ltl2tgba.org 26.1 KB
Newer Older
1
# -*- coding: utf-8 -*-
2
#+TITLE: =ltl2tgba=
3
4
#+SETUPFILE: setup.org
#+HTML_LINK_UP: tools.html
5
6

This tool translates LTL or PSL formulas into two kinds of Büchi
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
7
8
9
10
automata, or to monitors.  The default is to output Transition-based
Generalized Büchi Automata (hereinafter abbreviated TGBA), but more
traditional Büchi automata (BA) may be requested using the =-B=
option.
11
12
13
14
15
16

* TGBA and BA

Formulas to translate may be specified using [[file:ioltl.org][common input options for
LTL/PSL formulas]].

17
#+BEGIN_SRC sh :results verbatim :exports code
18
19
ltl2tgba -f 'Fa & GFb'
#+END_SRC
20
21
22
#+BEGIN_SRC sh :results verbatim :exports results
SPOT_DOTEXTRA= ltl2tgba -f 'Fa & GFb' --dot=
#+END_SRC
23
24
25
#+RESULTS:
#+begin_example
digraph G {
26
27
28
29
30
31
  rankdir=LR
  I [label="", style=invis, width=0]
  I -> 1
  0 [label="0"]
  0 -> 0 [label="b\n{0}"]
  0 -> 0 [label="!b"]
32
  1 [label="1"]
33
34
  1 -> 0 [label="a"]
  1 -> 1 [label="!a"]
35
36
37
38
39
40
41
42
43
}
#+end_example

Actually, because =ltl2tgba= is often used with a single formula
passed on the command line, the =-f= option can be omitted and any
command-line parameter that is not the argument of some option will be
assumed to be a formula to translate (this differs from [[file:ltlfilt.org][=ltlfilt=]],
where such parameters are assumed to be filenames).

44
=ltl2tgba= honors the [[file:aout.org][common options for selecting the output format]].
45
46
47
48
49
50
51
52
The default output format, as shown above, is [[http://http://www.graphviz.org/][GraphViz]]'s format.  This
can converted into a picture, or into vectorial format using =dot= or
=dotty=.  Typically, you could get a =pdf= of this TGBA using
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba "Fa & GFb" | dot -Tpdf > tgba.pdf
#+END_SRC
#+RESULTS:

53
The result would look like this (note that in this documentation
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
54
we use some [[file:aout.org][environment variables]] to produce a more colorful
55
output by default)
56
57
#+NAME: dotex
#+BEGIN_SRC sh :results verbatim :exports none
58
ltl2tgba "Fa & GFb"
59
60
61
62
#+END_SRC
#+RESULTS: dotex
#+begin_example
digraph G {
63
  rankdir=LR
64
65
66
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
67
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
68
69
70
  I [label="", style=invis, width=0]
  I -> 1
  0 [label="0"]
71
72
  0 -> 0 [label=<b<br/><font color="#5DA5DA">⓿</font>>]
  0 -> 0 [label=<!b>]
73
  1 [label="1"]
74
75
  1 -> 0 [label=<a>]
  1 -> 1 [label=<!a>]
76
77
78
79
80
81
82
83
84
85
}
#+end_example

#+BEGIN_SRC dot :file dotex.png :cmdline -Tpng :var txt=dotex :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:dotex.png]]

86
87
88
89
90
91
92
93
94
95
Characters like ⓿, ❶, etc. denotes the acceptance sets a transition
belongs to.  In this case, there is only one acceptance set, called
=0=, containing a single transition.  You may have many transitions in
the same acceptance set, and a transition may also belong to multiple
acceptance sets.  An infinite path through this automaton is accepting
iff it visit each acceptance set infinitely often.  Therefore, in the
above example, any accepted path will /necessarily/ leave the initial
state after a finite amount of steps, and then it will verify the
property =b= infinitely often.  It is also possible that an automaton
do not use any acceptance set at all, in which any run is accepting.
96
97
98
99
100

Here is a TGBA with multiple acceptance sets (we omit the call to
=dot= to render the output of =ltl2tgba= from now on):

#+NAME: dotex2
101
102
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba "GFa & GFb"
103
104
#+END_SRC
#+RESULTS: dotex2
105
106
107
#+begin_example
digraph G {
  rankdir=LR
108
109
110
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
111
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
112
113
114
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
115
116
117
118
  0 -> 0 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
  0 -> 0 [label=<!a &amp; !b>]
  0 -> 0 [label=<!a &amp; b<br/><font color="#F17CB0">❶</font>>]
  0 -> 0 [label=<a &amp; !b<br/><font color="#5DA5DA">⓿</font>>]
119
120
}
#+end_example
121
122
123
124
125
126
127

#+BEGIN_SRC dot :file dotex2.png :cmdline -Tpng :var txt=dotex2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2.png]]

128
The above TGBA has two acceptance sets: ⓿ and ❶.  The position of
129
130
these acceptance sets ensures that atomic propositions =a= and =b= must
be true infinitely often.
131
132
133
134

A Büchi automaton for the previous formula can be obtained with the
=-B= option:

135
#+NAME: dotex2ba
136
137
138
139
140
141
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B 'GFa & GFb'
#+END_SRC
#+RESULTS: dotex2ba
#+begin_example
digraph G {
142
  rankdir=LR
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
143
  node [shape="circle"]
144
145
146
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
147
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
148
149
  I [label="", style=invis, width=0]
  I -> 0
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
150
  0 [label="0", peripheries=2]
151
152
153
  0 -> 0 [label=<a &amp; b>]
  0 -> 1 [label=<!b>]
  0 -> 2 [label=<!a &amp; b>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
154
  1 [label="1"]
155
156
157
  1 -> 0 [label=<a &amp; b>]
  1 -> 1 [label=<!b>]
  1 -> 2 [label=<!a &amp; b>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
158
  2 [label="2"]
159
160
  2 -> 0 [label=<a>]
  2 -> 2 [label=<!a>]
161
162
163
164
165
166
167
168
169
}
#+end_example

#+BEGIN_SRC dot :file dotex2ba.png :cmdline -Tpng :var txt=dotex2ba :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2ba.png]]

Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
170
171
172
173
174
175
176
Although accepting states in the Büchi automaton are (traditionally)
pictured with double-lines, internally this automaton is still handled
as a TGBA with a single acceptance set such that the transitions
leaving the state are either all accepting, or all non-accepting.  You
can see this underlying TGBA if you pass the =--dot=t= option (the =t=
requests the use of transition-based acceptance as it is done
internally):
177

178
179
180
181
182
183
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba --dot=t -B 'GFa & GFb'
#+END_SRC

#+NAME: dotex2ba-t
#+BEGIN_SRC sh :results verbatim :exports none
184
ltl2tgba --dot=.t -B 'GFa & GFb'
185
186
187
188
189
190
#+END_SRC

#+RESULTS: dotex2ba-t
#+begin_example
digraph G {
  rankdir=LR
191
192
193
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
194
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
195
196
197
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
198
199
200
  0 -> 0 [label=<a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
  0 -> 1 [label=<!b<br/><font color="#5DA5DA">⓿</font>>]
  0 -> 2 [label=<!a &amp; b<br/><font color="#5DA5DA">⓿</font>>]
201
  1 [label="1"]
202
203
204
  1 -> 0 [label=<a &amp; b>]
  1 -> 1 [label=<!b>]
  1 -> 2 [label=<!a &amp; b>]
205
  2 [label="2"]
206
207
  2 -> 0 [label=<a>]
  2 -> 2 [label=<!a>]
208
209
210
211
212
213
214
215
216
217
218
}
#+end_example

#+BEGIN_SRC dot :file dotex2ba-t.png :cmdline -Tpng :var txt=dotex2ba-t :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2ba-t.png]]

As already discussed on the page about [[file:oaut.org][common output options]], various
options controls the output format of =ltl2tgba=:
219
220
221
222
223

#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/Output format:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
224
225
226
#+begin_example
  -8, --utf8                 enable UTF-8 characters in output (ignored with
                             --lbtt or --spin)
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
      --check[=PROP]         test for the additional property PROP and output
                             the result in the HOA format (implies -H).  PROP
                             may be any prefix of 'all' (default),
                             'unambiguous', or 'stutter-invariant'.
      --dot[=1|a|b|B|c|e|f(FONT)|h|n|N|o|r|R|s|t|v]
                             GraphViz's format (default).  Add letters for (1)
                             force numbered states, (a) acceptance display, (b)
                             acceptance sets as bullets, (B) bullets except for
                             Büchi/co-Büchi automata, (c) force circular
                             nodes, (e) force elliptic nodes, (f(FONT)) use
                             FONT, (h) horizontal layout, (v) vertical layout,
                             (n) with name, (N) without name, (o) ordered
                             transitions, (r) rainbow colors for acceptance
                             sets, (R) color acceptance sets by Inf/Fin, (s)
                             with SCCs, (t) force transition-based acceptance.
242
243
244
245
246
247
248
  -H, --hoaf[=i|s|t|m|l]     Output the automaton in HOA format.  Add letters
                             to select (i) use implicit labels for complete
                             deterministic automata, (s) prefer state-based
                             acceptance when possible [default], (t) force
                             transition-based acceptance, (m) mix state and
                             transition-based acceptance, (l) single-line
                             output
249
      --lbtt[=t]             LBTT's format (add =t to force transition-based
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
250
                             acceptance even on Büchi automata)
251
      --name=FORMAT          set the name of the output automaton
252
253
254
  -o, --output=FORMAT        send output to a file named FORMAT instead of
                             standard output.  The first automaton sent to a
                             file truncates it unless FORMAT starts with '>>'.
255
  -q, --quiet                suppress all normal output
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
256
257
258
  -s, --spin[=6|c]           Spin neverclaim (implies --ba).  Add letters to
                             select (6) Spin's 6.2.4 style, (c) comments on
                             states
259
260
      --stats=FORMAT         output statistics about the automaton
#+end_example
261

Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
262
Option =-8= can be used to improve the readability of the output
263
264
265
if your system can display UTF-8 correctly.

#+NAME: dotex2ba8
266
267
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B8 "GFa & GFb"
268
269
270
271
#+END_SRC
#+RESULTS: dotex2ba8
#+begin_example
digraph G {
272
  rankdir=LR
273
274
275
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
276
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
277
278
  I [label="", style=invis, width=0]
  I -> 0
279
  0 [label=<0<br/><font color="#5DA5DA">⓿</font>>]
280
281
282
  0 -> 0 [label=<a∧b>]
  0 -> 1 [label=<b̅>]
  0 -> 2 [label=<a̅∧b>]
283
  1 [label=<1>]
284
285
286
  1 -> 0 [label=<a∧b>]
  1 -> 1 [label=<b̅>]
  1 -> 2 [label=<a̅∧b>]
287
  2 [label=<2>]
288
289
  2 -> 0 [label=<a>]
  2 -> 2 [label=<a̅>]
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
}
#+end_example

#+BEGIN_SRC dot :file dotex2ba8.png :cmdline -Tpng :var txt=dotex2ba8 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:dotex2ba8.png]]

* Spin output

Using the =--spin= or =-s= option, =ltl2tgba= will produce a Büchi automaton
(the =-B= option is implied) as a never claim that can be fed to Spin.
=ltl2tgba -s= is therefore a drop-in replacement for =spin -f=.


#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -s 'GFa & GFb'
#+END_SRC
#+RESULTS:
#+begin_example
never { /* G(Fa & Fb) */
accept_init:
  if
  :: ((a) && (b)) -> goto accept_init
315
316
  :: ((!(b))) -> goto T0_S2
  :: ((!(a)) && (b)) -> goto T0_S3
317
318
319
  fi;
T0_S2:
  if
320
321
322
  :: ((a) && (b)) -> goto accept_init
  :: ((!(b))) -> goto T0_S2
  :: ((!(a)) && (b)) -> goto T0_S3
323
324
325
  fi;
T0_S3:
  if
326
327
  :: ((a)) -> goto accept_init
  :: ((!(a))) -> goto T0_S3
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
  fi;
}
#+end_example

Since Spin 6 extended its syntax to support arbitrary atomic
propositions, you may also need put the parser in =--lenient= mode to
support these:

#+BEGIN_SRC sh :results verbatim :exports both
ltl2tgba -s --lenient '(a < b) U (process[2]@ok)'
#+END_SRC
#+RESULTS:
: never { /* "a < b" U "process[2]@ok" */
: T0_init:
:   if
:   :: ((process[2]@ok)) -> goto accept_all
:   :: ((a < b) && (!(process[2]@ok))) -> goto T0_init
:   fi;
: accept_all:
:   skip
: }

* Do you favor deterministic or small automata?

The translation procedure can be controled by a few switches.  A first
set of options specifies the intent of the translation: whenever
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
354
355
possible, would you prefer a small automaton (=--small=) or a
deterministic (=--deterministic=) automaton?
356
357
358
359
360

#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/Translation intent:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
361
362
:   -a, --any                  no preference, do not bother making it small or
:                              deterministic
363
364
:   -C, --complete             output a complete automaton (combine with other
:                              intents)
365
366
:   -D, --deterministic        prefer deterministic automata
:       --small                prefer small automata (default)
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
367
368
:   -U, --unambiguous          output unambiguous automata (combine with other
:                              intents)
369

Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
370
371
372
373
The =--any= option tells the translator that it should attempt to
reduce or produce a deterministic result result: any automaton
denoting the given formula is OK.  This effectively disables
post-processings and speeds up the translation.
374

Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
375
376
377
378
379
With the =-D= or =--deterministic= option, the translator will
/attempt/ to produce a deterministic automaton, even if this requires
a lot of states.  =ltl2tgba= knows how to produce the minimal
deterministic Büchi automaton for any obligation property (this
includes safety properties).
380
381
382
383
384

With the =--small= option (the default), the translator will not
produce a deterministic automaton when it knows how to build smaller
automaton.

Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
385
386
387
388
Note that options =--deterministic= and =--small= express
/preferences/.  They certainly do /not/ guarantee that the output will
be deterministic, or will be the smallest automaton possible.

389
390
391
392
An example formula where the difference between =-D= and =--small= is
flagrant is =Ga|Gb|Gc=:

#+NAME: gagbgc1
393
#+BEGIN_SRC sh :results verbatim :exports code
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
394
ltl2tgba 'Ga|Gb|Gc'
395
396
397
398
#+END_SRC
#+RESULTS: gagbgc1
#+begin_example
digraph G {
399
  rankdir=LR
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
400
  node [shape="circle"]
401
402
403
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
404
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
405
406
  I [label="", style=invis, width=0]
  I -> 0
407
  0 [label=<0>]
408
409
410
  0 -> 1 [label=<a>]
  0 -> 2 [label=<b>]
  0 -> 3 [label=<c>]
411
  1 [label=<1>]
412
  1 -> 1 [label=<a>]
413
  2 [label=<2>]
414
  2 -> 2 [label=<b>]
415
  3 [label=<3>]
416
  3 -> 3 [label=<c>]
417
418
419
420
421
422
423
424
425
}
#+end_example

#+BEGIN_SRC dot :file gagbgc1.png :cmdline -Tpng :var txt=gagbgc1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:gagbgc1.png]]

426
#+NAME: gagbgc2
427
428
429
430
431
432
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -D 'Ga|Gb|Gc'
#+END_SRC
#+RESULTS: gagbgc2
#+begin_example
digraph G {
433
  rankdir=LR
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
434
  node [shape="circle"]
435
436
437
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
438
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
439
440
  I [label="", style=invis, width=0]
  I -> 6
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
441
  0 [label="0", peripheries=2]
442
  0 -> 0 [label=<c>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
443
  1 [label="1", peripheries=2]
444
445
446
  1 -> 0 [label=<!b &amp; c>]
  1 -> 1 [label=<b &amp; c>]
  1 -> 2 [label=<b &amp; !c>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
447
  2 [label="2", peripheries=2]
448
  2 -> 2 [label=<b>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
449
  3 [label="3", peripheries=2]
450
451
452
  3 -> 2 [label=<!a &amp; b>]
  3 -> 3 [label=<a &amp; b>]
  3 -> 5 [label=<a &amp; !b>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
453
  4 [label="4", peripheries=2]
454
455
456
  4 -> 0 [label=<!a &amp; c>]
  4 -> 4 [label=<a &amp; c>]
  4 -> 5 [label=<a &amp; !c>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
457
  5 [label="5", peripheries=2]
458
  5 -> 5 [label=<a>]
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
459
  6 [label="6", peripheries=2]
460
461
462
463
464
465
466
  6 -> 0 [label=<!a &amp; !b &amp; c>]
  6 -> 1 [label=<!a &amp; b &amp; c>]
  6 -> 2 [label=<!a &amp; b &amp; !c>]
  6 -> 3 [label=<a &amp; b &amp; !c>]
  6 -> 4 [label=<a &amp; !b &amp; c>]
  6 -> 5 [label=<a &amp; !b &amp; !c>]
  6 -> 6 [label=<a &amp; b &amp; c>]
467
468
469
470
471
472
473
474
475
476
477
478
479
}
#+end_example

#+BEGIN_SRC dot :file gagbgc2.png :cmdline -Tpng :var txt=gagbgc2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:gagbgc2.png]]

You can augment the number of terms in the disjunction to magnify the
difference.  For N terms, the =--small= automaton has N+1 states,
while the =--deterministic= automaton needs 2^N-1 states.

Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
480
Add the =-C= or =--complete= option if you want to obtain a complete
481
482
483
automaton, with a sink state capturing that rejected words that would
not otherwise have a run in the output automaton.

Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
Add the =-U= or =--unambiguous= option if you want unambiguous
automata to be produced.  An automaton is unambiguous if any word is
recognized by at most one accepting run of the automaton (however a
word can be rejected by multiple runs, so unambiguous automata can be
non-deterministic).

The following example is an ambiguous Büchi automaton, because the are
two ways to accept a run that repeats continuously the configuration
$\bar ab$.

#+NAME: ambig1
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B 'GFa -> GFb'
#+END_SRC
#+RESULTS: ambig1
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 1
  0 [label="0", peripheries=2]
  0 -> 0 [label=<!a>]
  1 [label="1"]
  1 -> 0 [label=<!a>]
  1 -> 1 [label=<1>]
  1 -> 2 [label=<b>]
  2 [label="2", peripheries=2]
  2 -> 2 [label=<b>]
  2 -> 3 [label=<!b>]
  3 [label="3"]
  3 -> 2 [label=<b>]
  3 -> 3 [label=<!b>]
}
#+end_example

#+BEGIN_SRC dot :file ambig1.png :cmdline -Tpng :var txt=ambig1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:ambig1.png]]

Here is an unambiguous automaton for the same formula, in which there
is only one run that recognizes this example word:

#+NAME: ambig2
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -B -U 'GFa -> GFb'
#+END_SRC

#+RESULTS: ambig2
#+begin_example
digraph G {
  rankdir=LR
  node [shape="circle"]
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
  I [label="", style=invis, width=0]
  I -> 0
  0 [label="0"]
  0 -> 1 [label=<!a &amp; !b>]
  0 -> 2 [label=<1>]
  0 -> 3 [label=<a | b>]
  0 -> 4 [label=<!a &amp; !b>]
  1 [label="1", peripheries=2]
  1 -> 1 [label=<!a &amp; !b>]
  2 [label="2", peripheries=2]
  2 -> 2 [label=<b>]
  2 -> 5 [label=<!b>]
  3 [label="3"]
  3 -> 1 [label=<!a &amp; !b>]
  3 -> 3 [label=<a | b>]
  3 -> 4 [label=<!a &amp; !b>]
  4 [label="4"]
  4 -> 3 [label=<a | b>]
  4 -> 4 [label=<!a &amp; !b>]
  5 [label="5"]
  5 -> 2 [label=<b>]
  5 -> 5 [label=<!b>]
}
#+end_example

#+BEGIN_SRC dot :file ambig2.png :cmdline -Tpng :var txt=ambig2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:ambig2.png]]


Unlike =--small= and =--deterministic= that express preferences,
options =--complete= and =--unambiguous= do guarantee that the output
will be complete and unambiguous.

584

585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
A last parameter that can be used to tune the translation is the amount
of pre- and post-processing performed.  These two steps can be adjusted
via a common set of switches:
#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/Optimization level:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
:       --high                 all available optimizations (slow, default)
:       --low                  minimal optimizations (fast)
:       --medium               moderate optimizations

Pre-processings are rewritings done on the LTL formulas, usually to
reduce its size, but mainly to put it in a form that will help the
translator (for instance =F(a|b)= is easier to translate than
=F(a)|F(b)=).  At =--low= level, only simple syntactic rewritings are
performed.  At =--medium= level, additional simplifications based on
syntactic implications are performed.  At =--high= level, language
containment is used instead of syntactic implications.

Post-processings are cleanups and simplifications of the automaton
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
605
606
produced by the core translator.  The algorithms used during
post-processing are
607
608
609
610
611
612
613
614
- SCC filtering: removing useless strongly connected components,
  and useless acceptance sets.
- direct simulation: merge states based on suffix inclusion.
- iterated simulations: merge states based on suffix inclusion,
  or prefix inclusion, in a loop.
- WDBA minimization: determinize and minimize automata representing
  obligation properties.
- degeneralization: convert a TGBA into a BA
Alexandre Duret-Lutz's avatar
Alexandre Duret-Lutz committed
615
- BA simulation (again direct or iterated)
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656

The chaining of these various algorithms depends on the selected
combination of optimization level (=--low=, =--medium=, =--high=),
translation intent (=--small=, =--deterministic=) and type of
automaton desired (=--tgba=, =--ba=).

A notable configuration is =--any --low=, which will produce a TGBA as
fast as possible.  In this case, post-processing is disabled, and only
syntactic rewritings are performed.  This can be used for
satisfiability checking, although in this context even building an
automaton is overkill (you only need an accepted run).

Finally, it should be noted that the default optimization options
(=--small --high=) are usually overkill.  =--low= will produce good
automata most of the time.  Most of pattern formulas of [[file:genltl.org][=genltl=]] will
be efficiently translated in this configuration (meaning that =--small
--high= will not produce a better automaton).  If you are planning to
generate automata for large family of pattern formulas, it makes sense
to experiment with the different settings on a small version of the
pattern, and select the lowest setting that satisfies your
expectations.

* Translating multiple formulas for statistics

If multiple formulas are given to =ltl2tgba=, the corresponding
automata will be output one after the other.  This is not very
convenient, since most of these output formats are not designed to
represent multiple automata, and tools like =dot= will only display
the first one.

One situation where passing many formulas to =ltl2tgba= is useful is
in combination with the =--stats=FORMAT= option.  This option will
output statistics about the translated automata instead of the
automata themselves.  The =FORMAT= string should indicate which
statistics should be output, and how they should be output using the
following sequence of characters (other characters are output as-is):

#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --help | sed -n '/^ *%/p'
#+END_SRC
#+RESULTS:
657
658
659
660
#+begin_example
  %%                         a single %
  %a                         number of acceptance sets
  %c                         number of SCCs
661
  %d                         1 if the output is deterministic, 0 otherwise
662
663
  %e                         number of edges
  %f                         the formula, in Spot's syntax
664
665
666
667
668
669
  %F                         name of the input file
  %L                         location in the input file
  %m                         name of the automaton
  %n                         number of nondeterministic states in output
  %p                         1 if the output is complete, 0 otherwise
  %r                         processing time (excluding parsing) in seconds
670
671
  %s                         number of states
  %t                         number of transitions
672
  %w                         one word accepted by the output automaton
673
#+end_example
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705

For instance we can study the size of the automata generated for the
right-nested =U= formulas as follows:

#+BEGIN_SRC sh :results verbatim :exports both
genltl --u-right=1..8 | ltl2tgba -F - --stats '%s states and %e edges for "%f"'
#+END_SRC
#+RESULTS:
: 2 states and 2 edges for "p1"
: 2 states and 3 edges for "p1 U p2"
: 3 states and 6 edges for "p1 U (p2 U p3)"
: 4 states and 10 edges for "p1 U (p2 U (p3 U p4))"
: 5 states and 15 edges for "p1 U (p2 U (p3 U (p4 U p5)))"
: 6 states and 21 edges for "p1 U (p2 U (p3 U (p4 U (p5 U p6))))"
: 7 states and 28 edges for "p1 U (p2 U (p3 U (p4 U (p5 U (p6 U p7)))))"
: 8 states and 36 edges for "p1 U (p2 U (p3 U (p4 U (p5 U (p6 U (p7 U p8))))))"

Here =-F -= means that formulas should be read from the standard input.

When computing the size of an automaton, we distinguish /transitions/
and /edges/.  An edge between two states is labeled by a Boolean
formula and may in fact represent several transitions labeled by
compatible Boolean assignment.

For instance if the atomic propositions are =x= and =y=, an edge labeled
by the formula =!x= actually represents two transitions labeled respectively
with =!x&y= and =!x&!y=.

Two automata with the same structures (states and edges) but differing
labels, may have a different count of transitions, e.g., if one has
more restricted labels.

706
707
708
[[file:csv.org][More examples of how to use =--stats= to create CSV
files are on a separate page]].

709
710
711
712
713
714
715
716
717
718
719
720
* Building Monitors

In addition to TGBA and BA, =ltl2tgba= can output /monitor/ using the
=-M= option.  These are finite automata that accept all prefixes of a
formula.  The idea is that you can use these automata to monitor a
system as it is running, and report a violation as soon as no
compatible outgoing transition exist.

=ltl2tgba -M= may output non-deterministic monitors while =ltl2tgba
-MD= (short for =--monitor --deterministic=) will output the minimal
deterministic monitor for the given formula.

721
#+NAME: monitor1
722
723
724
725
726
727
728
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba -M '(Xa & Fb) | Gc'
#+END_SRC

#+RESULTS: monitor1
#+begin_example
digraph G {
729
  rankdir=LR
730
731
732
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
733
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
734
735
  I [label="", style=invis, width=0]
  I -> 0
736
  0 [label=<0>]
737
738
  0 -> 1 [label=<1>]
  0 -> 3 [label=<c>]
739
  1 [label=<1>]
740
  1 -> 2 [label=<a>]
741
  2 [label=<2>]
742
  2 -> 2 [label=<1>]
743
  3 [label=<3>]
744
  3 -> 3 [label=<c>]
745
746
747
748
749
750
751
752
753
754
}
#+end_example

#+BEGIN_SRC dot :file monitor1.png :cmdline -Tpng :var txt=monitor1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:monitor1.png]]

755
#+NAME: monitor2
756
#+BEGIN_SRC sh :results verbatim :exports code
757
ltl2tgba -MD '(Xa & Fb) | Gc'
758
759
760
761
762
#+END_SRC

#+RESULTS: monitor2
#+begin_example
digraph G {
763
  rankdir=LR
764
765
766
  fontname="Lato"
  node [fontname="Lato"]
  edge [fontname="Lato"]
767
  node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
768
769
  I [label="", style=invis, width=0]
  I -> 3
770
  0 [label=<0>]
771
  0 -> 0 [label=<1>]
772
  1 [label=<1>]
773
  1 -> 0 [label=<a>]
774
  2 [label=<2>]
775
  2 -> 2 [label=<c>]
776
  3 [label=<3>]
777
778
  3 -> 1 [label=<!c>]
  3 -> 4 [label=<c>]
779
  4 [label=<4>]
780
781
  4 -> 0 [label=<a>]
  4 -> 2 [label=<!a &amp; c>]
782
783
784
785
786
787
788
789
790
791
792
793
}
#+end_example

#+BEGIN_SRC dot :file monitor2.png :cmdline -Tpng :var txt=monitor2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:monitor2.png]]

Because they accept all finite executions that could be extended to
match the formula, monitor cannot be used to check for eventualities
794
795
such as =F(a)=: indeed, any finite execution can be extended to match
=F(a)=.
796
797
798
799
800
801
802
803
804

#  LocalWords:  ltl tgba num toc PSL Büchi automata SRC GFb invis Acc
#  LocalWords:  ltlfilt filenames GraphViz vectorial pdf Tpdf dotex
#  LocalWords:  sed png cmdline Tpng txt iff GFa ba utf UTF lbtt Fb
#  LocalWords:  GraphViz's LBTT's neverclaim SPOT's init goto fi Gb
#  LocalWords:  controled Gc gagbgc disjunction pre rewritings SCC Xa
#  LocalWords:  WDBA determinize degeneralization satisfiability SCCs
#  LocalWords:  genltl nondeterministic eval setenv concat getenv
#  LocalWords:  setq