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ltlcross

ltlcross is a tool for cross-comparing the output of LTL-to-Büchi 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 are:

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:

Table of Contents

Formula selection

Formulas to translate should be specified using the 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:

  %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
  %N,%T,%D                   the output automaton as a Never claim, in LBTT's
                             or 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).

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

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

([](<>(a)))
Running [P0]: ltl2tgba -s '([](<>(a)))' >'lcr-o0-hvmySy'
Running [P1]: spin -f '([](<>(a)))' >'lcr-o1-HznRr3'
Running [N0]: ltl2tgba -s '(!([](<>(a))))' >'lcr-o0-QH1c8x'
Running [N1]: spin -f '(!([](<>(a))))' >'lcr-o1-JrUtU2'
Performing sanity checks and gathering statistics...

(X((a) U (b)))
Running [P0]: ltl2tgba -s '(X((a) U (b)))' >'lcr-o0-66J0Lx'
Running [P1]: spin -f '(X((a) U (b)))' >'lcr-o1-i4qfI2'
Running [N0]: ltl2tgba -s '(!(X((a) U (b))))' >'lcr-o0-eziSEx'
Running [N1]: spin -f '(!(X((a) U (b))))' >'lcr-o1-q0p6F2'
Performing sanity checks and gathering statistics...

No problem detected.

ltlcross can only read three 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 spin, ltl2ba, ltl3ba, or ltl2tgba --spin. These should be indicated using %N. The newer syntax introduced by Spin 6.24, using do instead of if, is also supported.
  • LBTT's format, which supports generalized Büchi automata with either state-based acceptance or transition-based acceptance. This output is used for instance by lbt, modella, or ltl2tgba --lbtt. These should be indicated using %T.
  • ltl2dsar's format, which support deterministic Rabin or Streett automata. After ltlcross reads such input, it immediately convert 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 >%N'
  • 'ltl2ba -f %s >%N'
  • 'ltl3ba -S -f %s >%N'
  • 'ltl3ba -S -M -f %s >%N' (more deterministic output)
  • 'modella -r12 -g -e %L %T'
  • '/path/to/script4lbtt.py %L %T' (script supplied by ltl2nba for its interface with LBTT)
  • 'ltl2tgba -s %s >%N' (smaller output, Büchi automaton)
  • 'ltl2tgba -s -D %s >%N' (more deterministic output, Büchi automaton)
  • 'ltl2tgba --lbtt %s >%T' (smaller output, TGBA)
  • 'ltl2tgba --lbtt -D %s >%T' (more deterministic output, TGBA)
  • 'lbt <%L >%T'
  • 'ltl2dstar --ltl2nba=spin:path/tp/ltl2tgba@-sD %L %D' (deterministic Rabin output)
  • 'ltl2dstar --automata=streett --ltl2nba=spin:path/tp/ltl2tgba@-sD %L %D' (deterministic Streett output)
  • 'ltl2dstar --ltl2nba=spin:path/tp/ltl2tgba@-sD %L - | dstar2tgba -s >%N' (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)
  • 'ltl3dra -f %s >%D'

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 two random formulas (where W and M operators have been rewritten away because they are not supported by spin and lbt).

randltl -n 2 a b |
ltlfilt --remove-wm |
ltlcross --csv=results.csv \
         'ltl2tgba -s %f >%N' \
         'spin -f %s >%N' \
         'lbt < %L >%T'
-:1: (G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1)))))))
Running [P0]: ltl2tgba -s '(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1)))))))' >'lcr-o0-wGKTcr'
Running [P1]: spin -f '([](((p0) U ((p0) && ([](p1)))) V (([](p1)) || ((p0) U ((p0) && ([](p1)))))))' >'lcr-o1-SxlqPW'
Running [P2]: lbt < 'lcr-i0-bMM9DV' >'lcr-o2-r95Zss'
Running [N0]: ltl2tgba -s '(!(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1))))))))' >'lcr-o0-rR0NQu'
Running [N1]: spin -f '(!([](((p0) U ((p0) && ([](p1)))) V (([](p1)) || ((p0) U ((p0) && ([](p1))))))))' >'lcr-o1-RzKV60'
Running [N2]: lbt < 'lcr-i0-IAOTEY' >'lcr-o2-ZWs3px'
Performing sanity checks and gathering statistics...

-:2: (!((G(F(p0))) -> (F(p1))))
Running [P0]: ltl2tgba -s '(!((G(F(p0))) -> (F(p1))))' >'lcr-o0-5PlknC'
Running [P1]: spin -f '(!((<>(p1)) || (!([](<>(p0))))))' >'lcr-o1-u27mU9'
Running [P2]: lbt < 'lcr-i1-8GxGT4' >'lcr-o2-3nbJrH'
Running [N0]: ltl2tgba -s '(G(F(p0))) -> (F(p1))' >'lcr-o0-p5Q8wM'
Running [N1]: spin -f '(<>(p1)) || (!([](<>(p0))))' >'lcr-o1-8ceh8j'
Running [N2]: lbt < 'lcr-i1-JMSqZe' >'lcr-o2-PYbHJR'
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","nonacc_scc","terminal_scc","weak_scc","strong_scc","nondet_states","nondet_aut","terminal_aut","weak_aut","strong_aut","product_states","product_transitions","product_scc"
"(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1)))))))","ltl2tgba -s %f >%N","ok",0,0.0250941,3,5,9,1,3,2,0,1,0,2,1,0,1,0,401,5168,3
"(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1)))))))","spin -f %s >%N","ok",0,0.00677223,6,13,18,1,3,2,0,0,1,6,1,0,0,1,999,14414,5
"(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1)))))))","lbt < %L >%T","ok",0,0.23047,8,41,51,1,3,2,0,0,1,8,1,0,0,1,1397,43175,5
"(!(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1))))))))","ltl2tgba -s %f >%N","ok",0,0.0280075,4,10,16,1,3,1,1,0,1,0,0,0,0,1,797,16411,3
"(!(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1))))))))","spin -f %s >%N","ok",0,0.0196859,7,24,63,1,4,2,1,0,1,6,1,0,0,1,1400,64822,4
"(!(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1))))))))","lbt < %L >%T","ok",0,0.00317245,39,286,614,3,28,26,1,0,1,33,1,0,0,1,7583,600472,4394
"(!((G(F(p0))) -> (F(p1))))","ltl2tgba -s %f >%N","ok",0,0.0226606,2,4,4,1,1,0,0,0,1,0,0,0,0,1,399,4130,1
"(!((G(F(p0))) -> (F(p1))))","spin -f %s >%N","ok",0,0.00185615,2,3,5,1,1,0,0,0,1,1,1,0,0,1,399,5174,1
"(!((G(F(p0))) -> (F(p1))))","lbt < %L >%T","ok",0,0.00195158,5,10,15,1,4,3,0,0,1,5,1,0,0,1,407,6333,9
"(G(F(p0))) -> (F(p1))","ltl2tgba -s %f >%N","ok",0,0.0227787,3,5,11,1,3,1,1,1,0,1,1,0,1,0,600,11305,3
"(G(F(p0))) -> (F(p1))","spin -f %s >%N","ok",0,0.00173696,3,5,14,1,3,1,1,1,0,1,1,0,1,0,600,14397,3
"(G(F(p0))) -> (F(p1))","lbt < %L >%T","ok",0,0.00187492,11,18,54,2,11,9,1,1,0,5,1,0,1,0,1245,25838,449

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 JSON output.

{
  "tool": [
    "ltl2tgba -s %f >%N",
    "spin -f %s >%N",
    "lbt < %L >%T"
  ],
  "formula": [
    "(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1)))))))",
    "(!(G(((p0) U ((p0) & (G(p1)))) R ((G(p1)) | ((p0) U ((p0) & (G(p1))))))))",
    "(!((G(F(p0))) -> (F(p1))))",
    "(G(F(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","product_states","product_transitions","product_scc"
  ],
  "inputs":  [ 0, 1 ],
  "results": [
    [ 0,0,"ok",0,0.0250941,3,5,9,1,3,2,0,1,0,2,1,0,1,0,401,5168,3 ],
    [ 0,1,"ok",0,0.00677223,6,13,18,1,3,2,0,0,1,6,1,0,0,1,999,14414,5 ],
    [ 0,2,"ok",0,0.23047,8,41,51,1,3,2,0,0,1,8,1,0,0,1,1397,43175,5 ],
    [ 1,0,"ok",0,0.0280075,4,10,16,1,3,1,1,0,1,0,0,0,0,1,797,16411,3 ],
    [ 1,1,"ok",0,0.0196859,7,24,63,1,4,2,1,0,1,6,1,0,0,1,1400,64822,4 ],
    [ 1,2,"ok",0,0.00317245,39,286,614,3,28,26,1,0,1,33,1,0,0,1,7583,600472,4394 ],
    [ 2,0,"ok",0,0.0226606,2,4,4,1,1,0,0,0,1,0,0,0,0,1,399,4130,1 ],
    [ 2,1,"ok",0,0.00185615,2,3,5,1,1,0,0,0,1,1,1,0,0,1,399,5174,1 ],
    [ 2,2,"ok",0,0.00195158,5,10,15,1,4,3,0,0,1,5,1,0,0,1,407,6333,9 ],
    [ 3,0,"ok",0,0.0227787,3,5,11,1,3,1,1,1,0,1,1,0,1,0,600,11305,3 ],
    [ 3,1,"ok",0,0.00173696,3,5,14,1,3,1,1,1,0,1,1,0,1,0,600,14397,3 ],
    [ 3,2,"ok",0,0.00187492,11,18,54,2,11,9,1,1,0,5,1,0,1,0,1245,25838,449 ]
  ]
}

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
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)))
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 & product_states & product_transitions & product_scc \\
ltl2tgba -s %f >%N &   4 &    0.0 &    3.0 &    6.0 &   10.0 &    1.0 &    2.5 &    1.0 &    0.5 &    0.5 &    0.5 &    0.8 &    0.5 &    0.0 &    0.5 &    0.5 &  549.2 & 9253.5 &    2.5 \\
spin -f %s >%N     &   4 &    0.0 &    4.5 &   11.2 &   25.0 &    1.0 &    2.8 &    1.2 &    0.5 &    0.2 &    0.8 &    3.5 &    1.0 &    0.0 &    0.2 &    0.8 &  849.5 & 24701.8 &    3.2 \\
lbt < %L >%T       &   4 &    0.1 &   15.8 &   88.8 &  183.5 &    1.8 &   11.5 &   10.0 &    0.5 &    0.2 &    0.8 &   12.8 &    1.0 &    0.0 &    0.2 &    0.8 & 2658.0 & 168954.5 & 1214.2 \\

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 (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. (Aside from parsing them and converting them to TGBA, Spot has no support for Rabin automata and Streett automata.)

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.

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.

Finally, 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.

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 >%N and ltl2tgba -s --deter %f >%N appear verbatim in the output:

ltlcross -f a -f Ga 'ltl2tgba -s --small %f >%N' 'ltl2tgba -s --deter %f >%N' --csv
"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 >%N","ok",0,0.0277752,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4092,2
"(a)","ltl2tgba -s --deter %f >%N","ok",0,0.0267896,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4092,2
"(!(a))","ltl2tgba -s --small %f >%N","ok",0,0.0248929,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4098,2
"(!(a))","ltl2tgba -s --deter %f >%N","ok",0,0.0234092,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4098,2
"(G(a))","ltl2tgba -s --small %f >%N","ok",0,0.0221926,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2023,1
"(G(a))","ltl2tgba -s --deter %f >%N","ok",0,0.0219565,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2023,1
"(!(G(a)))","ltl2tgba -s --small %f >%N","ok",0,0.0219226,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8166,2
"(!(G(a)))","ltl2tgba -s --deter %f >%N","ok",0,0.0222885,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8166,2

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:

ltlcross -f a -f Ga '{small} ltl2tgba -s --small %f >%N' '{deter} ltl2tgba -s --deter %f >%N' --csv
"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",0,0.0321048,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4092,2
"(a)","deter","ok",0,0.0296222,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4092,2
"(!(a))","small","ok",0,0.0278834,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4098,2
"(!(a))","deter","ok",0,0.0285633,2,2,3,1,2,1,1,0,0,0,0,1,0,0,201,4098,2
"(G(a))","small","ok",0,0.0242847,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2023,1
"(G(a))","deter","ok",0,0.0229045,1,1,1,1,1,0,0,1,0,0,0,0,1,0,200,2023,1
"(!(G(a)))","small","ok",0,0.0237626,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8166,2
"(!(G(a)))","deter","ok",0,0.0226834,2,3,4,1,2,1,1,0,0,0,0,1,0,0,400,8166,2

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 >%T' 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 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 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).

randltl -n -1 --tree-size 10..25 a b c | ltlcross --stop-on-error 'ltl2tgba --lbtt %f >%T' 'ltl3ba -f %s >%N'

--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.

Date: 2014-05-15T11:05+0200

Author: Alexandre Duret-Lutz

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