -s '0' >'lcr-o0-HIoc9n'
Running [P1]: spin -f 'false' >'lcr-o1-69Gi3B'
Running [P2]: lbt < 'lcr-i0-cYhaYP' >'lcr-o2-CZe2S3'
Running [N0]: ltl2tgba -s '1' >'lcr-o0-9YEpOh'
Running [N1]: spin -f 'true' >'lcr-o1-exCCOv'
Running [N2]: lbt < 'lcr-i0-DMSnRJ' >'lcr-o2-E0H9TX'
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Performing sanity checks and gathering statistics...

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-:2: !((1) U (F(!(p0))))
Running [P0]: ltl2tgba -s '!((1) U (F(!(p0))))' >'lcr-o0-ubhgZb'
Running [P1]: spin -f '!((true) U (<>(!(p0))))' >'lcr-o1-wBnwcq'
Running [P2]: lbt < 'lcr-i1-D3WcqE' >'lcr-o2-i3VTDS'
Running [N0]: ltl2tgba -s '(1) U (F(!(p0)))' >'lcr-o0-8F2eS6'
Running [N1]: spin -f '(true) U (<>(!(p0)))' >'lcr-o1-focrcl'
Running [N2]: lbt < 'lcr-i1-VgW9wz' >'lcr-o2-GbdTRN'
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Performing sanity checks and gathering statistics...

-:3: (1) U ((G(p0)) | (F(p1)))
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Running [P0]: ltl2tgba -s '(1) U ((G(p0)) | (F(p1)))' >'lcr-o0-jj8Ih2'
Running [P1]: spin -f '(true) U (([](p0)) || (<>(p1)))' >'lcr-o1-JarYMg'
Running [P2]: lbt < 'lcr-i2-yq4yiv' >'lcr-o2-Lw29NJ'
Running [N0]: ltl2tgba -s '!((1) U ((G(p0)) | (F(p1))))' >'lcr-o0-mHp6jY'
Running [N1]: spin -f '!((true) U (([](p0)) || (<>(p1))))' >'lcr-o1-pA7KVc'
Running [N2]: lbt < 'lcr-i2-ZxXHBr' >'lcr-o2-YadFhG'
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Performing sanity checks and gathering statistics...

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No problem detected.
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#+end_example

After this execution, the file =results.csv= contains the following:

#+BEGIN_SRC sh :results verbatim :exports results
cat results.csv
#+END_SRC
#+RESULTS:
#+begin_example
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"formula","tool","exit_status","exit_code","time","states","edges","transitions","acc","scc","nondet_states","nondet_aut","complete_aut","product_states","product_transitions","product_scc"
"0","ltl2tgba -s %f >%O","ok",0,0.0299298,1,1,0,1,1,0,0,0,1,0,1
"0","spin -f %s >%O","ok",0,0.00396246,2,2,1,1,2,0,0,0,1,0,1
"0","lbt < %L >%O","ok",0,0.00262385,1,0,0,0,1,0,0,0,1,0,1
"1","ltl2tgba -s %f >%O","ok",0,0.0261614,1,1,1,1,1,0,0,1,200,4199,1
"1","spin -f %s >%O","ok",0,0.0137128,2,2,2,1,2,0,0,1,201,4220,2
"1","lbt < %L >%O","ok",0,0.00792516,3,3,3,0,3,0,0,1,222,4653,23
"!((1) U (F(!(p0))))","ltl2tgba -s %f >%O","ok",0,0.043858,1,1,1,1,1,0,0,0,200,2059,1
"!((1) U (F(!(p0))))","spin -f %s >%O","ok",0,0.00202537,1,1,1,1,1,0,0,0,200,2059,1
"!((1) U (F(!(p0))))","lbt < %L >%O","ok",0,0.00331618,2,2,2,0,2,0,0,0,201,2071,2
"(1) U (F(!(p0)))","ltl2tgba -s %f >%O","ok",0,0.031689,2,3,4,1,2,0,0,1,400,8264,2
"(1) U (F(!(p0)))","spin -f %s >%O","ok",0,0.0026263,2,3,5,1,2,1,1,1,400,10337,2
"(1) U (F(!(p0)))","lbt < %L >%O","ok",0,0.00266354,7,13,22,2,7,4,1,1,1201,35191,604
"(1) U ((G(p0)) | (F(p1)))","ltl2tgba -s %f >%O","ok",0,0.0287222,3,5,11,1,3,1,1,0,600,11358,3
"(1) U ((G(p0)) | (F(p1)))","spin -f %s >%O","ok",0,0.00167423,4,8,24,1,4,2,1,0,800,24920,4
"(1) U ((G(p0)) | (F(p1)))","lbt < %L >%O","ok",0,0.0016987,9,17,52,2,9,4,1,0,1601,41559,805
"!((1) U ((G(p0)) | (F(p1))))","ltl2tgba -s %f >%O","ok",0,0.0308937,2,4,4,1,1,0,0,0,395,3964,1
"!((1) U ((G(p0)) | (F(p1))))","spin -f %s >%O","ok",0,0.0230605,6,18,17,1,4,5,1,0,592,8891,1
"!((1) U ((G(p0)) | (F(p1))))","lbt < %L >%O","ok",0,0.00280787,3,6,9,1,2,3,1,0,397,5957,2
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#+end_example

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This file can be loaded in any spreadsheet or statistical application.

Although we only supplied 2 random generated formulas, the output
contains 4 formulas because =ltlcross= had to translate the positive
and negative version of each.
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If we had used the option =--json=results.json= instead of (or in
addition to) =--cvs=results.csv=, the file =results.json= would have
contained the following [[http://www.json.org/][JSON]] output.
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#+BEGIN_SRC sh :results verbatim :exports results
cat results.json
#+END_SRC
#+RESULTS:
#+begin_example
{
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  "tool": [
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    "ltl2tgba -s %f >%O",
    "spin -f %s >%O",
    "lbt < %L >%O"
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  ],
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  "formula": [
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    "0",
    "1",
    "!((1) U (F(!(p0))))",
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    "(1) U (F(!(p0)))",
    "(1) U ((G(p0)) | (F(p1)))",
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    "!((1) U ((G(p0)) | (F(p1))))"
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  ],
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  "fields":  [
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  "formula","tool","exit_status","exit_code","time","states","edges","transitions","acc","scc","nondet_states","nondet_aut","complete_aut","product_states","product_transitions","product_scc"
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  ],
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  "inputs":  [ 0, 1 ],
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  "results": [
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    [ 0,0,"ok",0,0.0299298,1,1,0,1,1,0,0,0,1,0,1 ],
    [ 0,1,"ok",0,0.00396246,2,2,1,1,2,0,0,0,1,0,1 ],
    [ 0,2,"ok",0,0.00262385,1,0,0,0,1,0,0,0,1,0,1 ],
    [ 1,0,"ok",0,0.0261614,1,1,1,1,1,0,0,1,200,4199,1 ],
    [ 1,1,"ok",0,0.0137128,2,2,2,1,2,0,0,1,201,4220,2 ],
    [ 1,2,"ok",0,0.00792516,3,3,3,0,3,0,0,1,222,4653,23 ],
    [ 2,0,"ok",0,0.043858,1,1,1,1,1,0,0,0,200,2059,1 ],
    [ 2,1,"ok",0,0.00202537,1,1,1,1,1,0,0,0,200,2059,1 ],
    [ 2,2,"ok",0,0.00331618,2,2,2,0,2,0,0,0,201,2071,2 ],
    [ 3,0,"ok",0,0.031689,2,3,4,1,2,0,0,1,400,8264,2 ],
    [ 3,1,"ok",0,0.0026263,2,3,5,1,2,1,1,1,400,10337,2 ],
    [ 3,2,"ok",0,0.00266354,7,13,22,2,7,4,1,1,1201,35191,604 ],
    [ 4,0,"ok",0,0.0287222,3,5,11,1,3,1,1,0,600,11358,3 ],
    [ 4,1,"ok",0,0.00167423,4,8,24,1,4,2,1,0,800,24920,4 ],
    [ 4,2,"ok",0,0.0016987,9,17,52,2,9,4,1,0,1601,41559,805 ],
    [ 5,0,"ok",0,0.0308937,2,4,4,1,1,0,0,0,395,3964,1 ],
    [ 5,1,"ok",0,0.0230605,6,18,17,1,4,5,1,0,592,8891,1 ],
    [ 5,2,"ok",0,0.00280787,3,6,9,1,2,3,1,0,397,5957,2 ]
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  ]
}
#+end_example

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Here the =fields= table describes the columns of the =results= table.
The =inputs= tables lists the columns that are considered as inputs
for the experiments.  The values in the columns corresponding to the
fields =formula= and =tool= contains indices relative to the =formula=
and =tool= tables.  This format is more compact when dealing with lots
of translators and formulas, because they don't have to be repeated on
each line as in the CSV version.
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JSON data can be easily processed in any language.  For instance the
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following Python3 script averages each column (except the first four)
for each tool, and presents the results in a form that can almost be
copied into a LaTeX table (the =%= in the tool names have to be taken
care of).  Note that for simplicity we assume that the first two
columns are inputs, instead of reading the =inputs= field.
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#+BEGIN_SRC python :results output :exports both
#!/usr/bin/python3
import json
data = json.load(open('results.json'))
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datacols = range(4, len(data["fields"]))
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# Index results by tool
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results = { t:[] for t in range(0, len(data["tool"])) }
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for l in data["results"]:
  results[l[1]].append(l)
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# Average columns for each tool, and display them as a table
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print("%-18s & count & %s \\\\" % ("tool", " & ".join(data["fields"][4:])))
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for i in range(0, len(data["tool"])):
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  c = len(results[i])
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  sums = ["%6.1f" % (sum([x[j] for x in results[i]])/c)
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          for j in datacols]
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  print("%-18s & %3d & %s \\\\" % (data["tool"][i], c,
        " & ".join(sums)))
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#+END_SRC
#+RESULTS:
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: tool               & count & time & states & edges & transitions & acc & scc & nondet_states & nondet_aut & complete_aut & product_states & product_transitions & product_scc \\
: ltl2tgba -s %f >%O &   6 &    0.0 &    1.7 &    2.5 &    3.5 &    1.0 &    1.5 &    0.2 &    0.2 &    0.3 &  299.3 & 4974.0 &    1.5 \\
: spin -f %s >%O     &   6 &    0.0 &    2.8 &    5.7 &    8.3 &    1.0 &    2.5 &    1.3 &    0.5 &    0.3 &  365.7 & 8404.5 &    1.8 \\
: lbt < %L >%O       &   6 &    0.0 &    4.2 &    6.8 &   14.7 &    0.8 &    4.0 &    1.8 &    0.5 &    0.3 &  603.8 & 14905.2 &  239.5 \\
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Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
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The script =bench/ltl2tgba/sum.py= is a more evolved version of the
above script that generates two kinds of LaTeX tables.
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When computing such statistics, you should be aware that inputs for
which a tool failed to generate an automaton (e.g. it crashed, or it
was killed if you used =ltlcross='s =--timeout= option to limit run
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time) will appear as mostly empty lines in the CSV or JSON files,
since most statistics cannot be computed without an automaton...
Those lines with missing data can be omitted with the =--omit-missing=
option (this used to be the default up to Spot 1.2).

However data for bogus automata are still included: as shown below
=ltlcross= will report inconsistencies between automata as errors, but
it does not try to guess who is incorrect.
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** Description of the columns

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The number of column output in the CSV or JSON outputs depend on the
options passed to =ltlcross=. Additional columns will be output if
=--strength=, =--ambiguous=, =--automata=, or =--product=+N= are used.

Columns =formula= and =tool= contain the formula translated and the
command run to translate it.  In the CSV, these columns contain the
actual text.  In the JSON output, these column contains an index into
the =formula= and =tool= table declared separately.
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=exit_status= and =exit_code= are used to indicate if the translator
successfully produced an automaton, or if it failed.  On successful
translation, =exit_status= is equal to "=ok=" and =exit_code= is 0.
If the translation took more time than allowed with the =--timeout=
option, =exit_status= will contain "=timeout=" and =exit_code= will be
set to -1.  Other values are used to diagnose various issues: please
check the man-page for =ltlcross= for a list of them.

=time= obviously contains the time used by the translation.  Time is
measured with some high-resolution clock when available (that's
nanosecond accuracy under Linux), but because translator commands are
executed through a shell, it also includes the time to start a shell.
(This extra cost apply identically to all translators, so it is not unfair.)


All the values that follow will be missing if =exit_status= is not
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equal to "=ok=".  (You may instruct =ltlcross= not to output lines with
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such missing data with the option =--omit-missing=.)

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=states=, =edges=, =transitions=, =acc= are size measures for the
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automaton that was translated.  =acc= counts the number of acceptance
sets.  When building (degeneralized) Büchi automata, it will always be
=1=, so its value is meaningful only when evaluating translations to
generalized Büchi automata.  =edges= counts the actual number of edges
in the graph supporting the automaton; an edge (labeled by a Boolean
formula) might actually represent several transitions (each labeled by
assignment of all atomic propositions).  For instance in an automaton
where the atomic proposition are $a$ and $b$, one edge labeled by
$a\lor b$ actually represents three transitions $a b$, $a\bar b$, and
$\bar a b$.

The following picture displays two automata for the LTL formula =a U
b=.  They both have 2 states and 3 edges, however they differ in the
number of transitions (7 versus 8), because the initial self-loop is
more constrained in the first automaton.  A smaller number of
transition is therefore an indication of a more constrained automaton.

#+BEGIN_SRC dot :file edges.png :cmdline -Tpng :exports results
digraph G {
  0 [label="", style=invis, height=0]
  0 -> 1
  1 [label="A1"]
  1 -> 2 [label="b\n"]
  1 -> 1 [label="a & !b\n"]
  2 [label="B1", peripheries=2]
  2 -> 2 [label="1"]

  3 [label="", style=invis, height=0]
  3 -> 4
  4 [label="A2"]
  4 -> 5 [label="b\n"]
  4 -> 4 [label="a\n"]
  5 [label="B2", peripheries=2]
  5 -> 5 [label="1"]
}
#+END_SRC

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


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=scc= counts the number of strongly-connected components in the automaton.

If option =--strength= is passed to =ltlcross=, these SCCs are
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also partitioned on four sets based on their strengths:
- =nonacc_scc= for non-accepting SCCs (such as states A1 and A2 in the
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  previous picture).
- =terminal_scc= for accepting SCCs where all states or edges belong
  to the same acceptance sets, and that are complete (i.e., any state
  in a terminal SCC accepts the universal language).  States
  B1 and B2 in the previous picture are two terminal SCCs.
- =weak_scc= for accepting SCCs where all states or edges belong
  to the same acceptance sets, but that are not complete.
- =strong_scc= for accepting SCCs that are not weak.
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These SCC strengths can be used to compute the strength of the
automaton as a whole:
- an automaton is terminal if it contains only non-accepting or
  terminal SCCs,
- an automaton is weak if it it contains only non-accepting,
  terminal, or weak SCCs,
- an automaton is strong if it contains at least one strong SCC.

This classification is used to fill the =terminal_aut=, =weak_aut=,
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=strong_aut= columns with Boolean values (still only if option
=--strength= is passed).  Only one of these should contain =1=.  We
usually prefer terminal automata over weak automata, and weak automata
over strong automata, because the emptiness check of terminal (and
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weak) automata is easier.  When working with alternating automata, all
those strength-related columns will be empty, because the routines
used to compute those statistic do not yet support universal edges.
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=nondetstates= counts the number of non-deterministic states in the
automaton.  =nondeterministic= is a Boolean value indicating if the
automaton is not deterministic.  For instance in the previous picture
showing two automata for =a U b=, the first automaton is deterministic
(these two fields will contain 0), while the second automaton contain
a nondeterministic state (state A2 has two possible successors for the
assignment $ab$) and is therefore not deterministic.

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If option =--aumbiguous= was passed to =ltlcross=, the column
=ambiguous_aut= holds a Boolean indicating whether the automaton is
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ambiguous, i.e., if there exists a word that can be accepted by at
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least two different runs.  (This information is not yet available for
alternating automata.)
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=complete_aut= is a Boolean indicating whether the automaton is
complete.

Columns =product_states=, =product_transitions=, and =product_scc=
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count the number of state, transitions and strongly-connect components
in the product that has been built between the translated automaton
and a random model.  For a given formula, the same random model is of
course used against the automata translated by all tools.  Comparing
the size of these product might give another indication of the
"conciseness" of a translated automaton.

There is of course a certain "luck factor" in the size of the product.
Maybe some translator built a very dumb automaton, with many useless
states, in which just a very tiny part is translated concisely.  By
luck, the random model generated might synchronize with this tiny part
only, and ignore the part with all the useless states.  A way to
lessen this luck factor is to increase the number of products
performed against the translated automaton.  If option =--products=N=
is used, =N= products are builds instead of one, and the fields
=product_states=, =product_transitions=, and =product_scc= contain
average values.

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If the option =--products=+N= is used (with a =+= in front of the
number), then no average value is computed.  Instead, three columns
=product_states=, =product_transitions=, and =product_scc= are output
for each individual product (i.e., $3\times N$ columns are output).
This might be useful if you want to compute different kind of
statistic (e.g., a median instead of a mean) or if you want to build
scatter plots of all these products.

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Finally, if the =--automata= option was passed to =ltlcross=, the CSV
or JSON output will contain a column named =automaton= encoding each
produced automaton in the HOA format.

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** Changing the name of the translators

By default, the names used in the CSV and JSON output to designate the
translators are the command specified on the command line.

For instance in the following, =ltl2tgba= is run in two
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configurations, and the strings =ltl2tgba -s --small %f >%O= and
=ltl2tgba -s --deter %f >%O= appear verbatim in the output:
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#+BEGIN_SRC sh :results verbatim :exports both
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ltlcross -f a -f Ga 'ltl2tgba -s --small %f >%O' 'ltl2tgba -s --deter %f >%O' --csv
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#+END_SRC
#+RESULTS:
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: "formula","tool","exit_status","exit_code","time","states","edges","transitions","acc","scc","nonacc_scc","terminal_scc","weak_scc","strong_scc","nondet_states","nondet_aut","terminal_aut","weak_aut","strong_aut","product_states","product_transitions","product_scc"
: "(a)","ltl2tgba -s --small %f >%O","ok",0,0.043876,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4208,2
: "(a)","ltl2tgba -s --deter %f >%O","ok",0,0.0432137,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4208,2
: "(!(a))","ltl2tgba -s --small %f >%O","ok",0,0.0400653,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4205,2
: "(!(a))","ltl2tgba -s --deter %f >%O","ok",0,0.0450417,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4205,2
: "(G(a))","ltl2tgba -s --small %f >%O","ok",0,0.0429628,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2077,1
: "(G(a))","ltl2tgba -s --deter %f >%O","ok",0,0.0478663,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2077,1
: "(!(G(a)))","ltl2tgba -s --small %f >%O","ok",0,0.0436822,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8442,2
: "(!(G(a)))","ltl2tgba -s --deter %f >%O","ok",0,0.039919,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8442,2
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To present these results graphically, or even when analyzing these
data, it might be convenient to give each configured tool a shorter
name.  =ltlcross= supports the specification of such short names by
looking whether the command specification for a translator has the
form "={short name}actual command=".

For instance:
#+BEGIN_SRC sh :results verbatim :exports both
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ltlcross -f a -f Ga '{small} ltl2tgba -s --small %f >%O' '{deter} ltl2tgba -s --deter %f >%O' --csv
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#+END_SRC
#+RESULTS:
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: "formula","tool","exit_status","exit_code","time","states","edges","transitions","acc","scc","nonacc_scc","terminal_scc","weak_scc","strong_scc","nondet_states","nondet_aut","terminal_aut","weak_aut","strong_aut","product_states","product_transitions","product_scc"
: "(a)","small","ok",0,0.0433468,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4208,2
: "(a)","deter","ok",0,0.0429179,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4208,2
: "(!(a))","small","ok",0,0.0400513,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4205,2
: "(!(a))","deter","ok",0,0.040167,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4205,2
: "(G(a))","small","ok",0,0.0447826,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2077,1
: "(G(a))","deter","ok",0,0.0439892,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2077,1
: "(!(G(a)))","small","ok",0,0.0444007,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8442,2
: "(!(G(a)))","deter","ok",0,0.0396312,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8442,2
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* Detecting problems
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   :PROPERTIES:
   :CUSTOM_ID: checks
   :END:
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If a translator exits with a non-zero status code, or fails to output
an automaton =ltlcross= can read, and error will be displayed and the
result of the translation will be discarded.

Otherwise =ltlcross= performs the following checks on all translated
formulas ($P_i$ and $N_i$ designate respectively the translation of
positive and negative formulas by the ith translator).

  - Intersection check: $P_i\otimes N_j$ must be empty for all
    pairs of $(i,j)$.

    A single failing translator might generate a lot of lines of
    the form:

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    : error: P0*N1 is nonempty; both automata accept the infinite word
    :        cycle{p0 & !p1}
    : error: P1*N0 is nonempty; both automata accept the infinite word
    :        p0; !p1; cycle{p0 & p1}
    : error: P1*N1 is nonempty; both automata accept the infinite word
    :        p0; cycle{!p1 & !p0}
    : error: P1*N2 is nonempty; both automata accept the infinite word
    :        p0; !p1; cycle{p0 & p1}
    : error: P1*N3 is nonempty; both automata accept the infinite word
    :        p0; !p1; cycle{p0 & p1}
    : error: P1*N4 is nonempty; both automata accept the infinite word
    :        p0; cycle{!p1 & !p0}
    : error: P2*N1 is nonempty; both automata accept the infinite word
    :        p0; !p1; !p0; cycle{!p1 & !p0; p0 & !p1; !p1; !p1; p0 & !p1}
    : error: P3*N1 is nonempty; both automata accept the infinite word
    :        p0; !p1; !p1 & !p0; cycle{p0 & !p1}
    : error: P4*N1 is nonempty; both automata accept the infinite word
    :        p0; !p1; !p1 & !p0; cycle{p0 & !p1}
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    In this example, translator number =1= looks clearly faulty
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    (at least the other 4 translators do not contradict each other).

    Examples of infinite words that are accepted by both automata
    always have the form of a lasso: a (possibly empty) finite prefix
    followed by a cycle that should be repeated infinitely often.
    The cycle part is denoted by =cycle{...}=.
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  - Complemented intersection check.  If $P_i$ and $N_i$ are
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    deterministic, =ltlcross= builds their complements, $Comp(P_i)$
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    and $Comp(N_i)$, and then ensures that $Comp(P_i)\otimes
    Comp(N_i)$ is empty.  If only one of them is deterministic, for
    instance $P_i$, we check that $P_j\otimes Comp(P_i)$ for all $j
    \ne i$; likewise if it's $N_i$ that is deterministic.
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    By default this check is only done for deterministic automata,
    because complementation is relatively cheap is that case (at least
    it is cheap for simple acceptance conditions).  Using option
    =--determinize=, =ltlcross= can be instructed to perform
    complementation of non-deterministic automata as well, ensuring
    precise equivalence checks between all automata.  However be aware
    that this determinization + complementation may generate large
    automata.

    When validating a translator with =ltlcross= without using the
    =--determinize= option we highly recommend to include a translator
    with good deterministic output to augment test coverage.  Using
    '=ltl2tgba -lD %f >%O=' will produce deterministic automata for
    all obligation properties and many recurrence properties.  Using
    '=ltl2dstar --ltl2nba=spin:pathto/ltl2tgba@-Ds %L %O=' will
    systematically produce a deterministic Rabin automaton (that
    =ltlcross= can complement easily).
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  - Cross-comparison checks: for some state-space $S$,
    all $P_i\otimes S$ are either all empty, or all non-empty.
    Similarly all $N_i\otimes S$ are either all empty, or all non-empty.

    A cross-comparison failure could be displayed as:

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    : error: {P0,P2} disagree with {P1} when evaluating the state-space
    :        the following word(s) are not accepted by {P1}:
    :  P0 accepts: p0 & !p1 & !p2 & p3; p0 & p1 & !p2 & p3; p0 & p1 & p2 & p3; cycle{p0 & p1 & p2 & p3; p0 & p1 & !p2 & !p3; p0 & p1 & p2 & !p3; p0 & p1 & !p2 & !p3}
    :  P2 accepts: p0 & !p1 & !p2 & p3; cycle{p0 & p1 & !p2 & !p3; p0 & p1 & p2 & p3; p0 & p1 & !p2 & p3}
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    If =--products=N= is used with =N= greater than one, the number of
    the state-space is also printed.  This number is of no use by
    itself, except to explain why you may get multiple disagreement
    between the same sets of automata.

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    These products tests may sometime catch errors that were not
    captured by the first two tests if one non-deterministic automaton
    recognize less words than what it should.  If the input automata
    are deterministic or the =--determinize= option is used, this test
    is redundant and can be disabled.  (In fact, the =--determinize=
    option implies option =--product=0= to do so.)

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  - Consistency check:

    For each $i$, the products $P_i\otimes S$ and $N_i\otimes S$
    actually cover all states of $S$.  Because $S$ does not have any
    deadlock, any of its infinite path must be accepted by $P_i$ or
    $N_i$ (or both).

    An error in that case is displayed as

    : error: inconsistency between P1 and N1

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    If =--products=N= is used with =N= greater than one, the number of
    the state-space in which the inconsistency was detected is also
    printed.

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    This test may catch errors that were not captured by the first two
    tests if one non-deterministic automaton recognize less words than
    what it should.  If the input automata are deterministic or the
    =--determinize= option is used, this test is redundant and can be
    disabled.  (In fact, the =--determinize= option implies option
    =--product=0= to do so.)

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The above checks are similar to those that are performed by [[http://www.tcs.hut.fi/Software/lbtt/][LBTT]],
except for the complemented intersection check, which is only done in
=ltlcross=.
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If any problem was reported during the translation of one of the
formulas, =ltlcheck= will exit with an exit status of =1=.  Statistics
(if requested) are output nonetheless, and include any faulty
automaton as well.

* Miscellaneous options

** =--stop-on-error=

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The =--stop-on-error= option will cause =ltlcross= to abort on the
first detected error.  This include failure to start some translator,
read its output, or failure to passe the sanity checks.  Timeouts are
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allowed.

One use for this option is when =ltlcross= is used in combination with
=randltl= to check translators on an infinite stream of formulas.

For instance the following will cross-compare =ltl2tgba= against
=ltl3ba= until it finds an error, or your interrupt the command, or it
runs out of memory (the hash tables used by =randltl= and =ltlcross=
to remove duplicate formulas will keep growing).

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#+END_SRC

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** =--save-bogus=FILENAME=

The =--save-bogus=FILENAME= will save any formula for which an error
was detected (either some translation failed, or some problem was
detected using the resulting automata) in =FILENAME=.  Again, timeouts
are not considered to be errors, and therefore not reported in this
file.

The main use for this feature is in conjunction with =randltl='s
generation of random formulas.  For instance the following command
will run the translators on an infinite number of formulas, saving
any problematic formula in =bugs.ltl=.

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#+BEGIN_SRC sh :exports code :eval no
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randltl -n -1 --tree-size 10..25 a b c | ltlcross --save-bogus=bugs.ltl 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O'
760
761
762
763
764
#+END_SRC

You can periodically check the contents of =bugs.ltl=, and then run
=ltlcross= only on those formulas to look at the problems:

765
#+BEGIN_SRC sh :exports code :eval no
766
ltlcross -F bugs.ltl 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O'
767
768
#+END_SRC

769
** =--grind=FILENAME=
770

771
772
773
This option tells =ltlcross= that, when a problem is detected, it
should try to find a smaller formula that still exhibits the
problem.
774

775
776
777
778
779
780
781
Here is the procedure used:
   - internally list the mutations of the bogus formula and sort
     them by length (as [[file:ltlgrind.org][=ltlgrind --sort=]] would do)
   - process every mutation until one is found that exhibit the bug
   - repeat the process with this new formula, and again until a formula
     is found for which no mutation exhibit the bug
   - output that last formula in =FILENAME=
782

783
784
785
786
787
If =--save-bogus=OTHERFILENAME= is provided, every bogus formula found
during the process will be saved in =OTHERFILENAME=.

Example:
#+BEGIN_SRC sh :exports code :results verbatim
788
ltlcross -f '(G!b & (!c | F!a)) | (c & Ga & Fb)' "modella %L %O" \
789
790
791
  --save-bogus=bogus \
  --grind=bogus-grind
#+END_SRC
792
#+BEGIN_SRC sh :exports results :results verbatim
793
ltlcross -f '(G!b & (!c | F!a)) | (c & Ga & Fb)' "modella %L %O" \
794
795
796
  --save-bogus=bogus --grind=bogus-grind 2>&1
true
#+END_SRC
797
#+RESULTS:
798
799
#+begin_example
| & G ! p0 | ! p1 F ! p2 & & p1 G p2 F p0
800
801
Running [P0]: modella 'lcr-i0-1XJw2X' 'lcr-o0-p2nbUW'
Running [N0]: modella 'lcr-i0-lSPMMV' 'lcr-o0-3xEoFU'
802
803
804
805
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word
       cycle{!p0 & !p1}

806
807
808
809
Trying to find a bogus mutation of (G!b & (!c | F!a)) | (c & Ga & Fb)...
Mutation 1/22: & & p0 G p1 F p2
Running [P0]: modella 'lcr-i1-3PI0CT' 'lcr-o0-GxBDAS'
Running [N0]: modella 'lcr-i1-eOijzR' 'lcr-o0-wkIZxQ'
810
811
Performing sanity checks and gathering statistics...

812
813
814
Mutation 2/22: & G ! p0 | ! p1 F ! p2
Running [P0]: modella 'lcr-i2-6cYUzP' 'lcr-o0-3ihRBO'
Running [N0]: modella 'lcr-i2-6u8pEN' 'lcr-o0-phxZGM'
815
816
Performing sanity checks and gathering statistics...

817
818
819
Mutation 3/22: | G ! p0 & & p1 G p2 F p0
Running [P0]: modella 'lcr-i3-iwZZML' 'lcr-o0-LMB2SK'
Running [N0]: modella 'lcr-i3-r2g9ZJ' 'lcr-o0-HEfh7I'
820
821
822
823
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word
       cycle{!p0 & !p1}

824
825
826
827
Trying to find a bogus mutation of G!b | (c & Ga & Fb)...
Mutation 1/16: t
Running [P0]: modella 'lcr-i4-R8PbkI' 'lcr-o0-6ts7wH'
Running [N0]: modella 'lcr-i4-cv7EKG' 'lcr-o0-vVmdYF'
828
829
Performing sanity checks and gathering statistics...

830
831
832
Mutation 2/16: G ! p0
Running [P0]: modella 'lcr-i5-ZO3HcF' 'lcr-o0-UiydrE'
Running [N0]: modella 'lcr-i5-gOseGD' 'lcr-o0-CL4fVC'
833
834
Performing sanity checks and gathering statistics...

835
Mutation 3/16: & & p0 G p1 F p2
836
837
838
warning: This formula or its negation has already been checked.
         Use --allow-dups if it should not be ignored.

839
840
841
Mutation 4/16: | G ! p0 & p1 F p0
Running [P0]: modella 'lcr-i6-QTDjtB' 'lcr-o0-tFpmKA'
Running [N0]: modella 'lcr-i6-1lnX1z' 'lcr-o0-TWWyjz'
842
843
844
845
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word
       cycle{!p0 & !p1}

846
847
Trying to find a bogus mutation of G!b | (c & Fb)...
Mutation 1/10: t
848
849
850
warning: This formula or its negation has already been checked.
         Use --allow-dups if it should not be ignored.

851
Mutation 2/10: G ! p0
852
853
854
warning: This formula or its negation has already been checked.
         Use --allow-dups if it should not be ignored.

855
856
857
Mutation 3/10: & p0 F p1
Running [P0]: modella 'lcr-i7-CJTWjx' 'lcr-o0-DKbNEw'
Running [N0]: modella 'lcr-i7-tSts1v' 'lcr-o0-CGbWov'
858
859
Performing sanity checks and gathering statistics...

860
861
862
Mutation 4/10: | p0 G ! p1
Running [P0]: modella 'lcr-i8-6dtbPu' 'lcr-o0-gLnsfu'
Running [N0]: modella 'lcr-i8-Xn6gGt' 'lcr-o0-U7966s'
863
864
Performing sanity checks and gathering statistics...

865
866
867
Mutation 5/10: | G ! p0 F p0
Running [P0]: modella 'lcr-i9-Vh7hAs' 'lcr-o0-eYZt3r'
Running [N0]: modella 'lcr-i9-I6dfxr' 'lcr-o0-DoY00q'
868
869
Performing sanity checks and gathering statistics...

870
871
872
Mutation 6/10: | ! p0 & p1 F p0
Running [P0]: modella 'lcr-i10-j7fVvq' 'lcr-o0-aChQ0p'
Running [N0]: modella 'lcr-i10-bnpgwp' 'lcr-o0-KV3G1o'
873
874
Performing sanity checks and gathering statistics...

875
876
877
Mutation 7/10: | G p1 & p0 F p1
Running [P0]: modella 'lcr-i11-uUhFyo' 'lcr-o0-OlrF5n'
Running [N0]: modella 'lcr-i11-XHjeDn' 'lcr-o0-0CINan'
878
879
Performing sanity checks and gathering statistics...

880
881
882
Mutation 8/10: | & p0 p1 G ! p0
Running [P0]: modella 'lcr-i12-a7eJLm' 'lcr-o0-RUZFmm'
Running [N0]: modella 'lcr-i12-dobiYl' 'lcr-o0-LuVUzl'
883
884
Performing sanity checks and gathering statistics...

885
886
887
Mutation 9/10: | G ! p0 & p0 F p0
Running [P0]: modella 'lcr-i13-nFQYdl' 'lcr-o0-Mq84Rk'
Running [N0]: modella 'lcr-i13-CnPRwk' 'lcr-o0-7f4Fbk'
888
889
890
891
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word
       cycle{!p0}

892
893
Trying to find a bogus mutation of G!c | (c & Fc)...
Mutation 1/7: t
894
895
896
warning: This formula or its negation has already been checked.
         Use --allow-dups if it should not be ignored.

897
Mutation 2/7: G ! p0
898
899
900
warning: This formula or its negation has already been checked.
         Use --allow-dups if it should not be ignored.

901
902
903
Mutation 3/7: & p0 F p0
Running [P0]: modella 'lcr-i14-KsKPgj' 'lcr-o0-Qo3UXi'
Running [N0]: modella 'lcr-i14-FVPvFi' 'lcr-o0-Zh06mi'
904
905
Performing sanity checks and gathering statistics...

906
907
908
Mutation 4/7: | p0 G ! p0
Running [P0]: modella 'lcr-i15-JB045h' 'lcr-o0-EIL4Oh'
Running [N0]: modella 'lcr-i15-sYY8yh' 'lcr-o0-KVOdjh'
909
910
Performing sanity checks and gathering statistics...

911
Mutation 5/7: | G ! p0 F p0
912
913
914
warning: This formula or its negation has already been checked.
         Use --allow-dups if it should not be ignored.

915
916
917
Mutation 6/7: | ! p0 & p0 F p0
Running [P0]: modella 'lcr-i16-hoqdSg' 'lcr-o0-xghJEg'
Running [N0]: modella 'lcr-i16-5JvOrg' 'lcr-o0-SUdVeg'
918
919
Performing sanity checks and gathering statistics...

920
921
922
Mutation 7/7: | & p0 F p0 G p0
Running [P0]: modella 'lcr-i17-BNo92f' 'lcr-o0-0T8nRf'
Running [N0]: modella 'lcr-i17-J5v5Ff' 'lcr-o0-MOyNuf'
923
924
Performing sanity checks and gathering statistics...

925
Smallest bogus mutation found for (G!b & (!c | F!a)) | (c & Ga & Fb) is G!c | (c & Fc).
926
927
928
929

error: some error was detected during the above runs.
       Check file bogus for problematic formulas.
#+end_example
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946

#+BEGIN_SRC sh :exports both :results verbatim
cat bogus
#+END_SRC

#+RESULTS:
: (G!b & (!c | F!a)) | (c & Ga & Fb)
: G!b | (c & Ga & Fb)
: G!b | (c & Fb)
: G!c | (c & Fc)

#+BEGIN_SRC sh :exports both :results verbatim
cat bogus-grind
#+END_SRC

#+RESULTS:
: G!c | (c & Fc)
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965

** =--no-check=

The =--no-check= option disables all sanity checks, and only use the supplied
formulas in their positive form.

When checks are enabled, the negated formulas are intermixed with the
positives ones in the results.  Therefore the =--no-check= option can
be used to gather statistics about a specific set of formulas.


#  LocalWords:  ltlcross num toc LTL Büchi LBTT Testbench PSL SRC sed
#  LocalWords:  automata LBT LBTT's ltl tgba GFa lck iDGV sA FYp BYY
#  LocalWords:  ClVQg wyErP UNE dQ coM tH eHPoQy goto ba lbt modella
#  LocalWords:  lbtt csv json randltl ltlfilt wm eGEYaZ nYpFBX fGdZQ
#  LocalWords:  CPs kXiZZS ILLzR wU CcMCaQ IOckzW tsT RZ TJXmT jb XRO
#  LocalWords:  nxqfd hS vNItGg acc scc nondetstates nondeterministic
#  LocalWords:  cvs LaTeX datacols len ith otimes ltlcheck eval setq
#  LocalWords:  setenv concat getenv
966
967
968
969
970
971
** =--verbose=

The verbose option can be useful to troubleshoot problems or simply
follow the list of transformations and tests performed by =ltlcross=.

For instance here is what happens if we try to cross check =ltl2tgba=
972
973
974
and =ltl3ba -H1= on the formula =FGa=.  Note that =ltl2tgba= will
produce transition-based generalized Büchi automata, while =ltl3ba
-H1= produces co-Büchi alternating automata.
975
976

#+BEGIN_SRC sh :results verbatim :exports code
977
ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --determinize --verbose
978
979
980
#+END_SRC

#+BEGIN_SRC sh :results verbatim :exports results
981
ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --determinize --verbose 2>&1
982
983
984
985
986
#+END_SRC

#+RESULTS:
#+begin_example
F(G(a))
987
988
989
990
991
992
993
994
995
Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-Ak0bYx'
Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-5U1MyT'
Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-sX2kaf'
Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-4siKPA'
info: collected automata:
info:   P0	(2 st.,3 ed.,1 sets)
info:   N0	(1 st.,2 ed.,1 sets) deterministic complete
info:   P1	(2 st.,3 ed.,1 sets)
info:   N1	(3 st.,5 ed.,1 sets) univ-edges deterministic complete
996
Performing sanity checks and gathering statistics...
997
998
info: getting rid of universal edges...
info:   N1	(3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets)
999
1000
1001
1002
1003
1004
info: complementing non-deterministic automata via determinization...
info:   P0	(2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,2 sets)	Comp(P0)
info:   P1	(2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,2 sets)	Comp(P1)
info: getting rid of any Fin acceptance...
info:	Comp(P0)	(2 st.,4 ed.,2 sets) -> (3 st.,7 ed.,2 sets)
info:	Comp(N0)	(1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
1005
info:	     P1 	(2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
info:	Comp(P1)	(2 st.,4 ed.,2 sets) -> (4 st.,9 ed.,2 sets)
info:	Comp(N1)	(2 st.,4 ed.,1 sets) -> (3 st.,6 ed.,1 sets)
info: check_empty P0*N0
info: check_empty Comp(N0)*Comp(P0)
info: check_empty P0*N1
info: check_empty P1*N0
info: check_empty P1*N1
info: check_empty Comp(N1)*Comp(P1)

No problem detected.
#+end_example

First =FGa= and its negations =!FGa= are translated with the two
1019
tools, resulting in four automata: two positive automata =P0= and =P1=
1020
1021
for =FGa=, and two negative automata =N0= and =N1=.

1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
Some basic information about the collected automata are displayed.
For instance we can see that although =ltl3ba -H1= outputs co-Büchi
alternating automata, only automaton =N1= uses universal edges: the
automaton =P1= can be used like a non-alternating co-Büchi automaton.

=ltlcross= then proceeds to transform alternating automata (only weak
alternating automata are supported) into non-alternating automata.
Here only =N1= needs this transformation.

Then =ltlcross= computes the complement of these four
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
automata.  Since =P0= and =P1= are nondeterministic and the
=--determinize= option was given, a first pass determinize and
complete these two automata, creating =Comp(P0)= and =Comp(P1)=.

Apparently =N0= and =N1= are already deterministic, so their
complement could be obtained by just complementing their acceptance
condition (this is not written -- we only deduce so because they do
not appear in the list of automata that had to be determinized).

Now that =ltlcross= has four complemented automata, it has to make
sure they use only =Inf= acceptance because that is what our emptiness
check procedure can handle.  So there is a new pass over all automata,
rewriting them to get rid of any =Fin= acceptance.

After this preparatory work, it is time for actually comparing these
automata.  Together, the tests =P0*N0= and =Comp(N0)*Comp(P0)= ensure
that the automaton =N0= is really the complement of =P0=.  Similarly
=P1*N1= and =Comp(N1)*Comp(P1)= ensure that =N1= is the complement of
=P1=.  Finally =P0*N1= and =P1*N0= ensure that =P1= is equivalent to
=P0= and =N1= is equivalent to =N0=.



Note that if we had not used the =--determinize= option, the procedure
would look slightly more complex:

#+BEGIN_SRC sh :results verbatim :exports code
1059
ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --verbose
1060
1061
1062
#+END_SRC

#+BEGIN_SRC sh :results verbatim :exports results
1063
ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --verbose 2>&1
1064
1065
1066
1067
1068
#+END_SRC

#+RESULTS:
#+begin_example
F(G(a))
1069
1070
1071
1072
1073
1074
1075
1076
1077
Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-jD32mW'
Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-w6IJYI'
Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-dac1Av'
Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-OZL7fi'
info: collected automata:
info:   P0	(2 st.,3 ed.,1 sets)
info:   N0	(1 st.,2 ed.,1 sets) deterministic complete
info:   P1	(2 st.,3 ed.,1 sets)
info:   N1	(3 st.,5 ed.,1 sets) univ-edges deterministic complete
1078
Performing sanity checks and gathering statistics...
1079
1080
info: getting rid of universal edges...
info:   N1	(3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets)
1081
1082
info: getting rid of any Fin acceptance...
info:	Comp(N0)	(1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
1083
info:	     P1 	(2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
info:	Comp(N1)	(2 st.,4 ed.,1 sets) -> (3 st.,6 ed.,1 sets)
info: check_empty P0*N0
info: check_empty P0*N1
info: check_empty Comp(N0)*N1
info: check_empty P1*N0
info: check_empty Comp(N1)*N0
info: check_empty P1*N1
info: building state-space #0/1 of 200 states with seed 0
info: state-space has 4136 edges
info: building product between state-space and P0 (2 st., 3 ed.)
info:   product has 400 st., 8298 ed.
info:               2 SCCs
info: building product between state-space and P1 (2 st., 3 ed.)
info:   product has 400 st., 8298 ed.
info:               2 SCCs
info: building product between state-space and N0 (1 st., 2 ed.)
info:   product has 200 st., 4136 ed.
info:               1 SCCs
info: building product between state-space and N1 (2 st., 4 ed.)
info:   product has 400 st., 8272 ed.
info:               1 SCCs
info: cross_check {P0,P1}, state-space #0/1
info: cross_check {N0,N1}, state-space #0/1
info: consistency_check (P0,N0), state-space #0/1
info: consistency_check (P1,N1), state-space #0/1

No problem detected.
#+end_example

In this case, =ltlcross= does not have any complement automaton for
=P0= and =P1=, so it cannot make sure that =P0= and =P1= are
equivalent.  If we imagine for instance that =P0= has an empty
language, we can see that the six =check_empty= tests would still
succeed.

So =ltlcross= builds a random state-space of 200 states, synchronize
it with the four automata, and then performs additional checks
(=cross_check= and =consistency_check=) on these products as described
[[#checks][earlier]].  While these additional checks do not make a proof that =P0=
and =P1= are equivalent, they can catch some problems, and would
easily catch the case of an automaton with an empty language by
mistake.