715805fad3
Fixes #83. * src/bin/randaut.cc: Add option. * src/twaalgos/randomgraph.cc, src/twaalgos/randomgraph.hh: Honor it. * src/tests/randaut.test: Add tests. * doc/org/randaut.org: Document it.
458 lines
14 KiB
Org Mode
458 lines
14 KiB
Org Mode
# -*- coding: utf-8 -*-
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#+TITLE: =randaut=
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#+SETUPFILE: setup.org
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#+HTML_LINK_UP: tools.html
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The =randaut= tool generates random (connected) automata.
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By default, it will generate a random automaton with 10 states, no
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acceptance sets, and using a set of atomic propositions you have to
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supply.
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#+NAME: randaut1
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut a b
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#+END_SRC
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#+RESULTS: randaut1
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#+begin_example
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digraph G {
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rankdir=LR
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 8 [label=<!a & !b>]
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0 -> 4 [label=<!a & !b>]
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1 [label="1"]
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1 -> 2 [label=<a & !b>]
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1 -> 7 [label=<!a & !b>]
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1 -> 4 [label=<!a & b>]
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2 [label="2"]
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2 -> 2 [label=<!a & !b>]
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2 -> 0 [label=<a & !b>]
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2 -> 5 [label=<a & !b>]
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3 [label="3"]
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3 -> 6 [label=<!a & b>]
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4 [label="4"]
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4 -> 8 [label=<a & !b>]
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4 -> 2 [label=<!a & b>]
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4 -> 3 [label=<a & b>]
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4 -> 7 [label=<!a & b>]
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5 [label="5"]
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5 -> 9 [label=<!a & !b>]
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5 -> 3 [label=<a & !b>]
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5 -> 7 [label=<!a & !b>]
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6 [label="6"]
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6 -> 7 [label=<!a & !b>]
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6 -> 1 [label=<!a & b>]
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7 [label="7"]
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7 -> 3 [label=<!a & !b>]
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8 [label="8"]
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8 -> 8 [label=<!a & b>]
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9 [label="9"]
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9 -> 0 [label=<!a & b>]
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9 -> 6 [label=<!a & b>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut1.png :cmdline -Tpng :var txt=randaut1 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut1.png]]
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As for [[file:randltl.org][=randltl=]], you can supply a number of atomic propositions
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instead of giving a list of atomic propositions.
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* States and density
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The numbers of states can be controlled using the =-Q= option. This
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option will accept a range as argument, so for instance =-Q3..6= will
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generate an automaton with 3 to 6 states.
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The number of edges can be controlled using the =-d= (or
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=--density=) option. The argument should be a number between 0 and 1.
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In an automaton with $Q$ states and density $d$, the degree of each
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state will follow a normal distribution with mean $1+(Q-1)d$ and
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variance $(Q-1)d(1-d)$.
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In particular =-d0= will cause all states to have 1 successors, and
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=-d1= will cause all states to be interconnected.
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#+NAME: randaut2
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q3 -d0 2
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#+END_SRC
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#+RESULTS: randaut2
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#+begin_example
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digraph G {
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rankdir=LR
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 2 [label=<!p0 & !p1>]
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1 [label="1"]
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1 -> 1 [label=<!p0 & !p1>]
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2 [label="2"]
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2 -> 1 [label=<!p0 & !p1>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut2.png :cmdline -Tpng :var txt=randaut2 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut2.png]]
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#+NAME: randaut3
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q3 -d1 2
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#+END_SRC
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#+RESULTS: randaut3
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#+begin_example
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digraph G {
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rankdir=LR
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 2 [label=<!p0 & !p1>]
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0 -> 0 [label=<!p0 & !p1>]
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0 -> 1 [label=<!p0 & !p1>]
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1 [label="1"]
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1 -> 1 [label=<p0 & p1>]
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1 -> 2 [label=<p0 & !p1>]
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1 -> 0 [label=<p0 & !p1>]
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2 [label="2"]
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2 -> 1 [label=<!p0 & !p1>]
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2 -> 0 [label=<p0 & !p1>]
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2 -> 2 [label=<p0 & !p1>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut3.png :cmdline -Tpng :var txt=randaut3 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut3.png]]
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* Acceptance condition
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The generation of the acceptance sets abn is controlled with the following four parameters:
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- =-A ACCEPTANCE= (or =--acceptance=ACCEPTANCE=) controls both the acceptance condition,
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and the number of associated acceptance sets. The =ACCEPTANCE= argument is documented
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in =--help= as follows:
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#+BEGIN_SRC sh :results verbatim :exports results
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randaut --help | sed -n '/^ \(ACCEPTANCE\|RANGE\)/,/^$/p'
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#+END_SRC
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#+RESULTS:
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#+begin_example
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RANGE may have one of the following forms: 'INT', 'INT..INT', or '..INT'.
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In the latter case, the missing number is assumed to be 1.
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ACCEPTANCE may be either a RANGE (in which case generalized Büchi is
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assumed), or an arbitrary acceptance formula such as 'Fin(0)|Inf(1)&Fin(2)' in
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the same syntax as in the HOA format, or one of the following patterns:
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none
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all
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Buchi
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co-Buchi
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generalized-Buchi RANGE
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generalized-co-Buchi RANGE
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Rabin RANGE
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Streett RANGE
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generalized-Rabin INT RANGE RANGE ... RANGE
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parity (min|max|rand) (odd|even|rand) RANGE
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random RANGE
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random RANGE PROBABILITY
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The random acceptance condition uses each set only once, unless a probability
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(to reuse the set again every time it is used) is given.
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#+end_example
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When a range of the form $i..j$ is used, the actual value is taken as randomly
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between $i$ and $j$ (included).
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- =-a= (or =--acc-probability=) controls the probability that any
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transition belong to a given acceptance set.
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- =-S= (or =--state-based-acceptance=) requests that the automaton
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use state-based acceptance. In this case, =-a= is the probability
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that a /state/ belong to the acceptance set. (Because Spot only
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deals with transition-based acceptance internally, this options
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force all transitions leaving a state to belong to the same
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acceptance sets. But if the output format allows state-based
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acceptance, it will be used.)
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- =--colored= requests that each transition (of state if combined with =-S=)
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in the generated automaton should belong to exactly one set (in that
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case =-a= is ignored, and =-A= must be used to specify an acceptance
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condition with at least one set).
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In addition, =-B= (or =--ba=) is a shorthand for =-A1 -S=,
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ans =-s= (or =--spin=) implies =-B=.
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#+NAME: randaut4
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q3 -d0.5 -A3 -a0.5 2
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#+END_SRC
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#+RESULTS: randaut4
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#+begin_example
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digraph G {
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rankdir=LR
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 1 [label=<!p0 & !p1>]
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0 -> 0 [label=<!p0 & !p1<br/><font color="#FAA43A">❷</font>>]
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1 [label="1"]
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1 -> 2 [label=<!p0 & p1<br/><font color="#F17CB0">❶</font><font color="#FAA43A">❷</font>>]
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2 [label="2"]
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2 -> 2 [label=<!p0 & !p1<br/><font color="#5DA5DA">⓿</font>>]
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2 -> 1 [label=<p0 & !p1<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut4.png :cmdline -Tpng :var txt=randaut4 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut4.png]]
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#+NAME: randaut5
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q3 -d0.4 -B -a0.7 2
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#+END_SRC
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#+RESULTS: randaut5
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#+begin_example
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digraph G {
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rankdir=LR
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 2 [label=<!p0 & !p1>]
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1 [label="1"]
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1 -> 2 [label=<!p0 & p1>]
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2 [label="2", peripheries=2]
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2 -> 1 [label=<!p0 & !p1>]
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2 -> 0 [label=<p0 & !p1>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut5.png :cmdline -Tpng :var txt=randaut5 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut5.png]]
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#+NAME: randaut5b
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q6 -d0.4 -S -a.2 -A 'Streett 1..3' 2 --dot=.a
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#+END_SRC
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#+RESULTS: randaut5b
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#+begin_example
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digraph G {
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rankdir=LR
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label=<(Fin(<font color="#5DA5DA">⓿</font>) | Inf(<font color="#F17CB0">❶</font>)) & (Fin(<font color="#FAA43A">❷</font>) | Inf(<font color="#B276B2">❸</font>))>
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labelloc="t"
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node [shape="circle"]
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label=<0>]
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0 -> 2 [label=<!p0 & !p1>]
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0 -> 1 [label=<!p0 & !p1>]
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0 -> 3 [label=<!p0 & !p1>]
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1 [label=<1>]
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1 -> 5 [label=<!p0 & p1>]
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2 [label=<2<br/><font color="#5DA5DA">⓿</font><font color="#FAA43A">❷</font>>]
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2 -> 1 [label=<!p0 & p1>]
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2 -> 2 [label=<!p0 & !p1>]
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2 -> 4 [label=<p0 & p1>]
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3 [label=<3>]
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3 -> 2 [label=<!p0 & !p1>]
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3 -> 3 [label=<!p0 & p1>]
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4 [label=<4>]
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4 -> 0 [label=<!p0 & !p1>]
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4 -> 5 [label=<!p0 & p1>]
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5 [label=<5<br/><font color="#F17CB0">❶</font><font color="#FAA43A">❷</font>>]
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5 -> 1 [label=<p0 & p1>]
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5 -> 2 [label=<!p0 & !p1>]
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5 -> 3 [label=<p0 & p1>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut5b.png :cmdline -Tpng :var txt=randaut5b :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut5b.png]]
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For generating random parity automata you should use the option
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=--colored= to make sure each transition (or state in the following
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example) belong to exactly one acceptance set. Note that you can
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specify a precise parity acceptance such as =parity min even 3=, or
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give =randaut= some freedom, as in this example.
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#+NAME: randaut5c
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q10 -S --colored -A 'parity rand rand 3..4' 2 --dot=.a
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#+END_SRC
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#+RESULTS: randaut5c
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#+begin_example
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digraph G {
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rankdir=LR
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label=<Inf(<font color="#5DA5DA">⓿</font>) | (Fin(<font color="#F17CB0">❶</font>) & (Inf(<font color="#FAA43A">❷</font>) | Fin(<font color="#B276B2">❸</font>)))>
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labelloc="t"
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node [shape="circle"]
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label=<0<br/><font color="#F17CB0">❶</font>>]
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0 -> 2 [label=<!p0 & !p1>]
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0 -> 8 [label=<!p0 & !p1>]
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0 -> 0 [label=<p0 & !p1>]
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0 -> 6 [label=<!p0 & p1>]
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1 [label=<1<br/><font color="#B276B2">❸</font>>]
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1 -> 5 [label=<!p0 & !p1>]
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1 -> 9 [label=<!p0 & p1>]
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2 [label=<2<br/><font color="#FAA43A">❷</font>>]
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2 -> 4 [label=<p0 & p1>]
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2 -> 5 [label=<!p0 & !p1>]
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3 [label=<3<br/><font color="#5DA5DA">⓿</font>>]
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3 -> 6 [label=<p0 & !p1>]
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3 -> 1 [label=<!p0 & p1>]
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4 [label=<4<br/><font color="#5DA5DA">⓿</font>>]
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4 -> 6 [label=<!p0 & !p1>]
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4 -> 1 [label=<p0 & p1>]
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5 [label=<5<br/><font color="#5DA5DA">⓿</font>>]
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5 -> 0 [label=<!p0 & !p1>]
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5 -> 8 [label=<p0 & !p1>]
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5 -> 7 [label=<!p0 & !p1>]
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6 [label=<6<br/><font color="#5DA5DA">⓿</font>>]
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6 -> 2 [label=<!p0 & !p1>]
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6 -> 3 [label=<!p0 & !p1>]
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7 [label=<7<br/><font color="#FAA43A">❷</font>>]
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7 -> 3 [label=<!p0 & p1>]
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7 -> 1 [label=<!p0 & p1>]
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8 [label=<8<br/><font color="#5DA5DA">⓿</font>>]
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8 -> 3 [label=<!p0 & p1>]
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8 -> 4 [label=<p0 & !p1>]
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8 -> 2 [label=<p0 & !p1>]
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8 -> 0 [label=<!p0 & p1>]
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9 [label=<9<br/><font color="#F17CB0">❶</font>>]
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9 -> 0 [label=<p0 & p1>]
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9 -> 6 [label=<p0 & !p1>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut5c.png :cmdline -Tpng :var txt=randaut5c :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut5c.png]]
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* Determinism
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The output can only contain a single edge between two given states.
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By default, the label of this edge is a random assignment of all
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atomic propositions. Two edges leaving the same state may therefore
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have the same label.
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If the =-D= (or =--deterministic=) option is supplied, the labels
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are generated differently: once the degree $m$ of a state has been
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decided, the algorithm will compute a set of $m$ disjoint
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Boolean formulas over the given atomic propositions, such that the
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sum of all these formulas is $\top$. The resulting automaton is
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therefore deterministic and complete.
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#+NAME: randaut6
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -D -Q3 -d0.6 -A2 -a0.5 2
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#+END_SRC
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#+RESULTS: randaut6
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#+begin_example
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digraph G {
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rankdir=LR
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fontname="Lato"
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node [fontname="Lato"]
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edge [fontname="Lato"]
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node[style=filled, fillcolor="#ffffa0"]
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I [label="", style=invis, width=0]
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I -> 0
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0 [label="0"]
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0 -> 1 [label=<p0>]
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0 -> 2 [label=<!p0>]
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1 [label="1"]
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1 -> 2 [label=<p0<br/><font color="#F17CB0">❶</font>>]
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1 -> 0 [label=<!p0<br/><font color="#5DA5DA">⓿</font>>]
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2 [label="2"]
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2 -> 2 [label=<!p0 & p1<br/><font color="#5DA5DA">⓿</font>>]
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2 -> 0 [label=<!p0 & !p1<br/><font color="#F17CB0">❶</font>>]
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2 -> 1 [label=<p0<br/><font color="#5DA5DA">⓿</font>>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut6.png :cmdline -Tpng :var txt=randaut6 :exports results
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$txt
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#+END_SRC
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#+RESULTS:
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[[file:randaut6.png]]
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Note that in a deterministic automaton with $a$ atomic propositions,
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it is not possible to have states with more than $2^a$ successors. If
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the combination of =-d= and =-Q= allows the situation where a state
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can have more than $2^a$ successors, the degree will be clipped to
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$2^a$. When working with random deterministic automata over $a$
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atomic propositions, we suggest you always request more than $2^a$
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states.
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* Output formats
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The output format can be controlled using [[file:oaut.org][the common output options]]
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like =--hoaf=, =--dot==, =--lbtt=, and =--spin=. Note that =--spin=
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automatically implies =--ba=.
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* Generating a stream of automata
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Use option =-n= to specify a number of automata to build. A negative
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value will cause an infinite number of automata to be produced. This
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generation of multiple automata is probably useful only with =--hoaf=,
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when piped to another tool that can read this format and process
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automata in batches.
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