# -*- coding: utf-8 -*-
#+TITLE: =ltlcross=
#+DESCRIPTION: Spot command-line tool for cross-comparing the output of LTL translators.
#+INCLUDE: setup.org
#+HTML_LINK_UP: tools.html
#+PROPERTY: header-args:sh :results verbatim :exports both
#+PROPERTY: header-args:R :session :results output :exports both

=ltlcross= is a tool for cross-comparing the output of LTL-to-automata
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.

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:
    - 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
  - an option to *reduce counterexamples* by attempting to mutate and
    shorten troublesome formulas (option =--grind=),
  - 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").

Although =ltlcross= performs similar sanity checks as LBTT, it does
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
problems, maybe with a counterexample, but you will be on your own to
investigate and fix them (the =--grind= option may help you reduce the
problem to a shorter formula).

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
    will be denoted =P0=, =N0=, =P1=, =N1=, =P2=, =N2=.
  - Optionally build complemented automata denoted =Comp(P0)=, =Comp(N0)=, etc.
    (By default, this is done only for small automata, but see options =-D=,
    =--determinize-max-states= and =--determinize-max-edges=.)
  - Perform sanity checks between all these automata to detect any problem.
  - Optionally build the products of these automata with a random state-space (the same
    state-space for all translations).  (If the =--products=N= option is given,
    =N= products are performed instead.)
  - Gather statistics if requested.

* Formula selection

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.

* Configuring translators

** Translator specifications

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

#+BEGIN_SRC sh :exports results
  ltlcross --help | sed -n '/character sequences:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
:   %%                         a single %
:   %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
:   %O                         the automaton output in HOA, never claim, LBTT, or
:                              ltl2dstar's format

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

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

#+BEGIN_SRC sh :exports results
ltlcross -f 'GFa' -f 'X(a U b)' 'ltl2tgba -s %s >%O' 'spin -f %s >%O' 2>&1
#+END_SRC
#+RESULTS:
#+begin_example
([](<>(a)))
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'
Performing sanity checks and gathering statistics...

(X((a) U (b)))
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'
Performing sanity checks and gathering statistics...

No problem detected.
#+end_example

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.

=ltlcross= can only read four kinds of output:
  - Never claims (only if they are restricted to representing an
    automaton using =if=, =goto=, and =skip= statements) such as those
    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
    newer syntax introduced by Spin 6.24, using =do= instead of =if=,
    is also supported.
  - [[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
    --lbtt=.
  - Non-alternating automata in [[file:http://adl.github.io/hoaf/][the HOA format]] with any acceptance
    condition.
  - [[file:concepts.org::#property-flags][Weak]] alternating automata in [[file:http://adl.github.io/hoaf/][the HOA format]].
  - [[http://www.ltl2dstar.de/docs/ltl2dstar.html][=ltl2dstar='s format]], which supports deterministic Rabin or Streett
    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.)

Of course all configured tools need not use the same =%= sequences.
The following list shows some typical configurations for some existing
tools:

  - '=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)
  - '=modella -r12 -g -e %[MWei^]L %O='
  - '=/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
    its interface with LBTT)
  - '=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)
  - '=lbt <%L >%O='
  - '~ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD
    --output-format=hoa %[MW]L %O~' deterministic Rabin output in HOA, as
    supported since version 0.5.2 of =ltl2dstar=.
  - '~ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD --automata=streett
    --output-format=hoa %[MW]L %O~' deterministic Streett output in HOA,
    as supported since version 0.5.2 of =ltl2dstar=.
  - '=ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD %[MW]L %O=' (Rabin
    output in DSTAR format, as supported in older versions of
    =ltl2dstar=.
  - '=ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD %L - | dstar2tgba
    -s >%O=' (external conversion from Rabin to Büchi done by
    =dstar2tgba= for more reduction of the Büchi automaton than what
    =ltlcross= would provide)
  - '=java -jar Rabinizer.jar -ltl2dstar %[MW]F %O; mv %O.dst %O=' (Rabinizer
    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)
  - '~java -jar rabinizer3.1.jar -in=formula -silent -out=std -format=hoa -auto=tr %[MWRei^]f >%O~'
    (rabinizer 3.1 can output automata in the HOA format)
  - '=ltl3dra -f %s >%O=' (The HOA format is the default for =ltl3dra=.)
  - '=ltl3tela -f %s >%O=' (The HOA format is the default for =ltl3tela=.)

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

  delag                                    %f>%O
  lbt                                      <%L>%O
  ltl2ba                                   -f %s>%O
  ltl2(da|dgra|dpa|dra|ldba|na|nba|ngba)   %f>%O
  ltl2dstar                                --output-format=hoa %[MW]L %O
  ltl2tgba                                 -H %f>%O
  ltl3(ba|dra|hoa|tela)                    -f %s>%O
  modella                                  %[MWei^]L %O
  spin                                     -f %s>%O
  owl.* ltl2[bdeglnpr]+a\b                 -f %f>%O
  owl.* ltl2delta2\b                       -f %f
  owl.* ltl-utilities\b                    -f %f

Any {name} and directory component is skipped for the purpose of
matching those prefixes.  So for instance
  '{DRA} ~/mytools/ltl2dstar-0.5.2'
will be changed into
  '{DRA} ~/mytools/ltl2dstar-0.5.2 --output-format=hoa %[MW]L %O'
#+end_example

What this implies is that running =ltlcross ltl2ba ltl3ba ...= is
the same as running =ltlcross 'ltl2ba -f %s>%O' 'ltl3ba -f %s>%O' ...=

Because only the prefix of the actual command is checked, you can
still specify some options.  For instance =ltlcross 'ltl2tgba -D' ...=
is short for =ltlcross 'ltl2tgba -D -H %F>%O' ...=

** Trusted and untrusted translators

By default, all translators specified are not trusted.  This means
that =ltlcross= will cross-compare the output of all translators,
possibly yielding a quadratic number of tests.

It is possible to declare that certain translators should be trusted
by specifying them with the =--reference=COMMANDFMT= option.  This has
a few implications:
  - the automata output by reference translators are not tested
  - a pair of positive and negative reference automata are selected
    from the reference translators (the smallest automata, in case
    multiple references are available), and all other translators will
    only be compared to these reference automata.

Consequently, the number of test performed is now linear in the number
of untrusted references.  The easiest way to observe the effect of
=--reference= is to run the =ltlcross= with the [[#verbose][=--verbose= option]],
with and without some =--reference= translators.

* Detecting problems
   :PROPERTIES:
   :CUSTOM_ID: checks
   :END:

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:

    : 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}

    In this example, translator number =1= looks clearly faulty
    (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{...}=.

  - Complemented intersection check.  If $P_i$ and $N_i$ are
    deterministic or if they are small enough, =ltlcross= attempts to
    build their complements, $Comp(P_i)$ and $Comp(N_i)$.

    Complementation is not always attempted, especially when it
    requires a determinization-based construction.  The conditions
    specifying when the complement automata are constructed can be
    modified with the =--determinize-max-states=N= and
    =--determinize-max-edges=M= options, which abort the
    complementation if it would produce an automaton with more than
    =N= states (500 by default) or more than =M= edges (5000 by
    default).  Alternatively, use =--determinize= (a.k.a. =-D=) to
    force the complementation of all automata.

    If both complement automata could be computed, =ltlcross= ensures that
    $Comp(P_i)\otimes Comp(N_i)$ is empty.

    If only one automaton has been complemented, for instance $P_i$,
    =ltlcross= checks that $P_j\otimes Comp(P_i)$ for all $j \ne i$;
    likewise if it's $N_i$ that is deterministic.

    When validating a translator with =ltlcross= without using the
    =--determinize= option we highly recommend including a translator
    with good deterministic output to augment test coverage.  Using
    '=ltl2tgba -D %f >%O=' will produce deterministic automata for all
    obligation properties and many recurrence properties.  Using
    '=ltl2tgba -PD %f >%O=' will systematically produce a
    deterministic Parity automaton (that =ltlcross= can complement
    easily).

  - 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:

    : 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}

    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.

    These products tests may sometime catch errors that were not
    captured by the first two tests if one non-deterministic automaton
    recognize fewer words than what it should.  If the input automata
    are all 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.)

  - 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

    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.

    This test may catch errors that were not captured by the first two
    tests if one non-deterministic automaton recognize fewer 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.)

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=.

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.

* 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.

** CSV or JSON output (or both!)

The following compare =ltl2tgba=, =spin=, and =lbt= on three random
formulas (where =W= and =M= operators have been rewritten away because
they are not supported by =spin= and =lbt=).

#+BEGIN_SRC sh :results verbatim :exports code
randltl -n 3 --seed=5 a b |
ltlfilt --remove-wm |
ltlcross --csv=results.csv \
         'ltl2tgba -s %f >%O' \
         'spin -f %s >%O' \
         'lbt < %L >%O'
#+END_SRC
#+RESULTS:

#+BEGIN_SRC sh :results verbatim :exports results
randltl -n 3 --seed=5 a b | ltlfilt --remove-wm |
ltlcross --csv=results.csv --json=results.json \
         'ltl2tgba -s %f >%O' \
         'spin -f %s >%O' \
         'lbt < %L >%O' 2>&1
#+END_SRC

#+RESULTS:
#+begin_example
-:1: G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))
Running [P0]: ltl2tgba -s 'G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))' >'lcr-o0-UWmAOs'
Running [P1]: spin -f '[](!((!(p0)) && ((X(p0)) || (<>((p1) V (X(p0)))))))' >'lcr-o1-UALknk'
Running [P2]: lbt < 'lcr-i0-7JlcYb' >'lcr-o2-Yqa4y3'
Running [N0]: ltl2tgba -s '!(G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0))))))))' >'lcr-o0-O9vcaV'
Running [N1]: spin -f '!([](!((!(p0)) && ((X(p0)) || (<>((p1) V (X(p0))))))))' >'lcr-o1-jwQ2NM'
Running [N2]: lbt < 'lcr-i0-eDh8rE' >'lcr-o2-kjYd6v'
Performing sanity checks and gathering statistics...

-:2: F(p0)
Running [P0]: ltl2tgba -s 'F(p0)' >'lcr-o0-yDowOn'
Running [P1]: spin -f '<>(p0)' >'lcr-o1-zxU8yf'
Running [P2]: lbt < 'lcr-i1-l78Dj7' >'lcr-o2-NcC93Y'
Running [N0]: ltl2tgba -s '!(F(p0))' >'lcr-o0-5xUjPQ'
Running [N1]: spin -f '!(<>(p0))' >'lcr-o1-fD4OCI'
Running [N2]: lbt < 'lcr-i1-G0kdqA' >'lcr-o2-ZsPBds'
Performing sanity checks and gathering statistics...

-:3: (X(p0)) U ((!(p0)) & (X(p0)))
Running [P0]: ltl2tgba -s '(X(p0)) U ((!(p0)) & (X(p0)))' >'lcr-o0-I2461j'
Running [P1]: spin -f '(X(p0)) U ((!(p0)) && (X(p0)))' >'lcr-o1-IxFTSb'
Running [P2]: lbt < 'lcr-i2-02hRJ3' >'lcr-o2-yl6OAV'
Running [N0]: ltl2tgba -s '!((X(p0)) U ((!(p0)) & (X(p0))))' >'lcr-o0-yTDisN'
Running [N1]: spin -f '!((X(p0)) U ((!(p0)) && (X(p0))))' >'lcr-o1-CUj4lF'
Running [N2]: lbt < 'lcr-i2-xWC2fx' >'lcr-o2-aHi19o'
Performing sanity checks and gathering statistics...

No problem detected.
#+end_example

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

#+BEGIN_SRC sh :results output raw :exports results
sed 's/"//g
s/|/\\vert{}/g
s/--/@@html:--@@/g
1a\
|-|
s/^/| /
s/$/ |/
s/,/|/g
' results.csv
#+END_SRC

#+ATTR_HTML: :class csv-table
#+RESULTS:
| formula                                                  | tool               | exit_status | exit_code |        time | states | edges | transitions | acc | scc | nondet_states | nondet_aut | complete_aut | product_states | product_transitions | product_scc |
|----------------------------------------------------------+--------------------+-------------+-----------+-------------+--------+-------+-------------+-----+-----+---------------+------------+--------------+----------------+---------------------+-------------|
| G(!((!(p0)) & ((X(p0)) \vert{} (F((p1) R (X(p0)))))))    | ltl2tgba -s %f >%O | ok          |         0 |   0.0125294 |      2 |     3 |           3 |   1 |   2 |             0 |          0 |            0 |            400 |                6276 |           2 |
| G(!((!(p0)) & ((X(p0)) \vert{} (F((p1) R (X(p0)))))))    | spin -f %s >%O     | ok          |         0 |   0.0127994 |      7 |    21 |          35 |   1 |   2 |             7 |          1 |            0 |           1200 |               29442 |           4 |
| G(!((!(p0)) & ((X(p0)) \vert{} (F((p1) R (X(p0)))))))    | lbt < %L >%O       | ok          |         0 |  0.00127784 |      6 |    14 |          28 |   1 |   5 |             6 |          1 |            0 |           1000 |               20970 |         403 |
| !(G(!((!(p0)) & ((X(p0)) \vert{} (F((p1) R (X(p0)))))))) | ltl2tgba -s %f >%O | ok          |         0 |   0.0130485 |      3 |     5 |           6 |   1 |   3 |             0 |          0 |            1 |            600 |               12633 |           3 |
| !(G(!((!(p0)) & ((X(p0)) \vert{} (F((p1) R (X(p0)))))))) | spin -f %s >%O     | ok          |         0 |  0.00115759 |      6 |    13 |          32 |   1 |   5 |             4 |          1 |            0 |           1200 |               33622 |         204 |
| !(G(!((!(p0)) & ((X(p0)) \vert{} (F((p1) R (X(p0)))))))) | lbt < %L >%O       | ok          |         0 |  0.00120846 |     13 |    28 |          67 |   2 |  13 |             7 |          1 |            0 |           2400 |               59006 |        1604 |
| F(p0)                                                    | ltl2tgba -s %f >%O | ok          |         0 |   0.0124681 |      2 |     3 |           4 |   1 |   2 |             0 |          0 |            1 |            400 |                8264 |           2 |
| F(p0)                                                    | spin -f %s >%O     | ok          |         0 | 0.000771485 |      2 |     3 |           5 |   1 |   2 |             1 |          1 |            1 |            400 |               10323 |           2 |
| F(p0)                                                    | lbt < %L >%O       | ok          |         0 |  0.00188293 |      4 |     6 |          10 |   1 |   4 |             2 |          1 |            1 |            601 |               14487 |         203 |
| !(F(p0))                                                 | ltl2tgba -s %f >%O | ok          |         0 |   0.0125541 |      1 |     1 |           1 |   1 |   1 |             0 |          0 |            0 |            200 |                2073 |           1 |
| !(F(p0))                                                 | spin -f %s >%O     | ok          |         0 | 0.000802091 |      1 |     1 |           1 |   1 |   1 |             0 |          0 |            0 |            200 |                2073 |           1 |
| !(F(p0))                                                 | lbt < %L >%O       | ok          |         0 |  0.00127165 |      2 |     2 |           2 |   0 |   2 |             0 |          0 |            0 |            201 |                2081 |           2 |
| (X(p0)) U ((!(p0)) & (X(p0)))                            | ltl2tgba -s %f >%O | ok          |         0 |   0.0121934 |      3 |     3 |           4 |   1 |   3 |             0 |          0 |            0 |            208 |                4118 |           9 |
| (X(p0)) U ((!(p0)) & (X(p0)))                            | spin -f %s >%O     | ok          |         0 | 0.000795817 |      4 |     6 |           7 |   1 |   4 |             1 |          1 |            0 |            408 |                6159 |          10 |
| (X(p0)) U ((!(p0)) & (X(p0)))                            | lbt < %L >%O       | ok          |         0 |   0.0011121 |      6 |     7 |          10 |   1 |   6 |             1 |          1 |            0 |            484 |                7467 |          86 |
| !((X(p0)) U ((!(p0)) & (X(p0))))                         | ltl2tgba -s %f >%O | ok          |         0 |   0.0122315 |      3 |     4 |           5 |   1 |   3 |             0 |          0 |            0 |            208 |                4128 |           9 |
| !((X(p0)) U ((!(p0)) & (X(p0))))                         | spin -f %s >%O     | ok          |         0 | 0.000903485 |      4 |     8 |          10 |   1 |   3 |             2 |          1 |            0 |            800 |               20222 |         203 |
| !((X(p0)) U ((!(p0)) & (X(p0))))                         | lbt < %L >%O       | ok          |         0 |  0.00182082 |      9 |    17 |          23 |   0 |   9 |             4 |          1 |            0 |           1601 |               34445 |        1004 |

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.

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.

#+BEGIN_SRC sh :exports results :wrap SRC json
cat results.json
#+END_SRC
#+RESULTS:
#+begin_SRC json
{
  "tool": [
    "ltl2tgba -s %f >%O",
    "spin -f %s >%O",
    "lbt < %L >%O"
  ],
  "formula": [
    "G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))",
    "!(G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0))))))))",
    "F(p0)",
    "!(F(p0))",
    "(X(p0)) U ((!(p0)) & (X(p0)))",
    "!((X(p0)) U ((!(p0)) & (X(p0))))"
  ],
  "fields":  [
  "formula","tool","exit_status","exit_code","time","states","edges","transitions","acc","scc","nondet_states","nondet_aut","complete_aut","product_states","product_transitions","product_scc"
  ],
  "inputs":  [ 0, 1 ],
  "results": [
    [ 0,0,"ok",0,0.0132415,2,3,3,1,2,0,0,0,400,6276,2 ],
    [ 0,1,"ok",0,0.0129795,7,21,35,1,2,7,1,0,1200,29442,4 ],
    [ 0,2,"ok",0,0.00126994,6,14,28,1,5,6,1,0,1000,20970,403 ],
    [ 1,0,"ok",0,0.0130197,3,5,6,1,3,0,0,1,600,12633,3 ],
    [ 1,1,"ok",0,0.00115913,6,13,32,1,5,4,1,0,1200,33622,204 ],
    [ 1,2,"ok",0,0.00120161,13,28,67,2,13,7,1,0,2400,59006,1604 ],
    [ 2,0,"ok",0,0.012826,2,3,4,1,2,0,0,1,400,8264,2 ],
    [ 2,1,"ok",0,0.000675097,2,3,5,1,2,1,1,1,400,10323,2 ],
    [ 2,2,"ok",0,0.0011281,4,6,10,1,4,2,1,1,601,14487,203 ],
    [ 3,0,"ok",0,0.0122189,1,1,1,1,1,0,0,0,200,2073,1 ],
    [ 3,1,"ok",0,0.000746496,1,1,1,1,1,0,0,0,200,2073,1 ],
    [ 3,2,"ok",0,0.00124168,2,2,2,0,2,0,0,0,201,2081,2 ],
    [ 4,0,"ok",0,0.0130605,3,3,4,1,3,0,0,0,208,4118,9 ],
    [ 4,1,"ok",0,0.000788826,4,6,7,1,4,1,1,0,408,6159,10 ],
    [ 4,2,"ok",0,0.0012278,6,7,10,1,6,1,1,0,484,7467,86 ],
    [ 5,0,"ok",0,0.0124917,3,4,5,1,3,0,0,0,208,4128,9 ],
    [ 5,1,"ok",0,0.000906358,4,8,10,1,3,2,1,0,800,20222,203 ],
    [ 5,2,"ok",0,0.00129181,9,17,23,0,9,4,1,0,1601,34445,1004 ]
  ]
}
#+end_SRC

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.

JSON data can be easily processed in any language.  For instance the
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.

#+BEGIN_SRC python :results output :exports both
#!/usr/bin/python3
import json
data = json.load(open('results.json'))
datacols = range(4, len(data["fields"]))
# Index results by tool
results = { t:[] for t in range(0, len(data["tool"])) }
for l in data["results"]:
  results[l[1]].append(l)
# Average columns for each tool, and display them as a table
print("%-18s & count & %s \\\\" % ("tool", " & ".join(data["fields"][4:])))
for i in range(0, len(data["tool"])):
  c = len(results[i])
  sums = ["%6.1f" % (sum([x[j] for x in results[i]])/c)
          for j in datacols]
  print("%-18s & %3d & %s \\\\" % (data["tool"][i], c,
        " & ".join(sums)))
#+END_SRC
#+RESULTS:
: 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 &    2.3 &    3.2 &    3.8 &    1.0 &    2.3 &    0.0 &    0.0 &    0.3 &  336.0 & 6248.7 &    4.3 \\
: spin -f %s >%O     &   6 &    0.0 &    4.0 &    8.7 &   15.0 &    1.0 &    2.8 &    2.5 &    0.8 &    0.2 &  701.3 & 16973.5 &   70.7 \\
: lbt < %L >%O       &   6 &    0.0 &    6.7 &   12.3 &   23.3 &    0.8 &    6.5 &    3.3 &    0.8 &    0.2 & 1047.8 & 23076.0 &  550.3 \\

The script =bench/ltl2tgba/sum.py= is a more evolved version of the
above script that generates two kinds of LaTeX tables.

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
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.

** Description of the columns

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 columns contain an index into
the =formula= and =tool= table declared separately.

=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
equal to "=ok=".  (You may instruct =ltlcross= not to output lines with
such missing data with the option =--omit-missing=.)

=states=, =edges=, =transitions=, =acc= are size measures for the
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) [[file:concepts.org::#trans-edge][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
$\bar a b$.

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

If option =--strength= is passed to =ltlcross=, these SCCs are
also partitioned on four sets based on their strengths:
- =nonacc_scc= for non-accepting SCCs (such as states A1 and A2 in the
  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.

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 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=,
=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
weak) automata is easier.  When working with alternating automata, all
those strength-related columns will be empty, because the routines
used to compute those statistics do not yet support universal edges.

=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.

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

=complete_aut= is a Boolean indicating whether the automaton is
complete.

Columns =product_states=, =product_transitions=, and =product_scc=
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 products 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.

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.

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.

** 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
configurations, and the strings =ltl2tgba -s --small %f >%O= and
=ltl2tgba -s --deter %f >%O= appear verbatim in the output:

#+NAME: ltlcross-unnamed
#+BEGIN_SRC sh :exports code
ltlcross -f a -f Ga 'ltl2tgba -s --small %f >%O' 'ltl2tgba -s --deter %f >%O' --csv
#+END_SRC

#+BEGIN_SRC sh :results output raw :exports results :noweb yes
sed 's/"//g
s/|/\\vert{}/g
s/--/@@html:--@@/g
s/^/| /
s/$/ |/
s/,/|/g
$d
1a\
|-|
' <<EOF
<<ltlcross-unnamed()>>
EOF
#+END_SRC

#+ATTR_HTML: :class csv-table
#+RESULTS:
| formula | tool                                | exit_status | exit_code |      time | states | edges | transitions | acc | scc | nondet_states | nondet_aut | complete_aut | product_states | product_transitions | product_scc |
|---------+-------------------------------------+-------------+-----------+-----------+--------+-------+-------------+-----+-----+---------------+------------+--------------+----------------+---------------------+-------------|
| a       | ltl2tgba -s @@html:--@@small %f >%O | ok          |         0 |  0.045425 |      2 |     2 |           3 |   1 |   2 |             0 |          0 |            0 |            201 |                4144 |           2 |
| a       | ltl2tgba -s @@html:--@@deter %f >%O | ok          |         0 | 0.0452103 |      2 |     2 |           3 |   1 |   2 |             0 |          0 |            0 |            201 |                4144 |           2 |
| !(a)    | ltl2tgba -s @@html:--@@small %f >%O | ok          |         0 | 0.0475807 |      2 |     2 |           3 |   1 |   2 |             0 |          0 |            0 |            201 |                4149 |           2 |
| !(a)    | ltl2tgba -s @@html:--@@deter %f >%O | ok          |         0 | 0.0441754 |      2 |     2 |           3 |   1 |   2 |             0 |          0 |            0 |            201 |                4149 |           2 |
| G(a)    | ltl2tgba -s @@html:--@@small %f >%O | ok          |         0 | 0.0453961 |      1 |     1 |           1 |   1 |   1 |             0 |          0 |            0 |            200 |                2059 |           1 |
| G(a)    | ltl2tgba -s @@html:--@@deter %f >%O | ok          |         0 | 0.0467509 |      1 |     1 |           1 |   1 |   1 |             0 |          0 |            0 |            200 |                2059 |           1 |
| !(G(a)) | ltl2tgba -s @@html:--@@small %f >%O | ok          |         0 | 0.0459274 |      2 |     3 |           4 |   1 |   2 |             0 |          0 |            1 |            400 |                8264 |           2 |
| !(G(a)) | ltl2tgba -s @@html:--@@deter %f >%O | ok          |         0 |   0.04534 |      2 |     3 |           4 |   1 |   2 |             0 |          0 |            1 |            400 |                8264 |           2 |

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 :prologue "exec 2>&1"
genltl --and-f=1..5 |
ltlcross '{small} ltl2tgba -s --small %f >%O' \
         '{deter} ltl2tgba -s --deter %f >%O' --csv=ltlcross.csv
#+END_SRC

#+RESULTS:
#+begin_example
-:1: F(p1)
Running [P0: small]: ltl2tgba -s --small 'F(p1)' >'lcr-o0-gizrt5'
Running [P1: deter]: ltl2tgba -s --deter 'F(p1)' >'lcr-o1-E2Y2os'
Running [N0: small]: ltl2tgba -s --small '!(F(p1))' >'lcr-o0-LiBWmP'
Running [N1: deter]: ltl2tgba -s --deter '!(F(p1))' >'lcr-o1-ztzWmc'
Performing sanity checks and gathering statistics...

-:2: (F(p1)) & (F(p2))
Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2))' >'lcr-o0-DCyLpz'
Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2))' >'lcr-o1-MeEivW'
Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)))' >'lcr-o0-X5oXCj'
Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)))' >'lcr-o1-APddNG'
Performing sanity checks and gathering statistics...

-:3: (F(p1)) & (F(p2)) & (F(p3))
Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2)) & (F(p3))' >'lcr-o0-D8un13'
Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2)) & (F(p3))' >'lcr-o1-njT4hr'
Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)) & (F(p3)))' >'lcr-o0-YNmfBO'
Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)) & (F(p3)))' >'lcr-o1-2bGzWb'
Performing sanity checks and gathering statistics...

-:4: (F(p1)) & (F(p2)) & (F(p3)) & (F(p4))
Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4))' >'lcr-o0-rlrmnz'
Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4))' >'lcr-o1-KobkRW'
Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))' >'lcr-o0-5DFKnk'
Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))' >'lcr-o1-8IIcXH'
Performing sanity checks and gathering statistics...

-:5: (F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))
Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))' >'lcr-o0-h7sYE5'
Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))' >'lcr-o1-ypOBqt'
Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)))' >'lcr-o0-2rJtfR'
Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)))' >'lcr-o1-qK2p7e'
Performing sanity checks and gathering statistics...

No problem detected.
#+end_example

When =ltlcross= is running, the short name is now displayed on stderr
before the command.  Furthermore, the file =ltlcross.csv= now use the
short name in the =tool= column:

#+BEGIN_SRC sh :results output raw :exports results
sed 's/"//g
s/|/\\vert{}/g
s/--/@@html:--@@/g
s/^/| /
s/$/ |/
s/,/|/g
$d
1a\
|-|
' ltlcross.csv
#+END_SRC

#+ATTR_HTML: :class csv-table
#+RESULTS:
| formula                                            | tool  | exit_status | exit_code |      time | states | edges | transitions | acc | scc | nondet_states | nondet_aut | complete_aut | product_states | product_transitions | product_scc |
|----------------------------------------------------+-------+-------------+-----------+-----------+--------+-------+-------------+-----+-----+---------------+------------+--------------+----------------+---------------------+-------------|
| F(p1)                                              | small | ok          |         0 | 0.0143077 |      2 |     3 |           4 |   1 |   2 |             0 |          0 |            1 |            400 |                8272 |           3 |
| F(p1)                                              | deter | ok          |         0 | 0.0143547 |      2 |     3 |           4 |   1 |   2 |             0 |          0 |            1 |            400 |                8272 |           3 |
| !(F(p1))                                           | small | ok          |         0 | 0.0146721 |      1 |     1 |           1 |   1 |   1 |             0 |          0 |            0 |            200 |                2055 |           2 |
| !(F(p1))                                           | deter | ok          |         0 | 0.0145825 |      1 |     1 |           1 |   1 |   1 |             0 |          0 |            0 |            200 |                2055 |           2 |
| (F(p1)) & (F(p2))                                  | small | ok          |         0 | 0.0147275 |      4 |     9 |          16 |   1 |   4 |             0 |          0 |            1 |            798 |               16533 |           5 |
| (F(p1)) & (F(p2))                                  | deter | ok          |         0 | 0.0144936 |      4 |     9 |          16 |   1 |   4 |             0 |          0 |            1 |            798 |               16533 |           5 |
| !((F(p1)) & (F(p2)))                               | small | ok          |         0 | 0.0147704 |      3 |     5 |           7 |   1 |   3 |             0 |          0 |            0 |            598 |                7367 |           4 |
| !((F(p1)) & (F(p2)))                               | deter | ok          |         0 | 0.0146198 |      3 |     5 |           7 |   1 |   3 |             0 |          0 |            0 |            598 |                7367 |           4 |
| (F(p1)) & (F(p2)) & (F(p3))                        | small | ok          |         0 | 0.0153116 |      8 |    27 |          64 |   1 |   8 |             0 |          0 |            1 |           1587 |               33068 |          34 |
| (F(p1)) & (F(p2)) & (F(p3))                        | deter | ok          |         0 | 0.0156095 |      8 |    27 |          64 |   1 |   8 |             0 |          0 |            1 |           1587 |               33068 |          34 |
| !((F(p1)) & (F(p2)) & (F(p3)))                     | small | ok          |         0 |  0.015041 |      4 |     6 |          24 |   1 |   4 |             1 |          1 |            0 |            601 |                6171 |           4 |
| !((F(p1)) & (F(p2)) & (F(p3)))                     | deter | ok          |         0 | 0.0151199 |      7 |    19 |          37 |   1 |   7 |             0 |          0 |            0 |           1387 |               18792 |          33 |
| (F(p1)) & (F(p2)) & (F(p3)) & (F(p4))              | small | ok          |         0 | 0.0166691 |     16 |    81 |         256 |   1 |  16 |             0 |          0 |            1 |           2727 |               57786 |          74 |
| (F(p1)) & (F(p2)) & (F(p3)) & (F(p4))              | deter | ok          |         0 | 0.0161634 |     16 |    81 |         256 |   1 |  16 |             0 |          0 |            1 |           2727 |               57786 |          74 |
| !((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))           | small | ok          |         0 | 0.0167047 |      5 |     8 |          64 |   1 |   5 |             1 |          1 |            0 |            801 |                8468 |           5 |
| !((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))           | deter | ok          |         0 | 0.0160785 |     15 |    65 |         175 |   1 |  15 |             0 |          0 |            0 |           2527 |               37226 |          73 |
| (F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))    | small | ok          |         0 |  0.018485 |     32 |   243 |        1024 |   1 |  32 |             0 |          0 |            1 |           5330 |              114068 |         350 |
| (F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))    | deter | ok          |         0 | 0.0209914 |     32 |   243 |        1024 |   1 |  32 |             0 |          0 |            1 |           5330 |              114068 |         350 |
| !((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))) | small | ok          |         0 | 0.0196063 |      6 |    10 |         160 |   1 |   6 |             1 |          1 |            0 |           1000 |               10707 |           6 |

In this last example, we saved the CSV output to =ltlcross.csv= so we
can play with it in the next section.

** Working with these CSV files in R

The produced CSV should be directly readable by R's CSV input functions like
=read.csv()=, =readr::read_csv()=, or =data.table::fread()=.

#+BEGIN_SRC R
library(data.table)
dt <- fread('ltlcross.csv')
str(dt)
#+END_SRC
#+RESULTS:
#+begin_example
data.table 1.12.0  Latest news: r-datatable.com

Classes ‘data.table’ and 'data.frame':	20 obs. of  16 variables:
 $ formula            : chr  "F(p1)" "F(p1)" "!(F(p1))" "!(F(p1))" ...
 $ tool               : chr  "small" "deter" "small" "deter" ...
 $ exit_status        : chr  "ok" "ok" "ok" "ok" ...
 $ exit_code          : int  0 0 0 0 0 0 0 0 0 0 ...
 $ time               : num  0.0284 0.0283 0.0283 0.0282 0.029 ...
 $ states             : int  2 2 1 1 4 4 3 3 8 8 ...
 $ edges              : int  3 3 1 1 9 9 5 5 27 27 ...
 $ transitions        : int  4 4 1 1 16 16 7 7 64 64 ...
 $ acc                : int  1 1 1 1 1 1 1 1 1 1 ...
 $ scc                : int  2 2 1 1 4 4 3 3 8 8 ...
 $ nondet_states      : int  0 0 0 0 0 0 0 0 0 0 ...
 $ nondet_aut         : int  0 0 0 0 0 0 0 0 0 0 ...
 $ complete_aut       : int  1 1 0 0 1 1 0 0 1 1 ...
 $ product_states     : int  400 400 200 200 798 798 598 598 1587 1587 ...
 $ product_transitions: int  8272 8272 2055 2055 16533 16533 7367 7367 33068 33068 ...
 $ product_scc        : int  3 3 2 2 5 5 4 4 34 34 ...
 - attr(*, ".internal.selfref")=<externalptr>
#+end_example

Currently, the data frame shows one line per couple (formula, tool).
This makes comparing tools quite difficult, as their results are on
different lines.

A common transformation is to group the results of all tools on the
same line: using exactly one line per formula.  This is easily
achieved using =dcast()= from the =data.table= library.

#+BEGIN_SRC R
dt2 <- dcast(dt, formula ~ tool, value.var=names(dt)[-(1:2)], sep=".")
str(dt2)
#+END_SRC

#+RESULTS:
#+begin_example

Classes ‘data.table’ and 'data.frame':	10 obs. of  29 variables:
 $ formula                  : chr  "!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)))" "!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))" "!((F(p1)) & (F(p2)) & (F(p3)))" "!((F(p1)) & (F(p2)))" ...
 $ exit_status.deter        : chr  "ok" "ok" "ok" "ok" ...
 $ exit_status.small        : chr  "ok" "ok" "ok" "ok" ...
 $ exit_code.deter          : int  0 0 0 0 0 0 0 0 0 0
 $ exit_code.small          : int  0 0 0 0 0 0 0 0 0 0
 $ time.deter               : num  0.0172 0.0212 0.02 0.0191 0.0282 ...
 $ time.small               : num  0.0172 0.0221 0.0203 0.0195 0.0283 ...
 $ states.deter             : int  31 15 7 3 1 4 8 16 32 2
 $ states.small             : int  6 5 4 3 1 4 8 16 32 2
 $ edges.deter              : int  211 65 19 5 1 9 27 81 243 3
 $ edges.small              : int  10 8 6 5 1 9 27 81 243 3
 $ transitions.deter        : int  781 175 37 7 1 16 64 256 1024 4
 $ transitions.small        : int  160 64 24 7 1 16 64 256 1024 4
 $ acc.deter                : int  1 1 1 1 1 1 1 1 1 1
 $ acc.small                : int  1 1 1 1 1 1 1 1 1 1
 $ scc.deter                : int  31 15 7 3 1 4 8 16 32 2
 $ scc.small                : int  6 5 4 3 1 4 8 16 32 2
 $ nondet_states.deter      : int  0 0 0 0 0 0 0 0 0 0
 $ nondet_states.small      : int  1 1 1 0 0 0 0 0 0 0
 $ nondet_aut.deter         : int  0 0 0 0 0 0 0 0 0 0
 $ nondet_aut.small         : int  1 1 1 0 0 0 0 0 0 0
 $ complete_aut.deter       : int  0 0 0 0 0 1 1 1 1 1
 $ complete_aut.small       : int  0 0 0 0 0 1 1 1 1 1
 $ product_states.deter     : int  5130 2527 1387 598 200 798 1587 2727 5330 400
 $ product_states.small     : int  1000 801 601 598 200 798 1587 2727 5330 400
 $ product_transitions.deter: int  82897 37226 18792 7367 2055 16533 33068 57786 114068 8272
 $ product_transitions.small: int  10707 8468 6171 7367 2055 16533 33068 57786 114068 8272
 $ product_scc.deter        : int  349 73 33 4 2 5 34 74 350 3
 $ product_scc.small        : int  6 5 4 4 2 5 34 74 350 3
 - attr(*, ".internal.selfref")=<externalptr>
 - attr(*, "sorted")= chr "formula"
#+end_example

Using the above form, it is easy to compare two tools on some given
measurement, as we just need to plot two columns.  For example to
compare the number of states produced by the two configurations of
=ltl2tgba= for each formula, we just need to plot column
=dt2$state.small= against =dt2$state.deter=.

#+BEGIN_SRC R :results output graphics file :width 5 :height 5 :file ltlcross-r.svg
library(ggplot2)
ggplot(dt2, aes(x=states.small, y=states.deter)) +
   geom_abline(colour='white') + geom_point()
#+END_SRC

#+RESULTS:
[[file:ltlcross-r.svg]]


We should probably print the formulas for the cases where the two
sizes differ.

#+BEGIN_SRC R :results output graphics file :width 5 :height 5 :file ltlcross-r2.svg
ggplot(dt2, aes(x=states.small, y=states.deter)) +
   geom_abline(colour='white') + geom_point() +
   geom_text(data=subset(dt2, states.small != states.deter),
             aes(label=formula), hjust=0, nudge_x=.5)
#+END_SRC

#+RESULTS:
[[file:ltlcross-r2.svg]]

* Miscellaneous options

** =--stop-on-error=

The =--stop-on-error= option will cause =ltlcross= to abort on the
first detected error.  This includes failure to start some translator,
read its output, or failure to pass the sanity checks.  Timeouts are
allowed unless =--fail-on-time= is also given.

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).

#+BEGIN_SRC sh :exports code :eval no
randltl -n -1 --tree-size 10..25 a b c | ltlcross --stop-on-error 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O'
#+END_SRC

** =--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, unless =--fail-on-timeout= is given.

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=.

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

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

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

** =--grind=FILENAME=

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

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 exhibits the bug
   - output that last formula in =FILENAME=

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

Example:
#+BEGIN_SRC sh :prologue "exec 2>&1" :epilogue true
ltlcross -f '(G!b & (!c | F!a)) | (c & Ga & Fb)' "modella %L %O" \
  --save-bogus=bogus \
  --grind=bogus-grind
#+END_SRC
#+RESULTS:
#+begin_example
| & G ! p0 | ! p1 F ! p2 & & p1 G p2 F p0
Running [P0]: modella 'lcr-i0-Nc8B1P' 'lcr-o0-CDjvYF'
Running [N0]: modella 'lcr-i0-Io4LVv' 'lcr-o0-C482Sl'
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word:
       cycle{!p0 & !p1}

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-EmhjSb' 'lcr-o0-q1GzR1'
Running [N0]: modella 'lcr-i1-mwR1QR' 'lcr-o0-gEcuQH'
Performing sanity checks and gathering statistics...

Mutation 2/22: & G ! p0 | ! p1 F ! p2
Running [P0]: modella 'lcr-i2-4UoNQx' 'lcr-o0-W9W6Qn'
Running [N0]: modella 'lcr-i2-h5IDRd' 'lcr-o0-VDFaS3'
Performing sanity checks and gathering statistics...

Mutation 3/22: | G ! p0 & & p1 G p2 F p0
Running [P0]: modella 'lcr-i3-bkvvTT' 'lcr-o0-wMAQUJ'
Running [N0]: modella 'lcr-i3-qoYoWz' 'lcr-o0-ILwXXp'
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word:
       cycle{!p0 & !p1}

Trying to find a bogus mutation of G!b | (c & Ga & Fb)...
Mutation 1/16: t
Running [P0]: modella 'lcr-i4-avS30f' 'lcr-o0-MYCa45'
Running [N0]: modella 'lcr-i4-vJss7V' 'lcr-o0-ItCKaM'
Performing sanity checks and gathering statistics...

Mutation 2/16: G ! p0
Running [P0]: modella 'lcr-i5-TG7leC' 'lcr-o0-N6UXhs'
Running [N0]: modella 'lcr-i5-KwJJli' 'lcr-o0-kbRvp8'
Performing sanity checks and gathering statistics...

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

Mutation 4/16: | G ! p0 & p1 F p0
Running [P0]: modella 'lcr-i6-otaRtY' 'lcr-o0-bRLcyO'
Running [N0]: modella 'lcr-i6-3DMJCE' 'lcr-o0-v04gHu'
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word:
       cycle{!p0 & !p1}

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

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

Mutation 3/10: & p0 F p1
Running [P0]: modella 'lcr-i7-gKcHMk' 'lcr-o0-UPD7Ra'
Running [N0]: modella 'lcr-i7-4HUKX0' 'lcr-o0-Dpno3Q'
Performing sanity checks and gathering statistics...

Mutation 4/10: | p0 G ! p1
Running [P0]: modella 'lcr-i8-H6GH9G' 'lcr-o0-xyO1fx'
Running [N0]: modella 'lcr-i8-w3vxmn' 'lcr-o0-wgw3sd'
Performing sanity checks and gathering statistics...

Mutation 5/10: | G ! p0 F p0
Running [P0]: modella 'lcr-i9-vt8eA3' 'lcr-o0-982qHT'
Running [N0]: modella 'lcr-i9-qrbNOJ' 'lcr-o0-ceD9Vz'
Performing sanity checks and gathering statistics...

Mutation 6/10: | ! p0 & p1 F p0
Running [P0]: modella 'lcr-i10-6upQ3p' 'lcr-o0-EStxbg'
Running [N0]: modella 'lcr-i10-7nUoj6' 'lcr-o0-e4DgrW'
Performing sanity checks and gathering statistics...

Mutation 7/10: | & p1 F p0 G p0
Running [P0]: modella 'lcr-i11-ohXyzM' 'lcr-o0-bozRHC'
Running [N0]: modella 'lcr-i11-6wYkQs' 'lcr-o0-TCxOYi'
Performing sanity checks and gathering statistics...

Mutation 8/10: | & p0 p1 G ! p0
Running [P0]: modella 'lcr-i12-51Vd88' 'lcr-o0-uWKDhZ'
Running [N0]: modella 'lcr-i12-0OkfrP' 'lcr-o0-aEdRAF'
Performing sanity checks and gathering statistics...

Mutation 9/10: | G ! p0 & p0 F p0
Running [P0]: modella 'lcr-i13-vy57Kv' 'lcr-o0-lcfpVl'
Running [N0]: modella 'lcr-i13-D7SQ5b' 'lcr-o0-k8Hig2'
Performing sanity checks and gathering statistics...
error: P0*N0 is nonempty; both automata accept the infinite word:
       cycle{!p0}

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

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

Mutation 3/7: & p0 F p0
Running [P0]: modella 'lcr-i14-AvSorS' 'lcr-o0-AZkvCI'
Running [N0]: modella 'lcr-i14-Hd7LNy' 'lcr-o0-pM82Yo'
Performing sanity checks and gathering statistics...

Mutation 4/7: | p0 G ! p0
Running [P0]: modella 'lcr-i15-tygKaf' 'lcr-o0-YHFrm5'
Running [N0]: modella 'lcr-i15-GL9iyV' 'lcr-o0-riOaKL'
Performing sanity checks and gathering statistics...

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

Mutation 6/7: | ! p0 & p0 F p0
Running [P0]: modella 'lcr-i16-M0RHWB' 'lcr-o0-iVlf9r'
Running [N0]: modella 'lcr-i16-WD4Xli' 'lcr-o0-Ez6Gy8'
Performing sanity checks and gathering statistics...

Mutation 7/7: | G p0 & p0 F p0
Running [P0]: modella 'lcr-i17-F1BLLY' 'lcr-o0-Z9nQYO'
Running [N0]: modella 'lcr-i17-efo5bF' 'lcr-o0-fFzkpv'
Performing sanity checks and gathering statistics...

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

error: some error was detected during the above runs.
       Check file bogus for problematic formulas.
#+end_example

#+BEGIN_SRC sh
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
cat bogus-grind
#+END_SRC

#+RESULTS:
: G!c | (c & Fc)

** =--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.

** =--verbose=
   :PROPERTIES:
   :CUSTOM_ID: verbose
   :END:

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=
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.

#+BEGIN_SRC sh :prologue "export SPOT_HOA_TOLERANT=1; exec 2>&1"
ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --verbose
#+END_SRC

#+RESULTS:
#+begin_example
F(G(a))
Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-ltzvEc'
Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-dqnX28'
Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-wmIXr5'
Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-yJesT1'
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 complete
Performing sanity checks and gathering statistics...
info: getting rid of universal edges...
info:   N1	(3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets)
info: complementing automata...
info:   P0	(2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,1 sets)	Comp(P0)
info:   N0	(1 st.,2 ed.,1 sets) -> (1 st.,2 ed.,1 sets)	Comp(N0)
info:   P1	(2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,1 sets)	Comp(P1)
info:   N1	(2 st.,4 ed.,1 sets) -> (2 st.,4 ed.,1 sets)	Comp(N1)
info: getting rid of any Fin acceptance...
info:	Comp(N0)	(1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
info:	     P1 	(2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 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)
info: cross_checks and consistency_checks unnecessary

No problem detected.
#+end_example

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

Some basic information about the collected automata is 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
automata.

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 to actually compare 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 reduce =ltlcross='s ability to determinize
automata for complementation, the procedure
can look slightly more complex:

#+BEGIN_SRC sh :prologue "export SPOT_HOA_TOLERANT=1; exec 2>&1"
ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --determinize-max-states=1 --verbose
#+END_SRC

#+RESULTS:
#+begin_example
F(G(a))
Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-HHyVWR'
Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-scKnIH'
Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-6Wloux'
Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-MQ7Rin'
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 complete
Performing sanity checks and gathering statistics...
info: getting rid of universal edges...
info:   N1	(3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets)
info: complementing automata...
info:   P0	not complemented (more than 1 states required)
info:   N0	(1 st.,2 ed.,1 sets) -> (1 st.,2 ed.,1 sets)	Comp(N0)
info:   P1	not complemented (more than 1 states required)
info:   N1	(2 st.,4 ed.,1 sets) -> (2 st.,4 ed.,1 sets)	Comp(N1)
info: getting rid of any Fin acceptance...
info:	Comp(N0)	(1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
info:	     P1 	(2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
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: complements were not computed for some automata
info: continuing with cross_checks and consistency_checks
info: building state-space #1/1 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.

Here is the same example, if we declare that =ltl3ba= is a reference
implementation that should not be checked, and we just want to check
the output of =ltl2tgba= against this reference.  See how the number
of tests performed has been reduced.

#+BEGIN_SRC sh  :prologue "export SPOT_HOA_TOLERANT=1; exec 2>&1"
ltlcross -f 'FGa' ltl2tgba --reference 'ltl3ba -H1' --verbose
#+END_SRC

#+RESULTS:
#+begin_example
F(G(a))
Running [P0]: ltl3ba -H1 -f '<>([](a))'>'lcr-o0-bh9PHg'
Running [P1]: ltl2tgba -H 'F(G(a))'>'lcr-o1-LvvYEm'
Running [N0]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o0-bcUDEs'
Running [N1]: ltl2tgba -H '!(F(G(a)))'>'lcr-o1-Pw1REy'
info: collected automata:
info:   P0	(2 st.,3 ed.,1 sets)
info:   N0	(3 st.,5 ed.,1 sets) univ-edges complete
info:   P1	(2 st.,3 ed.,1 sets)
info:   N1	(1 st.,2 ed.,1 sets) deterministic complete
Performing sanity checks and gathering statistics...
info: getting rid of universal edges...
info:   N0	(3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets)
info: complementing automata...
info:   P1	(2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,1 sets)	Comp(P1)
info:   N1	(1 st.,2 ed.,1 sets) -> (1 st.,2 ed.,1 sets)	Comp(N1)
info: getting rid of any Fin acceptance...
info:	     P0 	(2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
info:	Comp(N1)	(1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets)
info: P0 and N0 assumed correct and used as references
info: check_empty P0*N1
info: check_empty P1*N0
info: check_empty P1*N1
info: check_empty Comp(N1)*Comp(P1)
info: cross_checks and consistency_checks unnecessary

No problem detected.
#+end_example

* Running =ltlcross= in parallel
   :PROPERTIES:
   :CUSTOM_ID: parallel
   :END:

  The =ltlcross= command itself has no built-in support for
  parallelization (patches welcome).  However, its interface makes it
  rather easy to parallelize =ltlcross= runs with third-party tools
  such as:

  - =xargs= from [[https://www.gnu.org/software/findutils/][GNU findutils]].  The [[https://www.gnu.org/software/findutils/manual/html_node/find_html/Controlling-Parallelism.html#Controlling-Parallelism][=-P n= option]] is a GNU extension
    to specify that n commands should be run in parallel.

    For instance the following command tests =ltl2tgba= and =ltl3ba=
    against 1000 formulas, running 8 formulas in parallel.

    #+begin_src sh :exports code
      randltl -n-1 3 | ltlfilt --relabel=pnn --unique -n1000 |
          xargs -P8 -I'{}' ltlcross -q --save-bogus='>>bugs.ltl' ltl2tgba ltl3ba -f '{}'
    #+end_src

    #+RESULTS:

    The above pipeline uses =randltl= to generate an infinite number
    of LTL formulas (=-n-1=) over three atomic propositions.  Those
    formules are then relabeled with =ltlfilt= (so that =a U b= and =b
    U a= both get mapped to the same =p0 U p1=) and filtered for
    duplicates (=--unique=).  This first 1000 formulas (=-n1000=) are
    then passed on to =xargs=. The command =xargs -I'{}' ltlcross...=
    takes each line of input, and executes the command =ltlcross...=
    with ={}= replaced by the input line.  The option =-P8= does this
    with 8 processes in parallel.  Here =ltlcross= is called with
    option =-q= to silence most its regular output as the 8 instances
    of =ltlcross= would be otherwise writing to the same terminal.
    With =-q=, only errors are displayed.  Additionally, =--save-bogus=
    is used to keep track of all formulas causing errors.  The =>>bugs.ltl=
    syntax means to open =bugs.ltl= in append mode, so that =bugs.ltl= does
    not get overwritten each time a new =ltlcross= instance finds a bug.

  - [[https://www.gnu.org/software/parallel/][GNU parallel]] or [[https://joeyh.name/code/moreutils/][moreutils's parallel]] can also be used similarly.

  - =make -j n= is another option: first convert the list of formulas
    into a =Makefile= that calls =ltlcross= for each of them.

    For instance here is how to build a makefile called =ltlcross.mk=
    testing =ltl2tgba= and =ltl3ba= against all formulas produced by
    =genltl --eh=, and gathering statistics from all runs in =all.csv=.

    #+NAME: ltlcross.mk
    #+begin_src sh :epilogue "cat ltlcross.mk" :wrap src makefile
    echo 'LTLCROSS=ltlcross -q ltl2tgba ltl3ba' > ltlcross.mk
    echo "ALL= $(echo $(genltl --eh --format="%F%L.csv"))" >> ltlcross.mk
    echo "all.csv: \$(ALL); cat \$(ALL) | sed -e 1n -e '/^\"formula\"/d' > \$@" >>ltlcross.mk
    genltl --eh --format="%F%L.csv:; \$(LTLCROSS) --csv=\$@ -f '%f'" >>ltlcross.mk
    #+end_src

    This creates =ltlcross.mk=:
    #+RESULTS: ltlcross.mk
    #+begin_src makefile
    LTLCROSS=ltlcross -q ltl2tgba ltl3ba
    ALL= eh-patterns1.csv eh-patterns2.csv eh-patterns3.csv eh-patterns4.csv eh-patterns5.csv eh-patterns6.csv eh-patterns7.csv eh-patterns8.csv eh-patterns9.csv eh-patterns10.csv eh-patterns11.csv eh-patterns12.csv
    all.csv: $(ALL); cat $(ALL) | sed -e 1n -e '/^"formula"/d' > $@
    eh-patterns1.csv:; $(LTLCROSS) --csv=$@ -f 'p0 U (p1 & Gp2)'
    eh-patterns2.csv:; $(LTLCROSS) --csv=$@ -f 'p0 U (p1 & X(p2 U p3))'
    eh-patterns3.csv:; $(LTLCROSS) --csv=$@ -f 'p0 U (p1 & X(p2 & F(p3 & XF(p4 & XF(p5 & XFp6)))))'
    eh-patterns4.csv:; $(LTLCROSS) --csv=$@ -f 'F(p0 & XGp1)'
    eh-patterns5.csv:; $(LTLCROSS) --csv=$@ -f 'F(p0 & X(p1 & XFp2))'
    eh-patterns6.csv:; $(LTLCROSS) --csv=$@ -f 'F(p0 & X(p1 U p2))'
    eh-patterns7.csv:; $(LTLCROSS) --csv=$@ -f 'FGp0 | GFp1'
    eh-patterns8.csv:; $(LTLCROSS) --csv=$@ -f 'G(p0 -> (p1 U p2))'
    eh-patterns9.csv:; $(LTLCROSS) --csv=$@ -f 'G(p0 & XF(p1 & XF(p2 & XFp3)))'
    eh-patterns10.csv:; $(LTLCROSS) --csv=$@ -f 'GFp0 & GFp1 & GFp2 & GFp3 & GFp4'
    eh-patterns11.csv:; $(LTLCROSS) --csv=$@ -f '(p0 U (p1 U p2)) | (p1 U (p2 U p0)) | (p2 U (p0 U p1))'
    eh-patterns12.csv:; $(LTLCROSS) --csv=$@ -f 'G(p0 -> (p1 U (Gp2 | Gp3)))'
    #+end_src

    This makefile could be executed for instance with =make -f
    ltlcross.mk -j 4=, where =-j 4= specifies that 4 processes can be
    executed in parallel.  Using different =csv= files for each
    process avoids potential race conditions that could occur if each
    instance of =ltlcross= was appending to the same file.  The =sed=
    command used while merging all =csv= files keeps the first header
    line (=1n=) while removing all subsequent ones (=/"formula"/d=).

#+BEGIN_SRC sh :results silent :exports results
rm -f results.csv results.json ltlcross.csv bogus-grind bogus ltlcross.mk
#+END_SRC

#  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 utf html args SCCs nonaccepting
#  LocalWords:  dstar's lcr hvzgTC iYh nzjV rqfB OUNHEn qzVvdx eUfHTG
#  LocalWords:  infixed LBT's dstar parsers MWei nba streett mv dst
#  LocalWords:  Rabinizer MWRei rabinizer dra tela COMMANDFMT delag
#  LocalWords:  da dgra dpa ldba untrusted ne UWmAOs UALknk JlcYb Yqa
#  LocalWords:  vcaV jwQ eDh rE kjYd yDowOn zxU yf Dj NcC xUjPQ fD yl
#  LocalWords:  OCI kdqA ZsPBds IxFTSb hRJ OAV yTDisN CUj lF xWC fx
#  LocalWords:  aHi ATTR nondet aut ok degeneralized lor nonacc noweb
#  LocalWords:  EOF genltl gizrt os LiBWmP ztzWmc DCyLpz MeEivW oXCj
#  LocalWords:  APddNG un njT YNmfBO bGzWb rlrmnz KobkRW DFKnk IIcXH
#  LocalWords:  sYE ypOBqt rJtfR qK stderr dt str SRC datatable chr
#  LocalWords:  ok num acc scc nondet aut attr dcast SRC dt sep str
#  LocalWords:  chr ok num acc scc nondet aut selfref attr chr tgba
#  LocalWords:  dt SRC ltlcross svg ggplot aes abline colour hjust ba
#  LocalWords:  randltl eval lbtt ltlgrind OTHERFILENAME Fb modella
#  LocalWords:  lcr Nc CDjvYF LVv Sl EmhjSb GzR mwR gEcuQH UoNQx Qn
#  LocalWords:  IDRd VDFaS bkvvTT wMAQUJ qoYoWz ILwXXp avS MYCa vJss
#  LocalWords:  ItCKaM TG leC UXhs KwJJli kbRvp dups otaRtY bRLcyO GH
#  LocalWords:  DMJCE gHu gKcHMk UPD HUKX Dpno xyO fx vxmn wgw sd vt
#  LocalWords:  eA qHT qrbNOJ ceD Vz upQ EStxbg nUoj DgrW ohXyzM Vd
#  LocalWords:  bozRHC wYkQs TCxOYi uWKDhZ OkfrP aEdRAF vy Kv lcfpVl
#  LocalWords:  Hig Fc AvSorS AZkvCI Hd LNy pM tygKaf YHFrm GL iyV WD
#  LocalWords:  riOaKL RHWB iVlf Xli Ez Gy BLLY nQYO efo bF fFzkpv MQ
#  LocalWords:  FGa ltzvEc dqnX wmIXr yJesT HHyVWR scKnIH Wloux Rin
#  LocalWords:  SCCs bh PHg LvvYEm bcUDEs Pw REy parallelization src
#  LocalWords:  parallelize xargs findutils ltlfilt pnn formules mk
#  LocalWords:  moreutils's Makefile makefile fread externalptr Gp XF
#  LocalWords:  XFp XGp FGp GFp
