ltlcross.org

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

=ltlcross= is a tool for cross-comparing the output of LTL-to-Büchi
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 are:
  - support for PSL formulas in addition to LTL
  - 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
  - statistics output in a format that can be more easily be post-processed,
  - more precise time measurement (LBTT was only precise to
    1/100 of a second, reporting most times as "0.00s"),
  - support for any type of acceptance condition,
  - additional intersection checks with the complement of any
    deterministic automaton produced by a translator.

Although =ltlcross= performs the same 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 conterexample, but you will be on your own to
investigate and fix them.

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.
  - Perform sanity checks between all these automata to detect any problem.
  - 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 no =-f= or =-F= option is given.

* 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
ltlcross --help | sed -n '/character sequences:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
:   %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,%D                      the automaton is output as either (%O) HOA/never
:                              claim/LBTT, or (%D) in 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 :results verbatim :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 :results verbatim :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

=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=]], [[http://sourceforge.net/projects/ltl3ba/][=ltl3ba=]], or =ltl2tgba --spin=.  These
    should be indicated using =%O=.  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=.  These should also be indicated using =%O=.
  - Non-alternating automata in [[file:http://adl.github.io/hoaf/][the HOA format]] with an acceptance
    condition that is is generalized-Büchi or inferior.  These
    should also be indicated using =%O=.
  - [[http://www.ltl2dstar.de/docs/ltl2dstar.html][=ltl2dstar='s format]], which supports deterministic Rabin or Streett
    automata.  After =ltlcross= reads such input, it immediately
    converts it into a Büchi automaton.  Rabin automata are converted
    to (degeneralized) Büchi automata and the conversion will preserve
    the determinism anytime a deterministic Büchi automaton exists for
    that property (this determinism is good for the complemented
    intersection check discussed below).  Streett automata are
    converted to non-deterministic TGBA, where generalized acceptance
    conditions are used to reduce the size of the automaton you would
    get by the classical conversion from Streett to Büchi.
    This kind of output (Rabin or Streett) should be indicated with =%D=.

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 %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 %s >%O=' (smaller output, Büchi automaton)
  - '=ltl2tgba -s -D %s >%O=' (more deterministic output, Büchi automaton)
  - '=ltl2tgba -H %s >%O=' (smaller output, TGBA)
  - '=ltl2tgba -H -D %s >%O=' (more deterministic output, TGBA)
  - '=lbt <%L >%O='
  - '~ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD
    --output-format=hoa %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 %L %O~' deterministic Streett output in HOA,
    as supported since version 0.5.2 of =ltl2dstar=.
  - '=ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD %L %D=' (Rabin
    output in DSTAR format, as supported in older version of
    =ltl2dstar=.  Note that when reading this format, =ltl2dstar=
    automatically converts the Rabin automaton into a Büchi automaton
    for historical reason.)
  - '=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 %F %D; mv %D.dst %D=' (Rabinizer
    uses the last =%D= argument as a prefix to which it always append =.dst=,
    so we have to rename =%D.dst= as =%D= so that =ltlcross= can find the file)
  - '=java -jar Rabinizer3.jar -hoa -formula %f; mv output.hoa %O=' (Rabinizer *3*
    always output in =output.hoa= when the formula comes from the command line.)
  - '=ltl3dra -f %s >%O=' (The HOA format is the default for =ltl2dra=.)

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.

  lbt          <%L>%O
  ltl2ba       -f %s>%O
  ltl2dstar    --output-format=hoa %L %O
  ltl2tgba     -H %f>%O
  ltl3ba       -f %s>%O
  ltl3dra      -f %f>%O
  modella      %L %O
  spin         -f %s>%O

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
  '{DRA} ~/mytools/ltl2dstar-0.5.2 --output-format=hoa %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' ...=

* 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 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 a b | ltlfilt --remove-wm |
ltlcross --csv=results.csv --json=results.json \
         'ltl2tgba -s %f >%O' \
         'spin -f %s >%O' \
         'lbt < %L >%O' --csv=results.csv 2>&1
#+END_SRC
#+RESULTS:
#+begin_example
-:1: (0)
Running [P0]: ltl2tgba -s '(0)' >'lcr-o0-HUCuLR'
Running [P1]: spin -f '(false)' >'lcr-o1-OEpUm3'
Running [P2]: lbt < 'lcr-i0-KCU1eG' >'lcr-o2-jzzGYe'
Running [N0]: ltl2tgba -s '(1)' >'lcr-o0-ppQ4cC'
Running [N1]: spin -f '(true)' >'lcr-o1-0OiIVN'
Running [N2]: lbt < 'lcr-i0-KZxSAq' >'lcr-o2-EcBREZ'
Performing sanity checks and gathering statistics...

-:2: (!((1) U (F(!(p0)))))
Running [P0]: ltl2tgba -s '(!((1) U (F(!(p0)))))' >'lcr-o0-3dS2an'
Running [P1]: spin -f '(!((true) U (<>(!(p0)))))' >'lcr-o1-SJop2y'
Running [P2]: lbt < 'lcr-i1-jbIWpb' >'lcr-o2-hBEeUK'
Running [N0]: ltl2tgba -s '(1) U (F(!(p0)))' >'lcr-o0-Hku1E8'
Running [N1]: spin -f '(true) U (<>(!(p0)))' >'lcr-o1-8sslEk'
Running [N2]: lbt < 'lcr-i1-M3MCMW' >'lcr-o2-f6kaEw'
Performing sanity checks and gathering statistics...

-:3: (1) U ((G(p0)) | (F(p1)))
Running [P0]: ltl2tgba -s '(1) U ((G(p0)) | (F(p1)))' >'lcr-o0-0Mu0SU'
Running [P1]: spin -f '(true) U (([](p0)) || (<>(p1)))' >'lcr-o1-Vhkn66'
Running [P2]: lbt < 'lcr-i2-tc2zLI' >'lcr-o2-iOQjkj'
Running [N0]: ltl2tgba -s '(!((1) U ((G(p0)) | (F(p1)))))' >'lcr-o0-eveiNH'
Running [N1]: spin -f '(!((true) U (([](p0)) || (<>(p1)))))' >'lcr-o1-7hg46T'
Running [N2]: lbt < 'lcr-i2-XwPNyv' >'lcr-o2-GuXns6'
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 verbatim :exports results
cat results.csv
#+END_SRC
#+RESULTS:
#+begin_example
"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","ambiguous_aut","product_states","product_transitions","product_scc"
"(0)","ltl2tgba -s %f >%O","ok",0,0.0329577,1,1,0,1,1,1,0,0,0,0,0,1,0,0,1,1,0,1
"(0)","spin -f %s >%O","ok",0,0.00219235,2,2,1,1,2,1,1,0,0,0,0,1,0,0,1,1,0,1
"(0)","lbt < %L >%O","ok",0,0.0026317,1,0,0,0,1,1,0,0,0,0,0,1,0,0,1,1,0,1
"(1)","ltl2tgba -s %f >%O","ok",0,0.0429693,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,200,4062,1
"(1)","spin -f %s >%O","ok",0,0.00305038,2,2,2,1,2,1,1,0,0,0,0,1,0,0,0,201,4083,2
"(1)","lbt < %L >%O","ok",0,0.00404441,3,3,3,0,3,2,1,0,0,0,0,1,0,0,0,222,4510,23
"(!((1) U (F(!(p0)))))","ltl2tgba -s %f >%O","ok",0,0.041792,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,200,2098,1
"(!((1) U (F(!(p0)))))","spin -f %s >%O","ok",0,0.00278539,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,200,2098,1
"(!((1) U (F(!(p0)))))","lbt < %L >%O","ok",0,0.00347608,2,2,2,0,2,1,0,1,0,0,0,0,1,0,0,201,2110,2
"(1) U (F(!(p0)))","ltl2tgba -s %f >%O","ok",0,0.046218,2,3,4,1,2,1,1,0,0,0,0,1,0,0,0,400,8356,2
"(1) U (F(!(p0)))","spin -f %s >%O","ok",0,0.00292604,2,3,5,1,2,1,1,0,0,1,1,1,0,0,1,400,10436,2
"(1) U (F(!(p0)))","lbt < %L >%O","ok",0,0.00420574,7,13,22,2,7,6,1,0,0,4,1,1,0,0,1,1201,35534,604
"(1) U ((G(p0)) | (F(p1)))","ltl2tgba -s %f >%O","ok",0,0.0395735,3,5,11,1,3,1,1,1,0,1,1,0,1,0,1,600,11104,4
"(1) U ((G(p0)) | (F(p1)))","spin -f %s >%O","ok",0,0.00349485,4,8,24,1,4,2,1,1,0,2,1,0,1,0,1,800,24284,4
"(1) U ((G(p0)) | (F(p1)))","lbt < %L >%O","ok",0,0.00316141,9,17,52,2,9,7,1,1,0,4,1,0,1,0,1,1601,40505,805
"(!((1) U ((G(p0)) | (F(p1)))))","ltl2tgba -s %f >%O","ok",0,0.0352196,2,4,4,1,1,0,0,0,1,0,0,0,0,1,0,398,3919,3
"(!((1) U ((G(p0)) | (F(p1)))))","spin -f %s >%O","ok",0,0.00995492,6,18,17,1,4,2,0,1,1,5,1,0,0,1,1,596,8720,4
"(!((1) U ((G(p0)) | (F(p1)))))","lbt < %L >%O","ok",0,0.0035013,3,6,9,1,2,1,0,0,1,3,1,0,0,1,1,399,5837,4
#+end_example

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 :results verbatim :exports results
cat results.json
#+END_SRC
#+RESULTS:
#+begin_example
{
  "tool": [
    "ltl2tgba -s %f >%O",
    "spin -f %s >%O",
    "lbt < %L >%O"
  ],
  "formula": [
    "(0)",
    "(1)",
    "(!((1) U (F(!(p0)))))",
    "(1) U (F(!(p0)))",
    "(1) U ((G(p0)) | (F(p1)))",
    "(!((1) U ((G(p0)) | (F(p1)))))"
  ],
  "fields":  [
  "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","ambiguous_aut","product_states","product_transitions","product_scc"
  ],
  "inputs":  [ 0, 1 ],
  "results": [
    [ 0,0,"ok",0,0.0329577,1,1,0,1,1,1,0,0,0,0,0,1,0,0,1,1,0,1 ],
    [ 0,1,"ok",0,0.00219235,2,2,1,1,2,1,1,0,0,0,0,1,0,0,1,1,0,1 ],
    [ 0,2,"ok",0,0.0026317,1,0,0,0,1,1,0,0,0,0,0,1,0,0,1,1,0,1 ],
    [ 1,0,"ok",0,0.0429693,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,200,4062,1 ],
    [ 1,1,"ok",0,0.00305038,2,2,2,1,2,1,1,0,0,0,0,1,0,0,0,201,4083,2 ],
    [ 1,2,"ok",0,0.00404441,3,3,3,0,3,2,1,0,0,0,0,1,0,0,0,222,4510,23 ],
    [ 2,0,"ok",0,0.041792,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,200,2098,1 ],
    [ 2,1,"ok",0,0.00278539,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,200,2098,1 ],
    [ 2,2,"ok",0,0.00347608,2,2,2,0,2,1,0,1,0,0,0,0,1,0,0,201,2110,2 ],
    [ 3,0,"ok",0,0.046218,2,3,4,1,2,1,1,0,0,0,0,1,0,0,0,400,8356,2 ],
    [ 3,1,"ok",0,0.00292604,2,3,5,1,2,1,1,0,0,1,1,1,0,0,1,400,10436,2 ],
    [ 3,2,"ok",0,0.00420574,7,13,22,2,7,6,1,0,0,4,1,1,0,0,1,1201,35534,604 ],
    [ 4,0,"ok",0,0.0395735,3,5,11,1,3,1,1,1,0,1,1,0,1,0,1,600,11104,4 ],
    [ 4,1,"ok",0,0.00349485,4,8,24,1,4,2,1,1,0,2,1,0,1,0,1,800,24284,4 ],
    [ 4,2,"ok",0,0.00316141,9,17,52,2,9,7,1,1,0,4,1,0,1,0,1,1601,40505,805 ],
    [ 5,0,"ok",0,0.0352196,2,4,4,1,1,0,0,0,1,0,0,0,0,1,0,398,3919,3 ],
    [ 5,1,"ok",0,0.00995492,6,18,17,1,4,2,0,1,1,5,1,0,0,1,1,596,8720,4 ],
    [ 5,2,"ok",0,0.0035013,3,6,9,1,2,1,0,0,1,3,1,0,0,1,1,399,5837,4 ]
  ]
}
#+end_example

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 & nonacc_scc & terminal_scc & weak_scc & strong_scc & nondet_states & nondet_aut & terminal_aut & weak_aut & strong_aut & ambiguous_aut & product_states & product_transitions & product_scc \\
: ltl2tgba -s %f >%O &   6 &    0.0 &    1.7 &    2.5 &    3.5 &    1.0 &    1.5 &    0.5 &    0.5 &    0.3 &    0.2 &    0.2 &    0.2 &    0.5 &    0.3 &    0.2 &    0.3 &  299.8 & 4923.2 &    2.0 \\
: spin -f %s >%O     &   6 &    0.0 &    2.8 &    5.7 &    8.3 &    1.0 &    2.5 &    1.2 &    0.7 &    0.5 &    0.2 &    1.3 &    0.5 &    0.5 &    0.3 &    0.2 &    0.7 &  366.3 & 8270.2 &    2.3 \\
: lbt < %L >%O       &   6 &    0.0 &    4.2 &    6.8 &   14.7 &    0.8 &    4.0 &    3.0 &    0.5 &    0.3 &    0.2 &    1.8 &    0.5 &    0.5 &    0.3 &    0.2 &    0.7 &  604.2 & 14749.3 &  239.8 \\

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

=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.

=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=.)

The columns =in_type=, =in_states=, =in_edges=, =in_transitions=,
=in_acc=, and =in_scc= are only output if one of the translator
produces Rabin or Streett automata in =ltl2dstar='s format (i.e., if
=%D= is used to specify one output filename for any translator).  In
that case these columns give the type (DRA or DSA) of the produced
automaton, as well as its size (states, edges, transitions, number of
acceptance pairs, and number of SCCs).  This input automaton is then
converted by =ltlcross= into a TGBA before being checked the result of
other translators, and all the following columns are measures of that
converted automaton.  This conversion to TGBA is historical.  When the
automata are produced in the HOA format (i.e., not in =ltl2dstar='s
format) then no conversion occur, and =ltlcross= works directly with
the given acceptance condition.

=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) 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]]


=scc= counts the number of strongly-connected components in the automaton.  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 SCCs that consist of a single state with an
  accepting self-loop labeled by true (such as states B1 and B2
  in the previous picture)
- =weak_scc= for non-terminal SCCs in which all cycles are accepting
- and =strong_scc= for accepting SCCs in which some cycles are not accepting.

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=,
=strong_aut= columns with Boolean values.  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.

=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.

=ambiguous_aut= is 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.

=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 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.

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:

#+BEGIN_SRC sh :results verbatim :exports both
ltlcross -f a -f Ga 'ltl2tgba -s --small %f >%O' 'ltl2tgba -s --deter %f >%O' --csv
#+END_SRC
#+RESULTS:
: "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

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
ltlcross -f a -f Ga '{small} ltl2tgba -s --small %f >%O' '{deter} ltl2tgba -s --deter %f >%O' --csv
#+END_SRC
#+RESULTS:
: "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

* Detecting problems

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 $P_j$ are
    deterministic, we =ltlcross= builds their complements, $Comp(P_i)$
    and $Comp(P_j)$, and then ensures that $Comp(P_i)\otimes
    Comp(P_j)$ 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.

    This check is only done for deterministic automata, because
    complementation is cheap is that case.  When validating a
    translator with =ltlcross=, 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@-sD %L %D=' is more expansive, but
    it will produce a deterministic Büchi automaton whenever one
    exists.

  - 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,P3,P4,P5,P6,P7,P8,P9} disagree with {P1} when evaluating the state-space

    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.

  - 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.

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.

* Miscellaneous options

** =--stop-on-error=

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
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).

#+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.

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 exhibit 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 :exports code :results verbatim
ltlcross -f '(G!b & (!c | F!a)) | (c & Ga & Fb)' "modella %L %O" \
  --save-bogus=bogus \
  --grind=bogus-grind
#+END_SRC
#+BEGIN_SRC sh :exports results :results verbatim
ltlcross -f '(G!b & (!c | F!a)) | (c & Ga & Fb)' "modella %L %O" \
  --save-bogus=bogus --grind=bogus-grind 2>&1
true
#+END_SRC
#+RESULTS:
#+begin_example
| & G ! p0 | ! p1 F ! p2 & & p1 G p2 F p0
Running [P0]: modella 'lcr-i0-1XJw2X' 'lcr-o0-p2nbUW'
Running [N0]: modella 'lcr-i0-lSPMMV' 'lcr-o0-3xEoFU'
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-i1-3PI0CT' 'lcr-o0-GxBDAS'
Running [N0]: modella 'lcr-i1-eOijzR' 'lcr-o0-wkIZxQ'
Performing sanity checks and gathering statistics...

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'
Performing sanity checks and gathering statistics...

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'
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-R8PbkI' 'lcr-o0-6ts7wH'
Running [N0]: modella 'lcr-i4-cv7EKG' 'lcr-o0-vVmdYF'
Performing sanity checks and gathering statistics...

Mutation 2/16: G ! p0
Running [P0]: modella 'lcr-i5-ZO3HcF' 'lcr-o0-UiydrE'
Running [N0]: modella 'lcr-i5-gOseGD' 'lcr-o0-CL4fVC'
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-QTDjtB' 'lcr-o0-tFpmKA'
Running [N0]: modella 'lcr-i6-1lnX1z' 'lcr-o0-TWWyjz'
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-CJTWjx' 'lcr-o0-DKbNEw'
Running [N0]: modella 'lcr-i7-tSts1v' 'lcr-o0-CGbWov'
Performing sanity checks and gathering statistics...

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

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'
Performing sanity checks and gathering statistics...

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'
Performing sanity checks and gathering statistics...

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'
Performing sanity checks and gathering statistics...

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'
Performing sanity checks and gathering statistics...

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'
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-KsKPgj' 'lcr-o0-Qo3UXi'
Running [N0]: modella 'lcr-i14-FVPvFi' 'lcr-o0-Zh06mi'
Performing sanity checks and gathering statistics...

Mutation 4/7: | p0 G ! p0
Running [P0]: modella 'lcr-i15-JB045h' 'lcr-o0-EIL4Oh'
Running [N0]: modella 'lcr-i15-sYY8yh' 'lcr-o0-KVOdjh'
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-hoqdSg' 'lcr-o0-xghJEg'
Running [N0]: modella 'lcr-i16-5JvOrg' 'lcr-o0-SUdVeg'
Performing sanity checks and gathering statistics...

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'
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 :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)

** =--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