# ltlcross

ltlcross is a tool for cross-comparing the output of LTL-to-automata translators. It is actually a Spot-based clone of 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 conterexample, 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 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:

%%                         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).

ltlcross -f 'GFa' -f 'X(a U b)' 'ltl2tgba -s %s >%O' 'spin -f %s >%O'


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.

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:

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 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.

ltlcross --list-shorthands

If a COMMANDFMT does not use any %-sequence, and starts with one of
the following words, then the string on the right is appended.

delag        %f>%O
lbt          <%L>%O
ltl2ba       -f %s>%O
ltl2da       %f>%O
ltl2dgra     %f>%O
ltl2dpa      %f>%O
ltl2dra      %f>%O
ltl2dstar    --output-format=hoa %[MW]L %O
ltl2ldba     %f>%O
ltl2na       %f>%O
ltl2nba      %f>%O
ltl2ngba     %f>%O
ltl2tgba     -H %f>%O
ltl3ba       -f %s>%O
ltl3dra      -f %s>%O
ltl3hoa      -f %f>%O
ltl3tela     -f %f>%O
modella      %[MWei^]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 be changed into
'{DRA} ~/mytools/ltl2dstar-0.5.2 --output-format=hoa %[MW]L %O'


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 option, with and without some --reference translators.

## 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 $$N_i$$ 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 to include 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 less 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 less words than what it should. If the input automata are deterministic or the --determinize option is used, this test is redundant and can be disabled. (In fact, the --determinize option implies option --product=0 to do so.)

The above checks are similar to those that are performed by 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).

randltl -n 3 a b |
ltlfilt --remove-wm |
ltlcross --csv=results.csv \
'ltl2tgba -s %f >%O' \
'spin -f %s >%O' \
'lbt < %L >%O'

-:1: 0
Running [P0]: ltl2tgba -s '0' >'lcr-o0-MWRc3C'
Running [P1]: spin -f 'false' >'lcr-o1-IHykCs'
Running [P2]: lbt < 'lcr-i0-cYTPbi' >'lcr-o2-0mcmL7'
Running [N0]: ltl2tgba -s '1' >'lcr-o0-4qenlX'
Running [N1]: spin -f 'true' >'lcr-o1-u5jYZM'
Running [N2]: lbt < 'lcr-i0-WwhWEC' >'lcr-o2-ukaVjs'
Performing sanity checks and gathering statistics...

-:2: !(F(!(p0)))
Running [P0]: ltl2tgba -s '!(F(!(p0)))' >'lcr-o0-atua0h'
Running [P1]: spin -f '!(<>(!(p0)))' >'lcr-o1-CEVXK7'
Running [P2]: lbt < 'lcr-i1-gxM8vX' >'lcr-o2-8MzkhN'
Running [N0]: ltl2tgba -s 'F(!(p0))' >'lcr-o0-g0I12C'
Running [N1]: spin -f '<>(!(p0))' >'lcr-o1-E0hlTs'
Running [N2]: lbt < 'lcr-i1-c6a1Ji' >'lcr-o2-ahTHA8'
Performing sanity checks and gathering statistics...

-:3: F((G(p0)) | (F(p1)))
Running [P0]: ltl2tgba -s 'F((G(p0)) | (F(p1)))' >'lcr-o0-s7QnuY'
Running [P1]: spin -f '<>(([](p0)) || (<>(p1)))' >'lcr-o1-00iXsO'
Running [P2]: lbt < 'lcr-i2-o2JXrE' >'lcr-o2-0t7Yqu'
Running [N0]: ltl2tgba -s '!(F((G(p0)) | (F(p1))))' >'lcr-o0-o55wqk'
Running [N1]: spin -f '!(<>(([](p0)) || (<>(p1))))' >'lcr-o1-UwpMua'
Running [N2]: lbt < 'lcr-i2-cZBdA0' >'lcr-o2-eNJFFQ'
Performing sanity checks and gathering statistics...

No problem detected.


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

formula
tool
exit_status
exit_code
time
states
edges
transitions
acc
scc
nondet_states
nondet_aut
complete_aut
product_states
product_transitions
product_scc
0 ltl2tgba -s %f >%O ok 0 0.0273999 1 1 0 1 1 0 0 0 1 0 1
0 spin -f %s >%O ok 0 0.00195725 2 2 1 1 2 0 0 0 1 0 1
0 lbt < %L >%O ok 0 0.00275329 1 0 0 0 1 0 0 0 1 0 1
1 ltl2tgba -s %f >%O ok 0 0.027268 1 1 1 1 1 0 0 1 200 4199 1
1 spin -f %s >%O ok 0 0.00188659 2 2 2 1 2 0 0 1 201 4220 2
1 lbt < %L >%O ok 0 0.00281626 3 3 3 0 3 0 0 1 222 4653 23
!(F(!(p0))) ltl2tgba -s %f >%O ok 0 0.0277402 1 1 1 1 1 0 0 0 200 2059 1
!(F(!(p0))) spin -f %s >%O ok 0 0.00199805 1 1 1 1 1 0 0 0 200 2059 1
!(F(!(p0))) lbt < %L >%O ok 0 0.00281056 2 2 2 0 2 0 0 0 201 2071 2
F(!(p0)) ltl2tgba -s %f >%O ok 0 0.0274115 2 3 4 1 2 0 0 1 400 8264 2
F(!(p0)) spin -f %s >%O ok 0 0.00194443 2 3 5 1 2 1 1 1 400 10337 2
F(!(p0)) lbt < %L >%O ok 0 0.00283659 4 6 10 1 4 2 1 1 601 14497 203
F((G(p0)) | (F(p1))) ltl2tgba -s %f >%O ok 0 0.0294767 3 5 11 1 3 1 1 0 600 11358 3
F((G(p0)) | (F(p1))) spin -f %s >%O ok 0 0.00233339 4 8 24 1 4 2 1 0 800 24920 4
F((G(p0)) | (F(p1))) lbt < %L >%O ok 0 0.00291947 9 17 52 2 9 4 1 0 1601 41559 805
!(F((G(p0)) | (F(p1)))) ltl2tgba -s %f >%O ok 0 0.0285971 2 4 4 1 1 0 0 0 395 3964 1
!(F((G(p0)) | (F(p1)))) spin -f %s >%O ok 0 0.00699975 2 3 5 1 1 1 1 0 396 4964 1
!(F((G(p0)) | (F(p1)))) lbt < %L >%O ok 0 0.00290436 3 6 9 1 2 3 1 0 397 5957 2

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 JSON output.

{
"tool": [
"ltl2tgba -s %f >%O",
"spin -f %s >%O",
"lbt < %L >%O"
],
"formula": [
"0",
"1",
"!(F(!(p0)))",
"F(!(p0))",
"F((G(p0)) | (F(p1)))",
"!(F((G(p0)) | (F(p1))))"
],
"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",0,0.0273999,1,1,0,1,1,0,0,0,1,0,1 ],
[ 0,1,"ok",0,0.00195725,2,2,1,1,2,0,0,0,1,0,1 ],
[ 0,2,"ok",0,0.00275329,1,0,0,0,1,0,0,0,1,0,1 ],
[ 1,0,"ok",0,0.027268,1,1,1,1,1,0,0,1,200,4199,1 ],
[ 1,1,"ok",0,0.00188659,2,2,2,1,2,0,0,1,201,4220,2 ],
[ 1,2,"ok",0,0.00281626,3,3,3,0,3,0,0,1,222,4653,23 ],
[ 2,0,"ok",0,0.0277402,1,1,1,1,1,0,0,0,200,2059,1 ],
[ 2,1,"ok",0,0.00199805,1,1,1,1,1,0,0,0,200,2059,1 ],
[ 2,2,"ok",0,0.00281056,2,2,2,0,2,0,0,0,201,2071,2 ],
[ 3,0,"ok",0,0.0274115,2,3,4,1,2,0,0,1,400,8264,2 ],
[ 3,1,"ok",0,0.00194443,2,3,5,1,2,1,1,1,400,10337,2 ],
[ 3,2,"ok",0,0.00283659,4,6,10,1,4,2,1,1,601,14497,203 ],
[ 4,0,"ok",0,0.0294767,3,5,11,1,3,1,1,0,600,11358,3 ],
[ 4,1,"ok",0,0.00233339,4,8,24,1,4,2,1,0,800,24920,4 ],
[ 4,2,"ok",0,0.00291947,9,17,52,2,9,4,1,0,1601,41559,805 ],
[ 5,0,"ok",0,0.0285971,2,4,4,1,1,0,0,0,395,3964,1 ],
[ 5,1,"ok",0,0.00699975,2,3,5,1,1,1,1,0,396,4964,1 ],
[ 5,2,"ok",0,0.00290436,3,6,9,1,2,3,1,0,397,5957,2 ]
]
}


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.

#!/usr/bin/python3
import 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)))

tool               & count & time & states & edges & transitions & acc & scc & nondet_states & nondet_aut & complete_aut & product_states & product_transitions & product_scc \\
ltl2tgba -s %f >%O &   6 &    0.0 &    1.7 &    2.5 &    3.5 &    1.0 &    1.5 &    0.2 &    0.2 &    0.3 &  299.3 & 4974.0 &    1.5 \\
spin -f %s >%O     &   6 &    0.0 &    2.2 &    3.2 &    6.3 &    1.0 &    2.0 &    0.7 &    0.5 &    0.3 &  333.0 & 7750.0 &    1.8 \\
lbt < %L >%O       &   6 &    0.0 &    3.7 &    5.7 &   12.7 &    0.7 &    3.5 &    1.5 &    0.5 &    0.3 &  503.8 & 11456.2 &  172.7 \\



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 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.)

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 $$\bar a b$$.

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 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 statistic 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 --aumbiguous 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 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:

ltlcross -f a -f Ga 'ltl2tgba -s --small %f >%O' 'ltl2tgba -s --deter %f >%O' --csv

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 --small %f >%O ok 0 0.0282574 2 2 3 1 2 0 0 0 201 4144 2
a ltl2tgba -s --deter %f >%O ok 0 0.0253441 2 2 3 1 2 0 0 0 201 4144 2
!(a) ltl2tgba -s --small %f >%O ok 0 0.0274906 2 2 3 1 2 0 0 0 201 4149 2
!(a) ltl2tgba -s --deter %f >%O ok 0 0.0254883 2 2 3 1 2 0 0 0 201 4149 2
G(a) ltl2tgba -s --small %f >%O ok 0 0.0276811 1 1 1 1 1 0 0 0 200 2059 1
G(a) ltl2tgba -s --deter %f >%O ok 0 0.0279394 1 1 1 1 1 0 0 0 200 2059 1
!(G(a)) ltl2tgba -s --small %f >%O ok 0 0.0282372 2 3 4 1 2 0 0 1 400 8264 2
!(G(a)) ltl2tgba -s --deter %f >%O ok 0 0.0280167 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, after

genltl --and-f=1..5 |
ltlcross '{small} ltl2tgba -s --small %f >%O' \
'{deter} ltl2tgba -s --deter %f >%O' --csv=ltlcross.csv


The file ltlcross.csv now contains:

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 0.0273228 2 3 4 1 2 0 0 1 400 8272 3
F(p1) deter ok 0 0.0276792 2 3 4 1 2 0 0 1 400 8272 3
!(F(p1)) small ok 0 0.025338 1 1 1 1 1 0 0 0 200 2055 2
!(F(p1)) deter ok 0 0.0275788 1 1 1 1 1 0 0 0 200 2055 2
(F(p1)) & (F(p2)) small ok 0 0.0284864 4 9 16 1 4 0 0 1 798 16533 5
(F(p1)) & (F(p2)) deter ok 0 0.0285247 4 9 16 1 4 0 0 1 798 16533 5
!((F(p1)) & (F(p2))) small ok 0 0.0262568 3 5 7 1 3 0 0 0 598 7367 4
!((F(p1)) & (F(p2))) deter ok 0 0.0280206 3 5 7 1 3 0 0 0 598 7367 4
(F(p1)) & (F(p2)) & (F(p3)) small ok 0 0.0303325 8 27 64 1 8 0 0 1 1587 33068 34
(F(p1)) & (F(p2)) & (F(p3)) deter ok 0 0.0300996 8 27 64 1 8 0 0 1 1587 33068 34
!((F(p1)) & (F(p2)) & (F(p3))) small ok 0 0.0313341 4 6 24 1 4 1 1 0 601 6171 4
!((F(p1)) & (F(p2)) & (F(p3))) deter ok 0 0.0300606 7 19 37 1 7 0 0 0 1387 18792 33
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) small ok 0 0.029071 16 81 256 1 16 0 0 1 2727 57786 74
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) deter ok 0 0.0278184 16 81 256 1 16 0 0 1 2727 57786 74
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4))) small ok 0 0.0300302 5 8 64 1 5 1 1 0 801 8468 5
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4))) deter ok 0 0.0324723 15 65 175 1 15 0 0 0 2527 37226 73
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)) small ok 0 0.0401099 32 243 1024 1 32 0 0 1 5330 114068 350
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)) deter ok 0 0.0319006 32 243 1024 1 32 0 0 1 5330 114068 350
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))) small ok 0 0.0319517 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().

library(data.table)
str(dt)

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 ...
$time : num 0.0273 0.0277 0.0253 0.0276 0.0285 ...$ 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")=<


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.

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


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  0.0382 0.0325 0.0301 0.028 0.0276 ...
$time.small : num 0.032 0.03 0.0313 0.0263 0.0253 ...$ 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
$product_scc.small : int 6 5 4 4 2 5 34 74 350 3 - attr(*, ".internal.selfref")=< - attr(*, "sorted")= chr "formula"  Using the above form, it is easy to compare two tools on some given measurement, as we just need to plot two columns. For example to compare the number of states produced by the two configurations of ltl2tgba for each formula, we just need to plot column dt2$state.small against dt2$state.deter. library(ggplot2) ggplot(dt2, aes(x=states.small, y=states.deter)) + geom_abline(colour='white') + geom_point()  We should probably print the formulas for the cases where the two sizes differ. 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)  ## 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 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). randltl -n -1 --tree-size 10..25 a b c | ltlcross --stop-on-error 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O'  ### --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. randltl -n -1 --tree-size 10..25 a b c | ltlcross --save-bogus=bugs.ltl 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O'  You can periodically check the contents of bugs.ltl, and then run ltlcross only on those formulas to look at the problems: ltlcross -F bugs.ltl 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O'  ### --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 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: ltlcross -f '(G!b & (!c | F!a)) | (c & Ga & Fb)' "modella %L %O" \ --save-bogus=bogus \ --grind=bogus-grind  | & G ! p0 | ! p1 F ! p2 & & p1 G p2 F p0 Running [P0]: modella 'lcr-i0-XLU69e' 'lcr-o0-H5Xj9p' Running [N0]: modella 'lcr-i0-90n58A' 'lcr-o0-jrmR8L' 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-BVSJ9W' 'lcr-o0-ig7Ca8' Running [N0]: modella 'lcr-i1-WpeQbj' 'lcr-o0-yYv4cu' Performing sanity checks and gathering statistics... Mutation 2/22: & G ! p0 | ! p1 F ! p2 Running [P0]: modella 'lcr-i2-2pxNeF' 'lcr-o0-4zsxgQ' Running [N0]: modella 'lcr-i2-wvxDi1' 'lcr-o0-qyuKkc' Performing sanity checks and gathering statistics... Mutation 3/22: | G ! p0 & & p1 G p2 F p0 Running [P0]: modella 'lcr-i3-25Iinn' 'lcr-o0-sFQRpy' Running [N0]: modella 'lcr-i3-UB2QsJ' 'lcr-o0-Ww2QvU' 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-iMdyz5' 'lcr-o0-o6egDg' Running [N0]: modella 'lcr-i4-oWFoHr' 'lcr-o0-gG5xLC' Performing sanity checks and gathering statistics... Mutation 2/16: G ! p0 Running [P0]: modella 'lcr-i5-ClT0PN' 'lcr-o0-KJyuUY' Running [N0]: modella 'lcr-i5-uypgZ9' 'lcr-o0-4P523k' 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-yREa9v' 'lcr-o0-A44ieH' Running [N0]: modella 'lcr-i6-qhwNjS' 'lcr-o0-yCPip3' 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-Qv6eve' 'lcr-o0-EficBp' Running [N0]: modella 'lcr-i7-O1wsHA' 'lcr-o0-08AJNL' Performing sanity checks and gathering statistics... Mutation 4/10: | p0 G ! p1 Running [P0]: modella 'lcr-i8-88jmUW' 'lcr-o0-YbXZ07' Running [N0]: modella 'lcr-i8-OJlW7i' 'lcr-o0-ohCTeu' Performing sanity checks and gathering statistics... Mutation 5/10: | G ! p0 F p0 Running [P0]: modella 'lcr-i9-AGRcmF' 'lcr-o0-8k1wtQ' Running [N0]: modella 'lcr-i9-4jC9A1' 'lcr-o0-Ai7MIc' Performing sanity checks and gathering statistics... Mutation 6/10: | ! p0 & p1 F p0 Running [P0]: modella 'lcr-i10-OD3KQn' 'lcr-o0-oYQJYy' Running [N0]: modella 'lcr-i10-cks16J' 'lcr-o0-S3UjfV' Performing sanity checks and gathering statistics... Mutation 7/10: | & p1 F p0 G p0 Running [P0]: modella 'lcr-i11-w59Xn6' 'lcr-o0-wYfDwh' Running [N0]: modella 'lcr-i11-yc2DFs' 'lcr-o0-K0CFOD' Performing sanity checks and gathering statistics... Mutation 8/10: | & p0 p1 G ! p0 Running [P0]: modella 'lcr-i12-KYC3XO' 'lcr-o0-iDxs7Z' Running [N0]: modella 'lcr-i12-MlZahb' 'lcr-o0-gFgUqm' Performing sanity checks and gathering statistics... Mutation 9/10: | G ! p0 & p0 F p0 Running [P0]: modella 'lcr-i13-CFpYAx' 'lcr-o0-kxs3KI' Running [N0]: modella 'lcr-i13-g50wVT' 'lcr-o0-Mcv154' 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-i14-0CdSgg' 'lcr-o0-oPKJrr' Running [N0]: modella 'lcr-i14-6JZTCC' 'lcr-o0-8934NN' Performing sanity checks and gathering statistics... Mutation 4/7: | p0 G ! p0 Running [P0]: modella 'lcr-i15-E7cAZY' 'lcr-o0-MNe6aa' Running [N0]: modella 'lcr-i15-4hoVml' 'lcr-o0-oAnLyw' 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-Q6bWKH' 'lcr-o0-ec78WS' Running [N0]: modella 'lcr-i16-kDpE93' 'lcr-o0-WRwamf' Performing sanity checks and gathering statistics... Mutation 7/7: | G p0 & p0 F p0 Running [P0]: modella 'lcr-i17-iL1Zyq' 'lcr-o0-MWaRLB' Running [N0]: modella 'lcr-i17-QWS0YM' 'lcr-o0-cgpbcY' 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.  cat bogus  (G!b & (!c | F!a)) | (c & Ga & Fb) G!b | (c & Ga & Fb) G!b | (c & Fb) G!c | (c & Fc)  cat bogus-grind  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 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. ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --verbose  F(G(a)) Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-SImDgp' Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-g3V2LA' Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-QltGiM' Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-yKMJUX' 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.  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 are displayed. For instance we can see that although ltl3ba -H1 outputs co-Büchi alternating automata, only automaton N1 uses universal edges: the automaton P1 can be used like a non-alternating co-Büchi automaton. ltlcross then proceeds to transform alternating automata (only weak alternating automata are supported) into non-alternating automata. Here only N1 needs this transformation. Then ltlcross computes the complement of these four 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: ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --determinize-max-states=1 --verbose  F(G(a)) Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-bIvAqp' Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-RaAWfB' Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-LQot6M' Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-bkUP1Y' 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 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.  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 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. ltlcross -f 'FGa' ltl2tgba --reference 'ltl3ba -H1' --verbose  F(G(a)) Running [P0]: ltl3ba -H1 -f '<>([](a))'>'lcr-o0-DwwBDy' Running [P1]: ltl2tgba -H 'F(G(a))'>'lcr-o1-DWKSLK' Running [N0]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o0-tLDlYW' Running [N1]: ltl2tgba -H '!(F(G(a)))'>'lcr-o1-fDBZb9' 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.  ## Running ltlcross in parallel 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 GNU findutils. The -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. randltl -n-1 3 | ltlfilt --relabel=pnn --unique -n1000 | xargs -P8 -I'{}' ltlcross -q --save-bogus='>>bugs.ltl' ltl2tgba ltl3ba -f '{}'  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. • GNU parallel or 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 ltl2tgbaand ltl3ba against all formulas produced by genltl --eh, and gathering statistics from all runs in all.csv. 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


This creates ltlcross.mk:

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)))'


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).