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532 lines
16 KiB
Org Mode
532 lines
16 KiB
Org Mode
# -*- coding: utf-8 -*-
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#+TITLE: =randaut=
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#+DESCRIPTION: Spot command-line tool for generating random ω-automata.
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#+INCLUDE: 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 --dot
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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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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 -> 8 [label=<!a & !b>]
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0 -> 3 [label=<!a & !b>]
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0 -> 4 [label=<!a & !b>]
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1 [label=<1>]
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1 -> 7 [label=<a & b>]
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1 -> 0 [label=<a & b>]
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1 -> 6 [label=<a & !b>]
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2 [label=<2>]
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2 -> 4 [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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2 -> 9 [label=<!a & b>]
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3 [label=<3>]
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3 -> 2 [label=<a & b>]
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3 -> 9 [label=<a & !b>]
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3 -> 3 [label=<a & !b>]
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4 [label=<4>]
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4 -> 0 [label=<!a & !b>]
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4 -> 7 [label=<!a & b>]
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5 [label=<5>]
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5 -> 3 [label=<a & !b>]
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5 -> 1 [label=<!a & b>]
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5 -> 7 [label=<!a & !b>]
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5 -> 9 [label=<!a & b>]
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5 -> 5 [label=<!a & !b>]
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6 [label=<6>]
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6 -> 8 [label=<a & b>]
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6 -> 5 [label=<a & !b>]
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6 -> 2 [label=<a & !b>]
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7 [label=<7>]
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7 -> 8 [label=<!a & !b>]
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7 -> 9 [label=<a & b>]
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7 -> 0 [label=<!a & b>]
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7 -> 1 [label=<!a & !b>]
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7 -> 4 [label=<a & b>]
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8 [label=<8>]
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8 -> 1 [label=<a & b>]
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8 -> 3 [label=<!a & b>]
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9 [label=<9>]
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9 -> 1 [label=<!a & b>]
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9 -> 3 [label=<a & !b>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut1.svg :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.svg]]
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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 =-e= (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 $e$, 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 $(Q-1)e(1-e)$.
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In particular =-e0= will cause all states to have 1 successors, and
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=-e1= 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 -e0 2 --dot
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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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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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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.svg :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.svg]]
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#+NAME: randaut3
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q3 -e1 2 --dot
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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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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 -> 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.svg :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.svg]]
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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 -e0.5 -A3 -a0.5 2 --dot
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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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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 -> 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 -> 1 [label=<!p0 & p1<br/><font color="#5DA5DA">⓿</font><font color="#F17CB0">❶</font>>]
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1 -> 2 [label=<!p0 & p1<br/><font color="#F17CB0">❶</font>>]
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2 [label="2"]
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2 -> 1 [label=<!p0 & p1<br/><font color="#5DA5DA">⓿</font>>]
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2 -> 0 [label=<p0 & p1<br/><font color="#5DA5DA">⓿</font>>]
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2 -> 2 [label=<!p0 & p1<br/><font color="#5DA5DA">⓿</font>>]
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}
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#+end_example
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#+BEGIN_SRC dot :file randaut4.svg :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.svg]]
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#+NAME: randaut5
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q3 -e0.4 -B -a0.7 2 --dot
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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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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 -> 1 [label=<!p0 & !p1>]
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0 -> 2 [label=<!p0 & !p1>]
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1 [label="1", peripheries=2]
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1 -> 1 [label=<!p0 & !p1>]
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1 -> 0 [label=<p0 & p1>]
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2 [label="2", peripheries=2]
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2 -> 0 [label=<p0 & p1>]
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2 -> 2 [label=<p0 & !p1>]
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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 randaut5.svg :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.svg]]
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#+NAME: randaut5b
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#+BEGIN_SRC sh :results verbatim :exports code
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randaut -Q6 -e0.4 -S -a.2 -A 'Streett 1..3' 2 --dot
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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.svg :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.svg]]
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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
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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.svg :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.svg]]
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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
|
|
decided, the algorithm will compute a set of $m$ disjoint
|
|
Boolean formulas over the given atomic propositions, such that the
|
|
sum of all these formulas is $\top$. The resulting automaton is
|
|
therefore deterministic and complete.
|
|
|
|
#+NAME: randaut6
|
|
#+BEGIN_SRC sh :results verbatim :exports code
|
|
randaut -D -Q3 -e0.6 -A2 -a0.5 2 --dot
|
|
#+END_SRC
|
|
|
|
#+RESULTS: randaut6
|
|
#+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=<p0>]
|
|
0 -> 2 [label=<!p0>]
|
|
1 [label="1"]
|
|
1 -> 2 [label=<p0<br/><font color="#F17CB0">❶</font>>]
|
|
1 -> 0 [label=<!p0<br/><font color="#5DA5DA">⓿</font>>]
|
|
2 [label="2"]
|
|
2 -> 2 [label=<!p0 & p1<br/><font color="#5DA5DA">⓿</font>>]
|
|
2 -> 0 [label=<!p0 & !p1<br/><font color="#F17CB0">❶</font>>]
|
|
2 -> 1 [label=<p0<br/><font color="#5DA5DA">⓿</font>>]
|
|
}
|
|
#+end_example
|
|
|
|
#+BEGIN_SRC dot :file randaut6.svg :var txt=randaut6 :exports results
|
|
$txt
|
|
#+END_SRC
|
|
#+RESULTS:
|
|
[[file:randaut6.svg]]
|
|
|
|
|
|
Note that in a deterministic automaton with $a$ atomic propositions,
|
|
it is not possible to have states with more than $2^a$ successors. If
|
|
the combination of =-e= and =-Q= allows the situation where a state
|
|
can have more than $2^a$ successors, the degree will be clipped to
|
|
$2^a$. When working with random deterministic automata over $a$
|
|
atomic propositions, we suggest you always request more than $2^a$
|
|
states.
|
|
|
|
* Output formats
|
|
|
|
The output format can be controlled using [[file:oaut.org][the common output options]]
|
|
like =--hoaf=, =--dot==, =--lbtt=, and =--spin=. Note that =--spin=
|
|
automatically implies =--ba=.
|
|
|
|
Automata are send to standard output by default, by you can use =-o=
|
|
to give a filename, or even a pattern for filenames. For instance the
|
|
following generates 20 automatas, but store them in different files
|
|
according to the acceptance condition. The format =%g= represent the
|
|
formula for the acceptance condition and would not make a nice
|
|
filename, but =%[s]g= is a short name for that acceptance condition
|
|
(its is replaced by "other" if Spot does not know better).
|
|
|
|
#+BEGIN_SRC sh :results verbatim :exports both
|
|
randaut -n20 -Q10 -A 'random 3' 2 -o 'randaut-%[s]g.hoa'
|
|
wc -l randaut-*.hoa
|
|
#+END_SRC
|
|
|
|
#+RESULTS:
|
|
: 222 randaut-Rabin-like.hoa
|
|
: 380 randaut-Streett-like.hoa
|
|
: 100 randaut-generalized-Buchi.hoa
|
|
: 249 randaut-other.hoa
|
|
: 951 total
|
|
|
|
#+BEGIN_SRC sh :results silent :exports results
|
|
rm -f rautaut-*.hoa
|
|
#+END_SRC
|
|
|
|
* Generating a stream of automata
|
|
|
|
Use option =-n= to specify a number of automata to build. A negative
|
|
value will cause an infinite number of automata to be produced. This
|
|
generation of multiple automata is useful when piped to another tool
|
|
that can process automata in batches.
|
|
|
|
Here is an example were we use [[file:autfilt.org][=autfilt=]] to scan an infinite stream
|
|
(=-n -1=) of random parity automata for the first automaton (=-n 1=)
|
|
that have exactly 5 SCCs of particular natures: we want 1 trivial SCC
|
|
(i.e. an SCC with no cycle), 1 rejecting SCC (an SCC without any
|
|
accepting SCCs), 2 inherently weak SCCs (SCCs contains only rejecting
|
|
cycles) among which one should be weak (all transitions should belong
|
|
to the same sets). This leaves us with one extra SCC that should
|
|
necessarily mix accepting and rejecting cycles.
|
|
|
|
(Note: the '=.=' argument passed to =--dot= below hides default
|
|
options discussed [[file:oaut.org::#default-dot][on another page]], while '=s=' causes SCCs to be
|
|
displayed.)
|
|
|
|
#+NAME: randaut7
|
|
#+BEGIN_SRC sh :results verbatim :exports code
|
|
randaut -n -1 --colored -A'parity min odd 4' a b |
|
|
autfilt --sccs=5 --trivial-sccs=1 --rejecting-sccs=1 \
|
|
--inherently-weak-sccs=2 --weak-sccs=1 -n 1 --dot=.s
|
|
#+END_SRC
|
|
|
|
#+BEGIN_SRC dot :file randaut7.svg :var txt=randaut7 :exports results
|
|
$txt
|
|
#+END_SRC
|
|
|
|
#+RESULTS:
|
|
[[file:randaut7.svg]]
|
|
|
|
You should be able to find each of the expected type of SCCs in the above picture.
|
|
The green rectangles mark the three SCCs that contain some accepting cycles.
|