# -*- coding: utf-8 -*- #+TITLE: Common output options for automata #+DESCRIPTION: Options for input and output of ω-automata in Spot's command-line tools #+SETUPFILE: setup.org #+HTML_LINK_UP: tools.html Spot supports different output syntaxes for automata. This page documents the options, common to all tools where it makes sense, that are used to specify how to output of automata. * Common output options All tools that can output automata implement the following options: #+BEGIN_SRC sh :results verbatim :exports results ltl2tgba --help | sed -n '/Output format:/,/^$/p' | sed '1d;$d' #+END_SRC #+RESULTS: #+begin_example -8, --utf8 enable UTF-8 characters in output (ignored with --lbtt or --spin) --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', 'stutter-invariant', or 'strength'. -d, --dot[=1|a|b|B|c|e|f(FONT)|h|n|N|o|r|R|s|t|v|+INT] GraphViz's format. 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, (+INT) add INT to all set numbers -H, --hoaf[=i|l|m|s|t|v] Output the automaton in HOA format (default). 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, (k) use state labels when possible, (l) single-line output, (v) verbose properties --lbtt[=t] LBTT's format (add =t to force transition-based acceptance even on Büchi automata) --name=FORMAT set the name of the output automaton -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 '>>'. -q, --quiet suppress all normal output -s, --spin[=6|c] Spin neverclaim (implies --ba). Add letters to select (6) Spin's 6.2.4 style, (c) comments on states --stats=FORMAT output statistics about the automaton #+end_example The main three output formats (that can also been used as input to some of the tools) are [[http://adl.github.io/hoaf/][HOA]] (used by default, or with =-H= or =--hoaf=), [[http://www.tcs.hut.fi/Software/lbtt/doc/html/Format-for-automata.html][LBTT]] (activated by =--lbtt=), or Spin [[http://spinroot.com/spin/Man/never.html][never claims]] (activated by =-s= or =--spin=). These three formats also support *streaming*, i.e., you can concatenate multiple automata (and even mix these three formats in the same stream), and the tools will be able to read and process them in sequence. The other possible outputs are [[http://www.graphviz.org/][GraphViz]] output (=-d= or =--dot=), various statistics (=--stats=), or nothing at all (=--quiet=). It may seem strange to ask a tool to not output anything, but it makes sense when only the exit status matters (for instance using [[file:autfilt.org][=autfilt=]] to check whether an input automaton has some property) or for timing purposes. * HOA output Details about supported features of the HOA format can be found on a [[file:hoa.org][separate page]]. The [[http://adl.github.io/hoaf/][HOA]] output should be the preferred format to use if you want to pass automata between different tools. Since Spot 1.99.7, it is the default output format, but you can explicitely request it using the =-H= parameter and this allows passing additional options to the HOA printer. Here is an example where [[file:ltl2tgba.org][=ltl2tgba=]] is used to construct two automata: one for =a U b= and one for =(Ga -> Gb) W c=. #+BEGIN_SRC sh :results verbatim :exports both :wrap SRC hoa ltl2tgba 'a U b' '(Ga -> Gb) W c' #+END_SRC #+RESULTS: #+BEGIN_SRC hoa HOA: v1 name: "a U b" States: 2 Start: 1 AP: 2 "a" "b" acc-name: Buchi Acceptance: 1 Inf(0) properties: trans-labels explicit-labels state-acc deterministic properties: stutter-invariant terminal --BODY-- State: 0 {0} [t] 0 State: 1 [1] 0 [0&!1] 1 --END-- HOA: v1 name: "(Gb | F!a) W c" States: 5 Start: 1 AP: 3 "b" "a" "c" acc-name: Buchi Acceptance: 1 Inf(0) properties: trans-labels explicit-labels trans-acc stutter-invariant --BODY-- State: 0 [0] 0 {0} State: 1 [0&1&!2] 0 [!1&!2] 1 {0} [1&!2] 2 [2] 3 State: 2 [!1&!2] 1 {0} [1&!2] 2 [!1&2] 3 [1&2] 4 State: 3 [t] 3 {0} State: 4 [!1] 3 [1] 4 --END-- #+END_SRC The above output contains two automata, named after the formulas they represent. Here is a picture of these two automata: #+NAME: hoafex #+BEGIN_SRC sh :results verbatim :exports none ltl2tgba --dot=.cn '(Ga -> Gb) W c' 'a U b' | dot | gvpack | perl -pe 's/\\\n//g;s/\\/\\\\/g;s/graph G/graph cluster/g' #+END_SRC #+RESULTS: hoafex #+begin_example digraph root { graph [bb="0,0,427,231.07", fontname=Lato, labelloc=t, lheight=0.21, rankdir=LR ]; node [fillcolor="#ffffa0", fontname=Lato, label="\\N", shape=circle, style=filled ]; edge [fontname=Lato]; subgraph cluster { graph [bb="", fontname=Lato, label=<(Gb | F!a) W c>, labelloc=t, lheight=0.21, lp="197.5,196.66", lwidth=1.19, rankdir=LR ]; node [fillcolor="#ffffa0", fontname=Lato, height="", label="\\N", pos="", shape=circle, style=filled, width="" ]; edge [fontname=Lato, label="", lp="", pos="" ]; I [height=0.013889, label="", pos="1,49.168", style=invis, width=0.013889]; 1 [height=0.5, label=1, pos="56,49.168", width=0.5]; I -> 1 [pos="e,37.942,49.324 1.1549,49.324 2.6725,49.324 15.097,49.324 27.628,49.324"]; 1 -> 1 [label=⓿>, lp="56,100.32", pos="e,62.379,66.362 49.621,66.362 48.319,76.182 50.445,85.324 56,85.324 59.472,85.324 61.604,81.753 62.398,76.677"]; 0 [height=0.5, label=0, pos="190,121.17", width=0.5]; 1 -> 0 [label=, lp="123,113.83", pos="e,172.99,115.19 70.127,60.572 76.491,65.727 84.391,71.704 92,76.324 115.21,90.42 143.57,103.1 163.61,111.38"]; 2 [height=0.5, label=2, pos="190,34.168", width=0.5]; 1 -> 2 [label=, lp="123,64.824", pos="e,175.09,44.492 73.8,53.268 93.402,57.17 126.62,61.596 154,54.324 158.19,53.213 162.39,51.47 166.37,49.467"]; 3 [height=0.5, label=3, pos="377,34.168", width=0.5]; 1 -> 3 [label=, lp="242,9.8246", pos="e,361.03,25.984 66.027,34.327 72.161,25.632 81.127,15.423 92,10.325 114.02,0 277.48,0.3418 312,7.3246 325.76,10.108 340.24,15.943 351.94,21.478"]; 0 -> 0 [label=⓿>, lp="190,172.33", pos="e,198.98,137.24 181.02,137.24 178.68,147.48 181.67,157.33 190,157.33 195.47,157.33 198.63,153.08 199.5,147.28"]; 2 -> 1 [label=⓿>, lp="123,35.324", pos="e,68.596,36.186 173.36,26.591 167.44,24.066 160.55,21.587 154,20.324 126.94,15.113 117.92,10.98 92,20.324 86.432,22.331 81.123,25.651 76.398,29.343"]; 2 -> 2 [label=, lp="190,77.824", pos="e,198.98,50.24 181.02,50.24 178.68,60.475 181.67,70.324 190,70.324 195.47,70.324 198.63,66.083 199.5,60.274"]; 2 -> 3 [label=, lp="294,105.83", pos="e,365.94,48.712 203.15,46.686 218.52,61.348 246.56,84.98 276,94.324 291.25,99.165 297.12,100.21 312,94.324 331.12,86.764 347.87,70.495 359.43,56.803"]; 4 [height=0.5, label=4, pos="294,34.168", width=0.5]; 2 -> 4 [label=, lp="242,41.824", pos="e,275.95,34.324 208.3,34.324 224.08,34.324 247.64,34.324 265.91,34.324"]; 3 -> 3 [label=<1
⓿>, lp="377,85.324", pos="e,384.03,50.989 369.97,50.989 368.41,60.949 370.75,70.324 377,70.324 381,70.324 383.4,66.477 384.2,61.093"]; 4 -> 3 [label=, lp="335.5,41.824", pos="e,358.85,34.324 312.18,34.324 322.81,34.324 336.69,34.324 348.8,34.324"]; 4 -> 4 [label=, lp="294,77.824", pos="e,301.03,50.989 286.97,50.989 285.41,60.949 287.75,70.324 294,70.324 298,70.324 300.4,66.477 301.2,61.093"]; } subgraph cluster_gv1 { graph [bb="", fontname=Lato, label=, labelloc=t, lheight=0.21, lp="81.5,88.5", lwidth=0.47, rankdir=LR ]; node [fillcolor="#ffffa0", fontname=Lato, height="", label="\\N", peripheries="", pos="", shape=circle, style=filled, width="" ]; edge [fontname=Lato, label="", lp="", pos="" ]; I_gv1 [height=0.013889, label="", pos="261,156.17", style=invis, width=0.013889]; "1_gv1" [height=0.5, label=1, pos="316,156.17", width=0.5]; I_gv1 -> "1_gv1" [pos="e,297.94,156.17 261.15,156.17 262.67,156.17 275.1,156.17 287.63,156.17"]; "1_gv1" -> "1_gv1" [label=, lp="316,199.67", pos="e,322.38,173.21 309.62,173.21 308.32,183.03 310.44,192.17 316,192.17 319.47,192.17 321.6,188.6 322.4,183.52"]; "0_gv1" [height=0.72222, label=0, peripheries=2, pos="401,156.17", width=0.72222]; "1_gv1" -> "0_gv1" [label=, lp="356.5,163.67", pos="e,379,156.17 334.2,156.17 344.16,156.17 357,156.17 368.7,156.17"]; "0_gv1" -> "0_gv1" [label=1, lp="401,203.67", pos="e,409.01,176.75 392.99,176.75 391.89,187.01 394.55,196.17 401,196.17 405.13,196.17 407.71,192.41 408.74,187.01"]; } } #+end_example #+BEGIN_SRC dot :file hoafex.png :cmdline -Tpng :var txt=hoafex :exports results $txt #+END_SRC #+RESULTS: [[file:hoafex.png]] The [[http://adl.github.io/hoaf/][HOA format]] supports both state and transition-based acceptance. Although Spot works only with transition-based acceptance, its output routines default to state-based acceptance whenever possible (this is the case in the first of these two automata) and use transition-based acceptance otherwise. You can change this behavior using =-Hs= (or =--hoaf=s=), =-Ht=, or =-Hm=. Option =s= corresponds to the default to use state-based acceptance whenever possible. Option =t= forces transition-based acceptance. For instance compare this output to the previous one: #+BEGIN_SRC sh :results verbatim :exports both :wrap SRC hoa ltl2tgba -Ht 'a U b' #+END_SRC #+RESULTS: #+BEGIN_SRC hoa HOA: v1 name: "a U b" States: 2 Start: 1 AP: 2 "a" "b" acc-name: Buchi Acceptance: 1 Inf(0) properties: trans-labels explicit-labels trans-acc deterministic properties: stutter-invariant terminal --BODY-- State: 0 [t] 0 {0} State: 1 [1] 0 [0&!1] 1 --END-- #+END_SRC Option =m= uses mixed acceptance, i.e, some states might use state-based acceptance while other will not: #+BEGIN_SRC sh :results verbatim :exports both :wrap SRC hoa ltl2tgba -Hm '(Ga -> Gb) W c' #+END_SRC #+RESULTS: #+BEGIN_SRC hoa HOA: v1 name: "(Gb | F!a) W c" States: 5 Start: 1 AP: 3 "b" "a" "c" acc-name: Buchi Acceptance: 1 Inf(0) properties: trans-labels explicit-labels stutter-invariant --BODY-- State: 0 {0} [0] 0 State: 1 [0&1&!2] 0 [!1&!2] 1 {0} [1&!2] 2 [2] 3 State: 2 [!1&!2] 1 {0} [1&!2] 2 [!1&2] 3 [1&2] 4 State: 3 {0} [t] 3 State: 4 [!1] 3 [1] 4 --END-- #+END_SRC It is also possible to output each automaton on a single line, in case the result should be used with line-based tools or embedded into a CSV file... Here is an example using both transition-based acceptance, and single-line output: #+BEGIN_SRC sh :results verbatim :exports both :wrap SRC hoa ltl2tgba -Htl 'a U b' '(Ga -> Gb) W c' #+END_SRC #+RESULTS: #+BEGIN_SRC hoa HOA: v1 name: "a U b" States: 2 Start: 1 AP: 2 "a" "b" acc-name: Buchi Acceptance: 1 Inf(0) properties: trans-labels explicit-labels trans-acc deterministic stutter-invariant terminal --BODY-- State: 0 [t] 0 {0} State: 1 [1] 0 [0&!1] 1 --END-- HOA: v1 name: "(Gb | F!a) W c" States: 5 Start: 1 AP: 3 "b" "a" "c" acc-name: Buchi Acceptance: 1 Inf(0) properties: trans-labels explicit-labels trans-acc stutter-invariant --BODY-- State: 0 [0] 0 {0} State: 1 [0&1&!2] 0 [!1&!2] 1 {0} [1&!2] 2 [2] 3 State: 2 [!1&!2] 1 {0} [1&!2] 2 [!1&2] 3 [1&2] 4 State: 3 [t] 3 {0} State: 4 [!1] 3 [1] 4 --END-- #+END_SRC Finally, version 1.1 of the HOA format can be specified using the =-H1.1= option. Version 1, which is currently the default, can also be requested explicitly using =-H1=. The main advantage of version 1.1, as far as Spot is concerned, is that some of negated properties can be transmitted. For instance, compare #+BEGIN_SRC sh :results verbatim :exports both :wrap SRC hoa ltl2tgba -f GFa -f FGa -H1 --check | grep -E '^(HOA|properties|name):' #+END_SRC #+RESULTS: #+BEGIN_SRC hoa HOA: v1 name: "GFa" properties: trans-labels explicit-labels trans-acc complete properties: deterministic stutter-invariant HOA: v1 name: "FGa" properties: trans-labels explicit-labels state-acc stutter-invariant properties: weak #+END_SRC versus #+BEGIN_SRC sh :results verbatim :exports both :wrap SRC hoa ltl2tgba -f GFa -f FGa -H1.1 --check | grep -E '^(HOA|properties|name):' #+END_SRC #+RESULTS: #+BEGIN_SRC hoa HOA: v1.1 name: "GFa" properties: trans-labels explicit-labels trans-acc complete properties: deterministic stutter-invariant !inherently-weak HOA: v1.1 name: "FGa" properties: trans-labels explicit-labels state-acc !complete properties: !deterministic !unambiguous stutter-invariant weak properties: !terminal #+END_SRC The =--check= option inspects the automata for additional properties such that their strength or whether they are stutter-invariant and unambiguous. You can see in this example that version 1.1 of the format carries additional negated properties that could not be represented in the first version. * LBTT output The [[http://www.tcs.hut.fi/Software/lbtt/doc/html/Format-for-automata.html][LBTT]] output has two flavors: state-based (which is used to output Büchi automata or monitors) or transition-based (for TGBA). #+BEGIN_SRC sh :results verbatim :exports both ltl2tgba --ba --lbtt 'p0 U p1' #+END_SRC #+RESULTS: : 2 1 : 0 1 -1 : 1 p1 : 0 & p0 ! p1 : -1 : 1 0 0 -1 : 1 t : -1 If you want to request transition-based output even for Büchi automata, use =--lbtt=t=. #+BEGIN_SRC sh :results verbatim :exports both ltl2tgba --ba --lbtt=t 'p0 U p1' #+END_SRC #+RESULTS: : 2 1t : 0 1 : 1 -1 p1 : 0 -1 & p0 ! p1 : -1 : 1 0 : 1 0 -1 t : -1 Note that the [[http://www.tcs.hut.fi/Software/lbtt/doc/html/Format-for-automata.html][LBTT]] output generalizes the format output by [[http://www.tcs.hut.fi/Software/maria/tools/lbt/][LBT]] with support for transition-based acceptance. Both formats however are restricted to atomic propositions of the form =p0=, =p1=, etc... In case other atomic propositions are used, Spot output them in double quotes. This other extension of the format is also supported by [[http://www.ltl2dstar.de/][ltl2dstar]]. #+BEGIN_SRC sh :results verbatim :exports both ltl2tgba --ba --lbtt 'a U b' #+END_SRC #+RESULTS: : 2 1 : 0 1 -1 : 1 "b" : 0 & "a" ! "b" : -1 : 1 0 0 -1 : 1 t : -1 * Spin output Spin [[http://spinroot.com/spin/Man/never.html][never claims]] can be requested using =-s= or =--spin=. They can only represent Büchi automata, so these options imply =--ba=. #+BEGIN_SRC sh :results verbatim :exports both ltl2tgba -s 'a U b' #+END_SRC #+RESULTS: : never { /* a U b */ : T0_init: : if : :: ((b)) -> goto accept_all : :: ((a) && (!(b))) -> goto T0_init : fi; : accept_all: : skip : } Recent versions of Spin (starting with Spin 6.2.4) output never claims in a slightly different style that can be requested using either =-s6= or =--spin=6=: #+BEGIN_SRC sh :results verbatim :exports both ltl2tgba -s6 'a U b' #+END_SRC #+RESULTS: : never { /* a U b */ : T0_init: : do : :: atomic { ((b)) -> assert(!((b))) } : :: ((a) && (!(b))) -> goto T0_init : od; : accept_all: : skip : } (Note that while Spot is able to read never claims that follow any of these two styles, it is not capable of interpreting an arbitrary piece of Promela syntax.) * Dot output The =-d= or =--dot= option causes automata to be output in GraphViz's format. #+BEGIN_SRC sh :results verbatim :exports code ltl2tgba '(Ga -> Gb) W c' -d #+END_SRC #+NAME: oaut-dot1 #+BEGIN_SRC sh :results verbatim :exports results SPOT_DOTEXTRA= ltl2tgba '(Ga -> Gb) W c' --dot= #+END_SRC #+RESULTS: oaut-dot1 #+begin_example digraph G { rankdir=LR node [shape="circle"] I [label="", style=invis, width=0] I -> 1 0 [label="0"] 0 -> 0 [label="b\n{0}"] 1 [label="1"] 1 -> 0 [label="a & b & !c"] 1 -> 1 [label="!a & !c\n{0}"] 1 -> 2 [label="a & !c"] 1 -> 3 [label="c"] 2 [label="2"] 2 -> 1 [label="!a & !c\n{0}"] 2 -> 2 [label="a & !c"] 2 -> 3 [label="!a & c"] 2 -> 4 [label="a & c"] 3 [label="3"] 3 -> 3 [label="1\n{0}"] 4 [label="4"] 4 -> 3 [label="!a"] 4 -> 4 [label="a"] } #+end_example ** Converting dot output to images or pdf This output should be processed with =dot= to be converted into a picture. For instance use =dot -Tpng= or =dot -Tpdf=. #+BEGIN_SRC dot :file oaut-dot1.png :cmdline -Tpng :var txt=oaut-dot1 :exports results $txt #+END_SRC #+RESULTS: [[file:oaut-dot1.png]] ** Customizing the dot output This output can be customized by passing optional characters to the =--dot= option. For instance =v= requests a vertical layout (instead of the default horizontal layout), =c= requests circle states, =s= causes strongly-connected components to be displayed, =n= causes the name (see below) of the automaton to be displayed, and =a= causes the acceptance condition to be shown as well. Option =b= causes sets to be ouput as bullets (e.g., ⓿ instead of ={0}=); option =r= (for rainbow) causes sets to be displayed in different colors, while option =R= also uses colors, but it chooses them depending on whether a set is used with Fin-acceptance, Inf-acceptance, or both. Option =C(COLOR)= can be used to color all states using =COLOR=, and the option =f(FONT)= is used to select a fontname: it is often necessary when =b= is used to ensure the characters ⓿, ❶, etc. are all selected from the same font. #+BEGIN_SRC sh :results verbatim :exports code ltl2tgba --dot=vcsna '(Ga -> Gb) W c' #+END_SRC #+RESULTS: #+begin_example digraph G { label="(Gb | F!a) W c\nInf(0)\n[Büchi]" labelloc="t" node [shape="circle"] edge[arrowhead=vee, arrowsize=.7] I [label="", style=invis, height=0] I -> 0 subgraph cluster_0 { color=green label="" 1 [label="1"] } subgraph cluster_1 { color=green label="" 2 [label="2"] } subgraph cluster_2 { color=red label="" 4 [label="4"] } subgraph cluster_3 { color=green label="" 0 [label="0"] 3 [label="3"] } 0 -> 0 [label="!a & !c\n{0}"] 0 -> 1 [label="c"] 0 -> 2 [label="a & b & !c"] 0 -> 3 [label="a & !c"] 1 -> 1 [label="1\n{0}"] 2 -> 2 [label="b\n{0}"] 3 -> 0 [label="!a & !c\n{0}"] 3 -> 1 [label="!a & c"] 3 -> 3 [label="a & !c"] 3 -> 4 [label="a & c"] 4 -> 1 [label="!a"] 4 -> 4 [label="a"] } #+end_example #+NAME: oaut-dot2 #+BEGIN_SRC sh :results verbatim :exports none SPOT_DOTEXTRA= ltl2tgba --dot=vcsna '(Ga -> Gb) W c' #+END_SRC #+RESULTS: oaut-dot2 #+begin_example digraph G { label="(Gb | F!a) W c\nInf(0)\n[Büchi]" labelloc="t" node [shape="circle"] I [label="", style=invis, height=0] I -> 0 subgraph cluster_0 { color=green label="" 1 [label="1"] } subgraph cluster_1 { color=green label="" 2 [label="2"] } subgraph cluster_2 { color=red label="" 4 [label="4"] } subgraph cluster_3 { color=green label="" 0 [label="0"] 3 [label="3"] } 0 -> 0 [label="!a & !c\n{0}"] 0 -> 1 [label="c"] 0 -> 2 [label="a & b & !c"] 0 -> 3 [label="a & !c"] 1 -> 1 [label="1\n{0}"] 2 -> 2 [label="b\n{0}"] 3 -> 0 [label="!a & !c\n{0}"] 3 -> 1 [label="!a & c"] 3 -> 3 [label="a & !c"] 3 -> 4 [label="a & c"] 4 -> 1 [label="!a"] 4 -> 4 [label="a"] } #+end_example #+BEGIN_SRC dot :file oaut-dot2.png :cmdline -Tpng :var txt=oaut-dot2 :exports results $txt #+END_SRC #+RESULTS: [[file:oaut-dot2.png]] The acceptance condition is displayed in the same way as in the [[http://adl.github.io/hoaf/][HOA format]]. Here =Inf(0)= means that runs are accepting if and only if they visit some the transitions in the set #0 infinitely often. For well known acceptance conditions (as Büchi in this case), their name is also displayed in bracket below. The strongly connected components are displayed using the following colors: - *green* components contain an accepting cycle - *red* components contain no accepting cycle - *black* components are trivial (i.e., they contain no cycle) - *gray* components are useless (i.e., they are non-accepting, and are only followed by non-accepting components) Here is an example involving all colors: #+NAME: oaut-dot3 #+BEGIN_SRC sh :results verbatim :exports none SPOT_DOTEXTRA= autfilt --dot=cas < 1 subgraph cluster_0 { color=grey label="" 5 [label="5"] 6 [label="6"] } subgraph cluster_1 { color=grey label="" 0 [label="0"] } subgraph cluster_2 { color=green label="" 8 [label="8"] 9 [label="9"] } subgraph cluster_3 { color=green label="" 7 [label="7"] } subgraph cluster_4 { color=black label="" 2 [label="2"] } subgraph cluster_5 { color=red label="" 4 [label="4"] } subgraph cluster_6 { color=green label="" 1 [label="1"] 3 [label="3"] } 0 -> 0 [label="a & b\n{0,1,2}"] 0 -> 0 [label="!a & !b\n{2}"] 0 -> 5 [label="a\n{2}"] 1 -> 4 [label="b"] 1 -> 3 [label="a & !b"] 2 -> 0 [label="a"] 2 -> 7 [label="b"] 3 -> 1 [label="a & b\n{0,1}"] 4 -> 4 [label="!b\n{1,2}"] 4 -> 2 [label="b"] 5 -> 6 [label="1\n{1}"] 6 -> 5 [label="1"] 7 -> 7 [label="!a & b\n{0,2}"] 7 -> 7 [label="a & b\n{0,1}"] 7 -> 8 [label="1"] 8 -> 8 [label="!a & b\n{0,2}"] 8 -> 9 [label="a & b\n{0,1}"] 9 -> 8 [label="!a & b\n{0,1}"] 9 -> 9 [label="a & b\n{0,2}"] } #+end_example #+BEGIN_SRC dot :file oaut-dot3.png :cmdline -Tpng :var txt=oaut-dot3 :exports results $txt #+END_SRC #+RESULTS: [[file:oaut-dot3.png]] <> The dot output can also be customized via two environment variables: - =SPOT_DOTDEFAULT= contains default arguments for the =--dot= option (for when it is used implicitly, or used as just =--dot= without argument). For instance after =export SPOT_DOTDEFAULT=vcsn=, using =--dot= is equivalent to =--dot=vcsn=. However using =--dot=xyz= (for any value of =xyz=, even empty) will ignore the =SPOT_DOTDEFAULT= variable. If the argument of =--dot= contains a dot character, then this dot is replaced by the contents of =SPOT_DOTDEFAULT=. So ~--dot=.a~ would be equivalent to =--dot=vcsna= with our example definition of =SPOT_DOTDEFAULT=. - =SPOT_DOTEXTRA= may contains an arbitrary string that will be emitted in the dot output before the first state. This can be used to modify any attribute. For instance (except for this page, where we had do demonstrate the various options of =--dot=, and a few pages where we show the =--dot= output verbatim) all the automata displayed in this documentation are generated with the following environment variables set: #+BEGIN_SRC sh :results verbatim :exports code export SPOT_DOTDEFAULT='Brf(Lato)C(#ffffa0)' export SPOT_DOTEXTRA='edge[arrowhead=vee, arrowsize=.7]' #+END_SRC ** Working with =dot2tex= :PROPERTIES: :CUSTOM_ID: dot2tex :END: The [[https://github.com/kjellmf/dot2tex][=dot2tex= program]] interacts with GraphViz to converts dot files into TeX figures. The layout is still done by tools provided by GraphViz (i.e. =dot=, =neato=, =circo=, ...) but the actual rendering is done using LaTeX with the TikZ or PSTricks packages. One advantage is that this allows embedding math formulas into the graph, something GraphViz alone cannot do. Another advantage is that you can then easily edit the LaTeX figure, for instance to add additional graphical elements. The =dot= formater of Spot has an option =x=, that is convenient to use with =dot2tex=. This option causes labels to be rendered as LaTeX mathematical formulas instead of ASCII text. #+BEGIN_SRC sh :exports code ltl2tgba 'p0 U p1' --dot=x | dot2tex --autosize --nominsize > out.tex #+END_SRC The above command should give you a LaTeX file that compiles to the following figure: #+BEGIN_SRC sh :results silent :exports results ltl2tgba 'p0 U p1' --dot=x | dot2tex --autosize --nominsize > dot2tex.tex latexmk --pdf dot2tex.tex convert -density 150 -trim dot2tex.pdf dot2tex.png latexmk -C dot2tex.tex rm -f dot2tex.tex #+END_SRC [[file:dot2tex.png]] Caveats: - =dot2tex= should be called with option =--autosize= in order to compute the size of each label before calling GraphViz to layout the graph. This is because GraphViz cannot compute the correct size of mathematical formulas. Unfortunately, the release of =dot2tex 2.9.0= contains a bug where sizes are intepreted as integers instead of floats. This can cause labels or shapes to disappear. This bug of =dot2tex= was fixed in 2014, but at the time of writing (summer 2017) no new release of =dot2tex= has been made. To work around this, make sure you install =dot2tex= from its git repository: #+BEGIN_SRC sh git clone https://github.com/kjellmf/dot2tex.git cd dot2tex sudo python setup.py install #+END_SRC - The default size of nodes seems slightly too big for our usage. Using =--nominsize= is just one way around it. Refer to the [[https://dot2tex.readthedocs.io/en/latest/][=dot2tex= manual]] for finer ways to set the node size. - Currently the =x= option of Spot's =--dot= output cannot yet be combined with the =r=, =R=, an =b= options used to display colored bullets. (Patches are welcome.) * Statistics :PROPERTIES: :CUSTOM_ID: stats :END: The =--stats= option takes format string parameter to specify what and how statistics should be output. Most tools support a common set of statistics about the output automaton (like =%s= for the number of states, =%t= for transitions, =%e= for edges, etc.). Additional statistics might be available depending on what the tool does (for instance [[file:autfilt.org][=autfilt=]] also uses capitaized letters =%S=, =%T=, and =%E= to display the same statistics about the input automaton). All the available statistics are displayed when a tool is run with =--help=. For instance here are the statistics available in [[file:randaut.org][=randaut=]]: #+BEGIN_SRC sh :results verbatim :exports results randaut --help | sed -n '/ sequences:/,/^$/p' | sed '1d;$d' #+END_SRC #+RESULTS: #+begin_example %% a single % %a number of acceptance sets %c, %[LETTERS]c number of SCCs; you may filter the SCCs to count using the following LETTERS, possibly concatenated: (a) accepting, (r) rejecting, (v) trivial, (t) terminal, (w) weak, (iw) inherently weak. Use uppercase letters to negate them. %d 1 if the output is deterministic, 0 otherwise %e number of edges %F seed number %g acceptance condition (in HOA syntax) %h the automaton in HOA format on a single line (use %[opt]h to specify additional options as in --hoa=opt) %L automaton number %m name of the automaton %n number of nondeterministic states in output %p 1 if the output is complete, 0 otherwise %r wall-clock time elapsed in seconds (excluding parsing) %R, %[LETTERS]R CPU time (excluding parsing), in seconds; Add LETTERS to restrict to(u) user time, (s) system time, (p) parent process, or (c) children processes. %s number of states %t number of transitions %w one word accepted by the output automaton #+end_example In most tools =%F= and =%L= are the input filename and line number, but as this makes no sense in =randaut=, these two sequences emit numbers related to the generation of automata. For instance let's generate 1000 random automata with 100 states and density 0.2, and just count the number of edges in each automaton. Then use =R= to summarize the distribution of these values: #+BEGIN_SRC sh :results verbatim :exports both randaut -e0.2 -Q100 -n1000 a --stats %e > size.csv Rscript -e "summary(read.csv('size.csv', header=FALSE, col.names='edges'))" #+END_SRC #+RESULTS: : edges : Min. :1939 : 1st Qu.:2056 : Median :2083 : Mean :2082 : 3rd Qu.:2107 : Max. :2233 For $Q=100$ states and density $D=0.2$ the expected degree of each state is $1+(Q-1)D = 1+99\times 0.2 = 20.8$, so the expected number of edges should be $20.8\times100=2080$. * Timing :PROPERTIES: :CUSTOM_ID: timing :END: Two of the statistics are related to time: =%r= displays wall-clock time, while =%R= displays CPU-time. #+BEGIN_SRC sh :results verbatim :exports both genltl --or-gf=1..8 | ltl2tgba --high --stats='%f,%r,%R' #+END_SRC #+RESULTS: : GFp1,0.000502296,0 : GF(p1 | p2),0.000796475,0 : GF(p1 | p2 | p3),0.00215579,0 : GF(p1 | p2 | p3 | p4),0.00441474,0 : GF(p1 | p2 | p3 | p4 | p5),0.00980961,0.01 : GF(p1 | p2 | p3 | p4 | p5 | p6),0.0255462,0.03 : GF(p1 | p2 | p3 | p4 | p5 | p6 | p7),0.121033,0.12 : GF(p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8),0.624101,0.62 Note that =%r= is implemented using the most precise clock available and usually has nano-second precision, while =%R= uses the =times()= system call (when available) and is usually only precise up to 1/100 of a second. However, as a wall-clock time, =%r= will also be affected by the load of the machine: if a machine is overloaded, or swapping a lot, you may notice a wall-clock time that is significantly higher than the CPU time measured by =%R=. Additional arguments may be passed to =%R= to select the time that must be output. By default, this the CPU-time spent in both user code and system calls. This can be restricted using one of =u= (user) or =s= (system). Also by default this includes the CPU-time for the current process and any of its children: adding =p= (parent) and =c= (children) will show only the selected time. Note that few tools actually execute other processes: [[file:autfilt.org][=autfilt=]] and [[file:ltl2tgba.org][=ltl2tgba=]] can do so when calling a SAT solver for [[file:satmin.org][SAT-based minimization]], and [[file:ltldo.org][=ltldo=]] will obviously call any listed tool. However in the case of =ltldo= the measured time is that of executing the other tools, so the result of =%[p]R= is likely to be always 0. Here is an example where we use =ltldo= to benchmark the (default) =--high= option of =ltl2tba= against the =--low= option, computing for each option the overall wall-clock time, CPU-time spent in =ltldo=, and CPU-time spent in =ltl2tgba=: #+BEGIN_SRC sh :results verbatim :exports both genltl --or-gf=1..8 | ltldo '{high}ltl2tgba' '{low}ltl2tgba --low' --stats='%T,%f,%r,%[p]R,%[c]R' #+END_SRC #+RESULTS: #+begin_example high,GFp1,0.0495443,0,0.02 low,GFp1,0.0427718,0,0.03 high,GFp1 | GFp2,0.0449237,0,0.03 low,GFp1 | GFp2,0.0429886,0,0.03 high,GFp1 | GFp2 | GFp3,0.0477704,0.01,0.03 low,GFp1 | GFp2 | GFp3,0.0294271,0,0.01 high,GFp1 | GFp2 | GFp3 | GFp4,0.0250874,0,0.02 low,GFp1 | GFp2 | GFp3 | GFp4,0.0203729,0,0.01 high,GFp1 | GFp2 | GFp3 | GFp4 | GFp5,0.0318887,0,0.03 low,GFp1 | GFp2 | GFp3 | GFp4 | GFp5,0.0207457,0,0.01 high,GFp1 | GFp2 | GFp3 | GFp4 | GFp5 | GFp6,0.0612968,0,0.05 low,GFp1 | GFp2 | GFp3 | GFp4 | GFp5 | GFp6,0.0145482,0,0.01 high,GFp1 | GFp2 | GFp3 | GFp4 | GFp5 | GFp6 | GFp7,0.130631,0,0.12 low,GFp1 | GFp2 | GFp3 | GFp4 | GFp5 | GFp6 | GFp7,0.0151502,0,0.01 high,GFp1 | GFp2 | GFp3 | GFp4 | GFp5 | GFp6 | GFp7 | GFp8,0.595865,0,0.59 low,GFp1 | GFp2 | GFp3 | GFp4 | GFp5 | GFp6 | GFp7 | GFp8,0.0160234,0,0.01 #+end_example * Naming automata Automata can be given names. This name can be output in the HOA format, but also in GraphViz output when =--dot=n= is given. By default, =ltl2tgba= will use the input formula as name. Other tools have no default name. This name can be changed using the =--name= option, that takes a format string similar to the one of =--stats=. #+NAME: oaut-name #+BEGIN_SRC sh :results verbatim :exports code ltl2tgba --name='TGBA for %f' --dot=n 'a U b' #+END_SRC #+RESULTS: oaut-name #+begin_example digraph G { rankdir=LR label="TGBA for a U b" labelloc="t" 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="1"] 1 [label="1"] 1 -> 0 [label="b"] 1 -> 1 [label="a & !b"] } #+end_example #+BEGIN_SRC dot :file oaut-name.png :cmdline -Tpng :var txt=oaut-name :exports results $txt #+END_SRC #+RESULTS: [[file:oaut-name.png]] If you have an automaton saved in the HOA format, you can extract its name using =autfilt --stats=%M input.hoa=. The =%M= escape sequence is replaced by the name of the input automaton. Here is a pipeline of commands that generates five LTL formulas $\varphi$ such that both $\varphi$ and $\lnot\varphi$ are translated into a 3-state TGBA by [[file:ltl2tgba.org][=ltl2tgba=]]. It starts by generating an infinite stream of random LTL formulas using =a= and =b= as atomic propositions, then it converts these formulas as TGBA (in the HOA format, therefore carrying the formula as name), filtering only the TGBA with 3 states and outputting =!(%M)= (that is the negation of the associated formula), translating the resulting formulas as TGBA, again retaining only the names (i.e. formulas) of the automata with 3 states, and finally restricting the output to the first 5 matches using =autfilt -n5=. #+BEGIN_SRC sh :results verbatim :exports both randltl -n -1 a b | ltl2tgba | autfilt --states=3 --stats='!(%M)' | ltl2tgba | autfilt --states=3 --stats=%M -n5 #+END_SRC #+RESULTS: : G(b | F(b & Fa)) : (!a | b | (!b & (b W Ga))) & (a | (!b & (b | (!b M F!a)))) : (!a | (!a R b)) & (a | (a U !b)) : !a & F((!a | FG!a) & (a | GFa)) : X(!b W a) Note that the above result can also be obtained without using =autfilt= and automata names. We can use the fact that =ltl2tgba --stats= can output the automaton size, and that =ltl2tgba= is also capable of [[file:csv.org][reading from a CSV file]] (=-F-/2= instructs =ltl2tgba= to read the standard input as if it was a CSV file, and to process its second column): #+BEGIN_SRC sh :results verbatim :exports both randltl -n -1 a b | # generate a stream of random LTL formulas ltl2tgba -F- --stats='%s,!(%f)' | # for each formula output "states,negated formula" grep '^3,' | # keep only formulas with 3 states ltl2tgba -F-/2 --stats='%s,%f' | # for each negated formula output "states,formula" grep '^3,' | # keep only negated formulas with 3 states head -n5 | cut -d, -f2 # return the five first formulas #+END_SRC #+RESULTS: : G(b | F(b & Fa)) : (!a | b | (!b & (b W Ga))) & (a | (!b & (b | (!b M F!a)))) : (!a | (!a R b)) & (a | (a U !b)) : !a & F((!a | FG!a) & (a | GFa)) : X(!b W a) Note that the =-F-= argument in the first call to =ltl2tgba= is superfluous as the tool default to reading from its standard input. But we put it there for symmetry with the second call. # LocalWords: num toc html syntaxes ltl tgba sed utf UTF lbtt SCCs # LocalWords: GraphViz's hoaf HOA LBTT's neverclaim ba SPOT's Gb cn # LocalWords: GraphViz autfilt acc Buchi hoafex gvpack perl pe bb # LocalWords: labelloc rankdir subgraph lp pos invis gv png cmdline # LocalWords: Tpng txt Hs Hm CSV Htl LBT dstar init goto fi Tpdf XF # LocalWords: oaut vcsn randaut nondeterministic filename csv hoa # LocalWords: varphi lnot GFb FG * Naming output By default, all output is sent to standard output, so you can either redirect it to a file, or pipe it to another program. You can also use the =--output= (a.k.a. =-o=) option to specify a filename where automata should be written. The advantage over a shell redirection, is that you may build a name using the same escape sequences as used by =--stats= and =--name=. For instance =%d= is replaced by 0 or 1 depending on whether the automaton is deterministic. We can generate 20 random automata, and output them in two files depending on their determinism: #+BEGIN_SRC sh :results verbatim :exports both randaut -n 20 -Q2 -e1 1 -o out-det%d.hoa autfilt -c out-det0.hoa # Count of non-deterministic automata autfilt -c out-det1.hoa # Count of deterministic automata #+END_SRC #+RESULTS: : 14 : 6 If you use this feature, beware that the output filename is only truncated once a first automaton is output to it: so if no automaton is output for a given filename, the existing file will be left untouched. For instance if we run the above commands again, but forcing [[file:randaut.org][=randaut=]] to output 20 *deterministic* automata, it may look like we produced more than 20 automata: #+BEGIN_SRC sh :results verbatim :exports both randaut -D -n 20 -Q2 -e1 1 -o out-det%d.hoa autfilt -c out-det0.hoa # Count of non-deterministic automata autfilt -c out-det1.hoa # Count of deterministic automata #+END_SRC #+RESULTS: : 14 : 20 This is because the =out-det0.hoa= file hasn't changed from the previous execution, while =out-det1.hoa= has been overwritten. In the case where you want to append to a file instead of overwriting it, prefix the output filename with =>>= as in : randaut -D -n 20 -Q2 1 -o '>>out-det%d.hoa' (You need the quotes so that the shell does not interpret =>>=.) #+BEGIN_SRC sh :results silent :exports results rm -f out-det0.hoa out-det1.hoa #+END_SRC