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# `ltlcross`

## Table of Contents

- Formula selection
- Configuring translators
  - Translator specifications
  - Trusted and untrusted translators
- Detecting problems
- Getting statistics
  - CSV or JSON output (or both!)
  - Description of the columns
  - Changing the name of the translators
  - Working with these CSV files in R
- Miscellaneous options
  - `--stop-on-error`
  - `--save-bogus=FILENAME`
  - `--grind=FILENAME`
  - `--no-check`
  - `--verbose`
- Running `ltlcross` in parallel

`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 counterexample, but you will be on your own to investigate and fix them (the `--grind` option may help you reduce the problem to a shorter formula).

The core of `ltlcross` is a loop that does the following steps:

- Input a formula
- Translate the formula and its negation using each configured translator. If there are 3 translators, the positive and negative translations will be denoted `P0`, `N0`, `P1`, `N1`, `P2`, `N2`.
- Optionally build complemented automata denoted `Comp(P0)`, `Comp(N0)`, etc. (By default, this is done only for small automata, but see options `-D`, `--determinize-max-states` and `--determinize-max-edges`.)
- Perform sanity checks between all these automata to detect any problem.
- Optionally build the products of these automata with a random state-space (the same state-space for all translations). (If the `--products=N` option is given, `N` products are performed instead.)
- Gather statistics if requested.

## Formula selection

Formulas to translate should be specified using the 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`.

    [](<>(a))
    Running [P0]: ltl2tgba -s '[](<>(a))' >'lcr-o0-V5nFJk'
    Running [P1]: spin -f '[](<>(a))' >'lcr-o1-nLbvk2'
    Running [N0]: ltl2tgba -s '!([](<>(a)))' >'lcr-o0-SaJ4ZO'
    Running [N1]: spin -f '!([](<>(a)))' >'lcr-o1-hlGIV1'
    Performing sanity checks and gathering statistics...

    X((a) U (b))
    Running [P0]: ltl2tgba -s 'X((a) U (b))' >'lcr-o0-iuww9Q'
    Running [P1]: spin -f 'X((a) U (b))' >'lcr-o1-MQ5FC6'
    Running [N0]: ltl2tgba -s '!(X((a) U (b)))' >'lcr-o0-AelA9K'
    Running [N1]: spin -f '!(X((a) U (b)))' >'lcr-o1-85uNLR'
    Performing sanity checks and gathering statistics...

    No problem detected.

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

`ltlcross` can only read four kinds of output:

- Never claims (only if they are restricted to representing an automaton using `if`, `goto`, and `skip` statements) such as those output by `spin`, `ltl2ba`, `ltl3ba`, or `ltl2tgba --spin`. 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`.
- Non-alternating automata in the HOA format with any acceptance condition.
- Weak alternating automata in the HOA format.
- `ltl2dstar`'s format, which supports deterministic Rabin or Streett automata.

Files in any of these format should be indicated with `%O`. (Past versions of `ltlcross` used different letters for each format, but the four parsers have been merged into a single one.)

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

- '`spin -f %s >%O`'
- '`ltl2ba -f %s >%O`'
- '`ltl3ba -M0 -f %s >%O`' (less deterministic output, can be smaller)
- '`ltl3ba -M1 -f %s >%O`' (more deterministic output)
- '`modella -r12 -g -e %[MWei^]L %O`'
- '`/path/to/script4lbtt.py %L %O`' (script supplied by 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 regexes, then the string on the right is appended.

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

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

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 including a translator with good deterministic output to augment test coverage. Using '`ltl2tgba -D %f >%O`' will produce deterministic automata for all obligation properties and many recurrence properties. Using '`ltl2tgba -PD %f >%O`' will systematically produce a deterministic Parity automaton (that `ltlcross` can complement easily).

- Cross-comparison checks: for some state-space \(S\), all \(P_i\otimes S\) are either all empty, or all non-empty. Similarly, all \(N_i\otimes S\) are either all empty, or all non-empty.

  A cross-comparison failure could be displayed as:

      error: {P0,P2} disagree with {P1} when evaluating the state-space
             the following word(s) are not accepted by {P1}:
       P0 accepts: p0 & !p1 & !p2 & p3; p0 & p1 & !p2 & p3; p0 & p1 & p2 & p3; cycle{p0 & p1 & p2 & p3; p0 & p1 & !p2 & !p3; p0 & p1 & p2 & !p3; p0 & p1 & !p2 & !p3}
       P2 accepts: p0 & !p1 & !p2 & p3; cycle{p0 & p1 & !p2 & !p3; p0 & p1 & p2 & p3; p0 & p1 & !p2 & p3}

  If `--products=N` is used with `N` greater than one, the number of the state-space is also printed. This number is of no use by itself, except to explain why you may get multiple disagreement between the same sets of automata.

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

- Consistency check:

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

  An error in that case is displayed as

      error: inconsistency between P1 and N1

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

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

The above checks are similar to those that are performed by 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 --seed=5 a b |
    ltlfilt --remove-wm |
    ltlcross --csv=results.csv \
             'ltl2tgba -s %f >%O' \
             'spin -f %s >%O' \
             'lbt < %L >%O'

    -:1: G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))
    Running [P0]: ltl2tgba -s 'G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))' >'lcr-o0-gwtiP9'
    Running [P1]: spin -f '[](!((!(p0)) && ((X(p0)) || (<>((p1) V (X(p0)))))))' >'lcr-o1-2MMZFA'
    Running [P2]: lbt < 'lcr-i0-smPrSw' >'lcr-o2-JWRWMb'
    Running [N0]: ltl2tgba -s '!(G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0))))))))' >'lcr-o0-ITtEGa'
    Running [N1]: spin -f '!([](!((!(p0)) && ((X(p0)) || (<>((p1) V (X(p0))))))))' >'lcr-o1-d3D9xQ'
    Running [N2]: lbt < 'lcr-i0-FSeVGv' >'lcr-o2-nkmJdb'
    Performing sanity checks and gathering statistics...

    -:2: F(p0)
    Running [P0]: ltl2tgba -s 'F(p0)' >'lcr-o0-QBuXsO'
    Running [P1]: spin -f '<>(p0)' >'lcr-o1-e2D1Kd'
    Running [P2]: lbt < 'lcr-i1-wgWXzu' >'lcr-o2-1erdOz'
    Running [N0]: ltl2tgba -s '!(F(p0))' >'lcr-o0-STwOew'
    Running [N1]: spin -f '!(<>(p0))' >'lcr-o1-QYNw9E'
    Running [N2]: lbt < 'lcr-i1-JKG7Vg' >'lcr-o2-6sBky7'
    Performing sanity checks and gathering statistics...

    -:3: (X(p0)) U ((!(p0)) & (X(p0)))
    Running [P0]: ltl2tgba -s '(X(p0)) U ((!(p0)) & (X(p0)))' >'lcr-o0-cFRCjj'
    Running [P1]: spin -f '(X(p0)) U ((!(p0)) && (X(p0)))' >'lcr-o1-y9W3Rp'
    Running [P2]: lbt < 'lcr-i2-bWZaDs' >'lcr-o2-faNOqx'
    Running [N0]: ltl2tgba -s '!((X(p0)) U ((!(p0)) & (X(p0))))' >'lcr-o0-pIO0he'
    Running [N1]: spin -f '!((X(p0)) U ((!(p0)) && (X(p0))))' >'lcr-o1-YvxXke'
    Running [N2]: lbt < 'lcr-i2-gCTBKR' >'lcr-o2-QKdbSr'
    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
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0))))))) | ltl2tgba -s %f >%O | ok | 0 | 0.0407812 | 2 | 3 | 3 | 1 | 2 | 0 | 0 | 0 | 400 | 6276 | 2
G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0))))))) | spin -f %s >%O | ok | 0 | 0.0642129 | 7 | 21 | 35 | 1 | 2 | 7 | 1 | 0 | 1200 | 29442 | 4
G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0))))))) | lbt < %L >%O | ok | 0 | 0.00383116 | 6 | 14 | 28 | 1 | 5 | 6 | 1 | 0 | 1000 | 20970 | 403
!(G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))) | ltl2tgba -s %f >%O | ok | 0 | 0.0410838 | 3 | 5 | 6 | 1 | 3 | 0 | 0 | 1 | 600 | 12633 | 3
!(G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))) | spin -f %s >%O | ok | 0 | 0.00394628 | 6 | 13 | 32 | 1 | 5 | 4 | 1 | 0 | 1200 | 33622 | 204
!(G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))) | lbt < %L >%O | ok | 0 | 0.00366486 | 13 | 28 | 67 | 2 | 13 | 7 | 1 | 0 | 2400 | 59006 | 1604
F(p0) | ltl2tgba -s %f >%O | ok | 0 | 0.039699 | 2 | 3 | 4 | 1 | 2 | 0 | 0 | 1 | 400 | 8264 | 2
F(p0) | spin -f %s >%O | ok | 0 | 0.00243053 | 2 | 3 | 5 | 1 | 2 | 1 | 1 | 1 | 400 | 10323 | 2
F(p0) | lbt < %L >%O | ok | 0 | 0.00356943 | 4 | 6 | 10 | 1 | 4 | 2 | 1 | 1 | 601 | 14487 | 203
!(F(p0)) | ltl2tgba -s %f >%O | ok | 0 | 0.0368359 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 200 | 2073 | 1
!(F(p0)) | spin -f %s >%O | ok | 0 | 0.00248508 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 200 | 2073 | 1
!(F(p0)) | lbt < %L >%O | ok | 0 | 0.00408954 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 201 | 2081 | 2
(X(p0)) U ((!(p0)) & (X(p0))) | ltl2tgba -s %f >%O | ok | 0 | 0.0401076 | 3 | 3 | 4 | 1 | 3 | 0 | 0 | 0 | 208 | 4118 | 9
(X(p0)) U ((!(p0)) & (X(p0))) | spin -f %s >%O | ok | 0 | 0.00255733 | 4 | 6 | 7 | 1 | 4 | 1 | 1 | 0 | 408 | 6159 | 10
(X(p0)) U ((!(p0)) & (X(p0))) | lbt < %L >%O | ok | 0 | 0.00333262 | 6 | 7 | 10 | 1 | 6 | 1 | 1 | 0 | 484 | 7467 | 86
!((X(p0)) U ((!(p0)) & (X(p0)))) | ltl2tgba -s %f >%O | ok | 0 | 0.0406445 | 3 | 4 | 5 | 1 | 3 | 0 | 0 | 0 | 208 | 4128 | 9
!((X(p0)) U ((!(p0)) & (X(p0)))) | spin -f %s >%O | ok | 0 | 0.00333652 | 4 | 8 | 10 | 1 | 4 | 2 | 1 | 0 | 800 | 20222 | 203
!((X(p0)) U ((!(p0)) & (X(p0)))) | lbt < %L >%O | ok | 0 | 0.00496907 | 9 | 17 | 23 | 0 | 9 | 4 | 1 | 0 | 1601 | 34445 | 1004

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

Although we only supplied 2 random generated formulas, the output contains 4 formulas because `ltlcross` had to translate the positive and negative version of each.

If we had used the option `--json=results.json` instead of (or in addition to) `--cvs=results.csv`, the file `results.json` would have contained the following JSON output.

    {
      "tool": [
        "ltl2tgba -s %f >%O",
        "spin -f %s >%O",
        "lbt < %L >%O"
      ],
      "formula": [
        "G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0)))))))",
        "!(G(!((!(p0)) & ((X(p0)) | (F((p1) R (X(p0))))))))",
        "F(p0)",
        "!(F(p0))",
        "(X(p0)) U ((!(p0)) & (X(p0)))",
        "!((X(p0)) U ((!(p0)) & (X(p0))))"
      ],
      "fields":  [
      "formula","tool","exit_status","exit_code","time","states","edges","transitions","acc","scc","nondet_states","nondet_aut","complete_aut","product_states","product_transitions","product_scc"
      ],
      "inputs":  [ 0, 1 ],
      "results": [
        [ 0,0,"ok",0,0.0407812,2,3,3,1,2,0,0,0,400,6276,2 ],
        [ 0,1,"ok",0,0.0642129,7,21,35,1,2,7,1,0,1200,29442,4 ],
        [ 0,2,"ok",0,0.00383116,6,14,28,1,5,6,1,0,1000,20970,403 ],
        [ 1,0,"ok",0,0.0410838,3,5,6,1,3,0,0,1,600,12633,3 ],
        [ 1,1,"ok",0,0.00394628,6,13,32,1,5,4,1,0,1200,33622,204 ],
        [ 1,2,"ok",0,0.00366486,13,28,67,2,13,7,1,0,2400,59006,1604 ],
        [ 2,0,"ok",0,0.039699,2,3,4,1,2,0,0,1,400,8264,2 ],
        [ 2,1,"ok",0,0.00243053,2,3,5,1,2,1,1,1,400,10323,2 ],
        [ 2,2,"ok",0,0.00356943,4,6,10,1,4,2,1,1,601,14487,203 ],
        [ 3,0,"ok",0,0.0368359,1,1,1,1,1,0,0,0,200,2073,1 ],
        [ 3,1,"ok",0,0.00248508,1,1,1,1,1,0,0,0,200,2073,1 ],
        [ 3,2,"ok",0,0.00408954,2,2,2,0,2,0,0,0,201,2081,2 ],
        [ 4,0,"ok",0,0.0401076,3,3,4,1,3,0,0,0,208,4118,9 ],
        [ 4,1,"ok",0,0.00255733,4,6,7,1,4,1,1,0,408,6159,10 ],
        [ 4,2,"ok",0,0.00333262,6,7,10,1,6,1,1,0,484,7467,86 ],
        [ 5,0,"ok",0,0.0406445,3,4,5,1,3,0,0,0,208,4128,9 ],
        [ 5,1,"ok",0,0.00333652,4,8,10,1,4,2,1,0,800,20222,203 ],
        [ 5,2,"ok",0,0.00496907,9,17,23,0,9,4,1,0,1601,34445,1004 ]
      ]
    }

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 & nondet_states & nondet_aut & complete_aut & product_states & product_transitions & product_scc \\
    ltl2tgba -s %f >%O &   6 &    0.0 &    2.3 &    3.2 &    3.8 &    1.0 &    2.3 &    0.0 &    0.0 &    0.3 &  336.0 & 6248.7 &    4.3 \\
    spin -f %s >%O     &   6 &    0.0 &    4.0 &    8.7 &   15.0 &    1.0 &    3.0 &    2.5 &    0.8 &    0.2 &  701.3 & 16973.5 &   70.7 \\
    lbt < %L >%O       &   6 &    0.0 &    6.7 &   12.3 &   23.3 &    0.8 &    6.5 &    3.3 &    0.8 &    0.2 & 1047.8 & 23076.0 &  550.3 \\

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

When computing such statistics, you should be aware that inputs for which a tool failed to generate an automaton (e.g. it crashed, or it was killed if you used `ltlcross`'s `--timeout` option to limit run time) will appear as mostly empty lines in the CSV or JSON files, since most statistics cannot be computed without an automaton… Those lines with missing data can be omitted with the `--omit-missing` option (this used to be the default up to Spot 1.2).

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

### Description of the columns

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

Columns `formula` and `tool` contain the formula translated, and the command run to translate it. In the CSV, these columns contain the actual text. In the JSON output, these columns contain an index into the `formula` and `tool` table declared separately.

`exit_status` and `exit_code` are used to indicate if the translator successfully produced an automaton, or if it failed. On successful translation, `exit_status` is equal to "`ok`" and `exit_code` is 0. If the translation took more time than allowed with the `--timeout` option, `exit_status` will contain "`timeout`" and `exit_code` will be set to -1. Other values are used to diagnose various issues: please check the man-page for `ltlcross` for a list of them.

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

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

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

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

If option `--strength` is passed to `ltlcross`, these SCCs are also partitioned on four sets based on their strengths:

- `nonacc_scc` for non-accepting SCCs (such as states A1 and A2 in the previous picture).
- `terminal_scc` for accepting SCCs where all states or edges belong to the same acceptance sets, and that are complete (i.e., any state in a terminal SCC accepts the universal language). States B1 and B2 in the previous picture are two terminal SCCs.
- `weak_scc` for accepting SCCs where all states or edges belong to the same acceptance sets, but that are not complete.
- `strong_scc` for accepting SCCs that are not weak.

These SCC strengths can be used to compute the strength of the automaton as a whole:

- an automaton is terminal if it contains only non-accepting or terminal SCCs,
- an automaton is weak if it contains only non-accepting, terminal, or weak SCCs,
- an automaton is strong if it contains at least one strong SCC.

This classification is used to fill the `terminal_aut`, `weak_aut`, `strong_aut` columns with Boolean values (still only if option `--strength` is passed). Only one of these should contain `1`. We usually prefer terminal automata over weak automata, and weak automata over strong automata, because the emptiness check of terminal (and weak) automata is easier. When working with alternating automata, all those strength-related columns will be empty, because the routines used to compute those statistics do not yet support universal edges.

`nondetstates` counts the number of non-deterministic states in the automaton. `nondeterministic` is a Boolean value indicating if the automaton is not deterministic. For instance in the previous picture showing two automata for `a U b`, the first automaton is deterministic (these two fields will contain 0), while the second automaton contain a nondeterministic state (state A2 has two possible successors for the assignment \(ab\)) and is therefore not deterministic.

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

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

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

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

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

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

### Changing the name of the translators

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

For instance in the following, `ltl2tgba` is run in two configurations, and the strings `ltl2tgba -s --small %f >%O` and `ltl2tgba -s --deter %f >%O` appear verbatim in the output:

    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 | 0.049789 | 2 | 2 | 3 | 1 | 2 | 0 | 0 | 0 | 201 | 4144 | 2
a | ltl2tgba -s --deter %f >%O | ok | 0 | 0.0451396 | 2 | 2 | 3 | 1 | 2 | 0 | 0 | 0 | 201 | 4144 | 2
!(a) | ltl2tgba -s --small %f >%O | ok | 0 | 0.0409285 | 2 | 2 | 3 | 1 | 2 | 0 | 0 | 0 | 201 | 4149 | 2
!(a) | ltl2tgba -s --deter %f >%O | ok | 0 | 0.0386534 | 2 | 2 | 3 | 1 | 2 | 0 | 0 | 0 | 201 | 4149 | 2
G(a) | ltl2tgba -s --small %f >%O | ok | 0 | 0.0375853 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 200 | 2059 | 1
G(a) | ltl2tgba -s --deter %f >%O | ok | 0 | 0.0374146 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 200 | 2059 | 1
!(G(a)) | ltl2tgba -s --small %f >%O | ok | 0 | 0.0441821 | 2 | 3 | 4 | 1 | 2 | 0 | 0 | 1 | 400 | 8264 | 2
!(G(a)) | ltl2tgba -s --deter %f >%O | ok | 0 | 0.0393874 | 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:

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

    -:1: F(p1)
    Running [P0: small]: ltl2tgba -s --small 'F(p1)' >'lcr-o0-3vnYE9'
    Running [P1: deter]: ltl2tgba -s --deter 'F(p1)' >'lcr-o1-WFzS5K'
    Running [N0: small]: ltl2tgba -s --small '!(F(p1))' >'lcr-o0-33Nlkl'
    Running [N1: deter]: ltl2tgba -s --deter '!(F(p1))' >'lcr-o1-6j97uo'
    Performing sanity checks and gathering statistics...

    -:2: (F(p1)) & (F(p2))
    Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2))' >'lcr-o0-xIQ6tk'
    Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2))' >'lcr-o1-NqrpKz'
    Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)))' >'lcr-o0-RJJ4dN'
    Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)))' >'lcr-o1-wBExQ8'
    Performing sanity checks and gathering statistics...

    -:3: (F(p1)) & (F(p2)) & (F(p3))
    Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2)) & (F(p3))' >'lcr-o0-GW5ZVq'
    Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2)) & (F(p3))' >'lcr-o1-tHgJp6'
    Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)) & (F(p3)))' >'lcr-o0-9MnLIv'
    Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)) & (F(p3)))' >'lcr-o1-86epxw'
    Performing sanity checks and gathering statistics...

    -:4: (F(p1)) & (F(p2)) & (F(p3)) & (F(p4))
    Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4))' >'lcr-o0-iPkx4O'
    Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4))' >'lcr-o1-4KZokN'
    Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))' >'lcr-o0-aIFU76'
    Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))' >'lcr-o1-OJyR0C'
    Performing sanity checks and gathering statistics...

    -:5: (F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))
    Running [P0: small]: ltl2tgba -s --small '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))' >'lcr-o0-A8fYbl'
    Running [P1: deter]: ltl2tgba -s --deter '(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))' >'lcr-o1-xGjaPz'
    Running [N0: small]: ltl2tgba -s --small '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)))' >'lcr-o0-38mTDw'
    Running [N1: deter]: ltl2tgba -s --deter '!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)))' >'lcr-o1-2BpnXH'
    Performing sanity checks and gathering statistics...

    No problem detected.

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

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 | 0.037841 | 2 | 3 | 4 | 1 | 2 | 0 | 0 | 1 | 400 | 8272 | 3
F(p1) | deter | ok | 0 | 0.0373372 | 2 | 3 | 4 | 1 | 2 | 0 | 0 | 1 | 400 | 8272 | 3
!(F(p1)) | small | ok | 0 | 0.0405419 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 200 | 2055 | 2
!(F(p1)) | deter | ok | 0 | 0.0386705 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 200 | 2055 | 2
(F(p1)) & (F(p2)) | small | ok | 0 | 0.0410353 | 4 | 9 | 16 | 1 | 4 | 0 | 0 | 1 | 798 | 16533 | 5
(F(p1)) & (F(p2)) | deter | ok | 0 | 0.0378562 | 4 | 9 | 16 | 1 | 4 | 0 | 0 | 1 | 798 | 16533 | 5
!((F(p1)) & (F(p2))) | small | ok | 0 | 0.0408338 | 3 | 5 | 7 | 1 | 3 | 0 | 0 | 0 | 598 | 7367 | 4
!((F(p1)) & (F(p2))) | deter | ok | 0 | 0.0399733 | 3 | 5 | 7 | 1 | 3 | 0 | 0 | 0 | 598 | 7367 | 4
(F(p1)) & (F(p2)) & (F(p3)) | small | ok | 0 | 0.0401245 | 8 | 27 | 64 | 1 | 8 | 0 | 0 | 1 | 1587 | 33068 | 34
(F(p1)) & (F(p2)) & (F(p3)) | deter | ok | 0 | 0.0402408 | 8 | 27 | 64 | 1 | 8 | 0 | 0 | 1 | 1587 | 33068 | 34
!((F(p1)) & (F(p2)) & (F(p3))) | small | ok | 0 | 0.0431423 | 4 | 6 | 24 | 1 | 4 | 1 | 1 | 0 | 601 | 6171 | 4
!((F(p1)) & (F(p2)) & (F(p3))) | deter | ok | 0 | 0.0403026 | 7 | 19 | 37 | 1 | 7 | 0 | 0 | 0 | 1387 | 18792 | 33
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) | small | ok | 0 | 0.0431836 | 16 | 81 | 256 | 1 | 16 | 0 | 0 | 1 | 2727 | 57786 | 74
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) | deter | ok | 0 | 0.0440727 | 16 | 81 | 256 | 1 | 16 | 0 | 0 | 1 | 2727 | 57786 | 74
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4))) | small | ok | 0 | 0.0432125 | 5 | 8 | 64 | 1 | 5 | 1 | 1 | 0 | 801 | 8468 | 5
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4))) | deter | ok | 0 | 0.0412843 | 15 | 65 | 175 | 1 | 15 | 0 | 0 | 0 | 2527 | 37226 | 73
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)) | small | ok | 0 | 0.0511993 | 32 | 243 | 1024 | 1 | 32 | 0 | 0 | 1 | 5330 | 114068 | 350
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)) | deter | ok | 0 | 0.0512913 | 32 | 243 | 1024 | 1 | 32 | 0 | 0 | 1 | 5330 | 114068 | 350
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))) | small | ok | 0 | 0.0480072 | 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)
    dt <- fread('ltlcross.csv')
    str(dt)

    data.table 1.17.0 using 12 threads (see ?getDTthreads).  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  0.0378 0.0373 0.0405 0.0387 0.041 ...
     $ states             : int  2 2 1 1 4 4 3 3 8 8 ...
     $ edges              : int  3 3 1 1 9 9 5 5 27 27 ...
     $ transitions        : int  4 4 1 1 16 16 7 7 64 64 ...
     $ acc                : int  1 1 1 1 1 1 1 1 1 1 ...
     $ scc                : int  2 2 1 1 4 4 3 3 8 8 ...
     $ nondet_states      : int  0 0 0 0 0 0 0 0 0 0 ...
     $ nondet_aut         : int  0 0 0 0 0 0 0 0 0 0 ...
     $ complete_aut       : int  1 1 0 0 1 1 0 0 1 1 ...
     $ product_states     : int  400 400 200 200 798 798 598 598 1587 1587 ...
     $ product_transitions: int  8272 8272 2055 2055 16533 16533 7367 7367 33068 33068 ...
     $ product_scc        : int  3 3 2 2 5 5 4 4 34 34 ...
     - attr(*, ".internal.selfref")=<externalptr>

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  0.0502 0.0413 0.0403 0.04 0.0387 ...
     $ time.small               : num  0.048 0.0432 0.0431 0.0408 0.0405 ...
     $ states.deter             : int  31 15 7 3 1 4 8 16 32 2
     $ states.small             : int  6 5 4 3 1 4 8 16 32 2
     $ edges.deter              : int  211 65 19 5 1 9 27 81 243 3
     $ edges.small              : int  10 8 6 5 1 9 27 81 243 3
     $ transitions.deter        : int  781 175 37 7 1 16 64 256 1024 4
     $ transitions.small        : int  160 64 24 7 1 16 64 256 1024 4
     $ acc.deter                : int  1 1 1 1 1 1 1 1 1 1
     $ acc.small                : int  1 1 1 1 1 1 1 1 1 1
     $ scc.deter                : int  31 15 7 3 1 4 8 16 32 2
     $ scc.small                : int  6 5 4 3 1 4 8 16 32 2
     $ nondet_states.deter      : int  0 0 0 0 0 0 0 0 0 0
     $ nondet_states.small      : int  1 1 1 0 0 0 0 0 0 0
     $ nondet_aut.deter         : int  0 0 0 0 0 0 0 0 0 0
     $ nondet_aut.small         : int  1 1 1 0 0 0 0 0 0 0
     $ complete_aut.deter       : int  0 0 0 0 0 1 1 1 1 1
     $ complete_aut.small       : int  0 0 0 0 0 1 1 1 1 1
     $ product_states.deter     : int  5130 2527 1387 598 200 798 1587 2727 5330 400
     $ product_states.small     : int  1000 801 601 598 200 798 1587 2727 5330 400
     $ product_transitions.deter: int  82897 37226 18792 7367 2055 16533 33068 57786 114068 8272
     $ product_transitions.small: int  10707 8468 6171 7367 2055 16533 33068 57786 114068 8272
     $ product_scc.deter        : int  349 73 33 4 2 5 34 74 350 3
     $ product_scc.small        : int  6 5 4 4 2 5 34 74 350 3
     - attr(*, ".internal.selfref")=<externalptr>
     - attr(*, "sorted")= chr "formula"

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 includes failure to start some translator, read its output, or failure to pass the sanity checks. Timeouts are allowed unless `--fail-on-time` is also given.

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

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

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

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

Example:

    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-a1Qkeq' 'lcr-o0-2e5xNN'
    Running [N0]: modella 'lcr-i0-jUXDwM' 'lcr-o0-xuBcZb'
    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-qcCXWZ' 'lcr-o0-aP1tCU'
    Running [N0]: modella 'lcr-i1-5LtGzV' 'lcr-o0-0Xt395'
    Performing sanity checks and gathering statistics...

    Mutation 2/22: & G ! p0 | ! p1 F ! p2
    Running [P0]: modella 'lcr-i2-DUPezj' 'lcr-o0-I76WDE'
    Running [N0]: modella 'lcr-i2-TevA5C' 'lcr-o0-R0E5o7'
    Performing sanity checks and gathering statistics...

    Mutation 3/22: | G ! p0 & & p1 G p2 F p0
    Running [P0]: modella 'lcr-i3-wIKRba' 'lcr-o0-Rfu5WJ'
    Running [N0]: modella 'lcr-i3-RH31Fq' 'lcr-o0-UxFeJ9'
    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-OryCOB' 'lcr-o0-2eB6h1'
    Running [N0]: modella 'lcr-i4-CHADa0' 'lcr-o0-UvKd8q'
    Performing sanity checks and gathering statistics...

    Mutation 2/16: G ! p0
    Running [P0]: modella 'lcr-i5-jkMX3B' 'lcr-o0-N1C7Hc'
    Running [N0]: modella 'lcr-i5-n43Ebs' 'lcr-o0-Mn4ZbF'
    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-ckiPHa' 'lcr-o0-VoEIre'
    Running [N0]: modella 'lcr-i6-P42vbZ' 'lcr-o0-N2jhlS'
    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-NlpyOl' 'lcr-o0-xco4Iq'
    Running [N0]: modella 'lcr-i7-RdwpjM' 'lcr-o0-MDoalx'
    Performing sanity checks and gathering statistics...

    Mutation 4/10: | p0 G ! p1
    Running [P0]: modella 'lcr-i8-cZJjY0' 'lcr-o0-iCMYaN'
    Running [N0]: modella 'lcr-i8-bX3xvM' 'lcr-o0-lSmH5l'
    Performing sanity checks and gathering statistics...

    Mutation 5/10: | G ! p0 F p0
    Running [P0]: modella 'lcr-i9-O7XjCb' 'lcr-o0-4DV7rF'
    Running [N0]: modella 'lcr-i9-uZI8A3' 'lcr-o0-HFUmhp'
    Performing sanity checks and gathering statistics...

    Mutation 6/10: | ! p0 & p1 F p0
    Running [P0]: modella 'lcr-i10-FZNVR0' 'lcr-o0-EMnP3F'
    Running [N0]: modella 'lcr-i10-rvC4AX' 'lcr-o0-89vyHv'
    Performing sanity checks and gathering statistics...

    Mutation 7/10: | & p1 F p0 G p0
    Running [P0]: modella 'lcr-i11-NlSkvh' 'lcr-o0-dqsjYy'
    Running [N0]: modella 'lcr-i11-iRg0wW' 'lcr-o0-XaGsyJ'
    Performing sanity checks and gathering statistics...

    Mutation 8/10: | & p0 p1 G ! p0
    Running [P0]: modella 'lcr-i12-CL8Beh' 'lcr-o0-vAJZ9u'
    Running [N0]: modella 'lcr-i12-DzT3ZL' 'lcr-o0-un040W'
    Performing sanity checks and gathering statistics...

    Mutation 9/10: | G ! p0 & p0 F p0
    Running [P0]: modella 'lcr-i13-JMayF5' 'lcr-o0-crGLJZ'
    Running [N0]: modella 'lcr-i13-uuDqco' 'lcr-o0-meyhGM'
    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-SJDOfY' 'lcr-o0-8H2u8z'
    Running [N0]: modella 'lcr-i14-bydLwe' 'lcr-o0-cByro8'
    Performing sanity checks and gathering statistics...

    Mutation 4/7: | p0 G ! p0
    Running [P0]: modella 'lcr-i15-nnG5P2' 'lcr-o0-s4i8Yc'
    Running [N0]: modella 'lcr-i15-mQZx9V' 'lcr-o0-epU3GL'
    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-i16-5ZLVrk' 'lcr-o0-APOfwO'
    Running [N0]: modella 'lcr-i16-slHxSM' 'lcr-o0-SC5EZK'
    Performing sanity checks and gathering statistics...

    Mutation 7/7: | G p0 & p0 F p0
    Running [P0]: modella 'lcr-i17-oI6mox' 'lcr-o0-Rmr6pS'
    Running [N0]: modella 'lcr-i17-Dh6HIV' 'lcr-o0-gSnVHB'
    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-o0-2VtW8I'
    Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-sWd2CR'
    Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-Dkl2SA'
    Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-VOxazX'
    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: 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 is displayed. For instance, we can see that although `ltl3ba -H1` outputs co-Büchi alternating automata, only automaton `N1` uses universal edges: the automaton `P1` can be used like a non-alternating co-Büchi automaton.

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

Then `ltlcross` computes the complement of these four automata.

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

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

Note that if we reduce `ltlcross`'s ability to determinize automata for complementation, the procedure can look slightly more complex:

    ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --determinize-max-states=1 --verbose

    F(G(a))
    Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-PsFNJN'
    Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-rbhOfq'
    Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-kLUSXs'
    Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-Rmwsug'
    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:   P0info:   N0	(1 st.,2 ed.,1 sets) -> (1 st.,2 ed.,1 sets)	Comp(N0)
    info:   P1info:   N1	(2 st.,4 ed.,1 sets) -> (2 st.,4 ed.,1 sets)	Comp(N1)
    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-LLklOk'
    Running [P1]: ltl2tgba -H 'F(G(a))'>'lcr-o1-H1G02f'
    Running [N0]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o0-KIDHj9'
    Running [N1]: ltl2tgba -H '!(F(G(a)))'>'lcr-o1-6FhpvA'
    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: 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 `ltl2tgba` and `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`).