69dcff5ab0
We used to set PATH in emacs, but because babel executes "sh" via shell-command, the configuration of the main shell may supersedes ours. * doc/org/ltl2tgba.org, doc/org/ltl2tgta.org, doc/org/ltlcross.org, doc/org/ltlfilt.org, doc/org/randltl.org, doc/org/tools.org, doc/org/genltl.org: Move all local-file variable to... * doc/org/.dir-locals.el: ... here. And also set the PATH in org-babel-sh-command. * doc/org/init.el.in: Set the PATH in org-babel-sh-command.
320 lines
10 KiB
Org Mode
320 lines
10 KiB
Org Mode
#+TITLE: =randltl=
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#+EMAIL spot@lrde.epita.fr
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#+OPTIONS: H:2 num:nil toc:t
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#+LINK_UP: file:tools.html
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This tool generates random formulas. By default, it will generate one
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random LTL formula using atomic propositions supplied on the
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command-line. It can be instructed to generate random Boolean or PSL
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formulas instead, but let us first focus on LTL generation.
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For instance to obtain one random LTL formula over the propositions
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=a=, =b=, or =c=, use:
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl a b c
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#+END_SRC
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#+RESULTS:
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: G(Gb W (Gb M c))
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Note that the result does not always use all atomic propositions.
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The syntax of the formula output can be changed using the
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[[file:ioltl.org][common output options]]:
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#+BEGIN_SRC sh :results verbatim :exports results
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randltl --help | sed -n '/Output options:/,/^$/p' | sed '1d;$d'
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#+END_SRC
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#+RESULTS:
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: -8, --utf8 output using UTF-8 characters
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: -l, --lbt output in LBT's syntax
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: -p, --full-parentheses output fully-parenthesized formulas
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: -s, --spin output in Spin's syntax
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: --spot output in Spot's syntax (default)
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: --wring output in Wring's syntax
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When you select Spin's or Wring's syntax, operators =W= and =M= are
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automatically rewritten using =U= and =R= (written =V= for Spin).
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When you select LBT's syntax, you should name you atomic propositions
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like =p0=, =p1=, etc... (Atomic proposition named differently will be
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output by Spot in double-quotes, but this is not supported by LBT.)
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -l p1 p2 p3
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randltl -8 p1 p2 p3
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randltl -s p1 p2 p3
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randltl --wring p1 p2 p3
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#+END_SRC
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#+RESULTS:
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: G W G p2 M G p2 p3
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: □(□p2 W (□p2 M p3))
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: []((p3 U (p3 && []p2)) V ([]p2 || (p3 U (p3 && []p2))))
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: (G(((p3=1) U ((p3=1) * (G(p2=1)))) R ((G(p2=1)) + ((p3=1) U ((p3=1) * (G(p2=1)))))))
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As you might guess from the above result, for a given set of atomic
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propositions (and on the same computer) the generated formula will
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always be the same. This is because each time =randltl= starts, it
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initialize the seed of the random number generator to =0=. This seed
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can be changed using the =--seed= option. For instance the following
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three commands:
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl a b c
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randltl --seed=123 a b c
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randltl --seed=0 a b c
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#+END_SRC
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Will give three formulas in which the first and last are identical:
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#+RESULTS:
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: G(Gb W (Gb M c))
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: X((c <-> F(a M Xc)) -> a)
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: G(Gb W (Gb M c))
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When generating random formulas, we usually want large quantity of
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them. Rather than running =randltl= several times with different
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seeds, we can use the =-n= option to specify a number of formulas to
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produce.
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -n 5 a b c
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#+END_SRC
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#+RESULTS:
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: G(Gb W (Gb M c))
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: !(GFb -> Fa)
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: (((c xor Xc) R c) R Gc) & !c
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: X(1 U Xb) M Fb
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: !XFb U !(Xa W 0)
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By default =randltl= will never output the same formula twice (this
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can be changed with the =--allow-dups= option), so it may generate
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more formulas internally than it eventually prints. To ensure
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termination, for each output formula the number of ignored (because
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duplicated) random formulas that are generated is limited to 100000.
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Therefore in some situations, most likely when generating small
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formulas, with few atomic proposition, you may see =randltl= stop
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before the requested number of formulas has been output with an error
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message.
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If the integer passed to =-n= is negative, =randltl= will attempt to
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generate as many formulas as it can. This is most useful when
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=randltl= is piped to =ltlfilt= to select random formulas matching a
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certain criterion, as we shall see later.
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Besides the list of atomic propositions (=a b c= in our example) and
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the seed, several other parameters control the generation of the
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random formulas.
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Initially, the random generator selects a tree size for the formula.
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The default size is =15=, but it can be changed using the =--tree-size=
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option. For instance in the following, for each formula the tree size
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will be chosen randomly in the range =22..30=.
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -n 5 a b c --tree-size=22..30
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#+END_SRC
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#+RESULTS:
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: ((Fc W 0) M 1) & ((a | XXc) M (b W Fb))
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: (!(c | (a R Xb)) <-> (Xc R (a & c))) -> !a
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: 0
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: X(Xc R GX(1 U Xb)) & GX(b M Xc)
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: (!b -> ((c xor Fa) W b)) W (!Fb U a)
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The tree size is just the number of nodes in the syntax tree of the
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formula during its construction. However because Spot automatically
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applies some /trivial simplifications/ during the construction of its
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formulas (e.g., =F(F(a)= is reduced to =F(a)=, =a&0= to =0=, etc.),
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the actual size of the formula output may be smaller than the
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tree size specified.
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It is pretty common to obtain the formulas =0= or =1= among the first
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formulas output, since many random formulas trivially simplify to
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these. However because duplicate formulas are suppressed by default,
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they shall only occur once.
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Stronger simplifications may be requested using the =-r= option, that
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implements many rewritings that helps Spot translators algorithms (so
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beware that using =-r= reduces the randomness of the output).
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -n 5 a b c --tree-size=22..30 -r
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#+END_SRC
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#+RESULTS:
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: GFc & ((a | XXc) M Fb)
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: !a | (!c & (!a U X!b)) | (a & c & Xc)
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: 0
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: XG(b M Xc)
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: (((!c & Fa) | (c & G!a)) W b) W (G!b U a)
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The generator build the syntax tree recursively from its root, by
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considering all operators that could be used for a given tree size (for
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example a tree-size of 2 disables binary operators). A /priority/ is
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associated to each operator, and the probability of this operator
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being selected is this priority over the sum of the priorities of all
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considered operators. The default priorities for each operator can
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be seen with =--dump-priorities=:
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl a b c --dump-priorities
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#+END_SRC
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#+RESULTS:
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#+begin_example
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Use --ltl-priorities to set the following LTL priorities:
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ap 3
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false 1
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true 1
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not 1
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F 1
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G 1
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X 1
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equiv 1
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implies 1
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xor 1
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R 1
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U 1
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W 1
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M 1
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and 1
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or 1
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#+end_example
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Where =ap= stands for /atomic propositions/ (=a=, =b=, =c=). In this
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example, when the generator builds a leaf of the syntax tree (i.e., a
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subformula of tree-size 1), it must ignore all operators, and chose
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between =ap=, =false=, or =true=, and the odds of choosing =ap= is
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3/(3+1+1).
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As indicated at the top of the output, these priorities can be changed
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using the =--ltl-priorities= options. The most common scenario is to
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disable some of the operators by giving them a null priority. The
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following example disables 6 operators, and augments the priority of
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=U= to 3 to favor its occurrence.
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -n 5 a b c --ltl-priorities 'xor=0,implies=0,equiv=0,W=0,M=0,X=0,U=3'
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#+END_SRC
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#+RESULTS:
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: F(Fb U (c & Gb))
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: !(Fb U Fa)
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: ((Gc U c) U c) U Fc
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: 0
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: 1
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When using =-r= to simplify generated formulas, beware that these
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rewritings may use operators that you disabled during the initial
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random generation: you may obtain a formula that uses =W= even if =W=
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has a null priority. (You can use =ltlfilt= to rewrite these
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operators if that is a problem.)
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If the =--weak-fairness= option is used, for each random formula
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generated, a weak-fairness formula of the form =GF(a) & GF(b) & GF(c)=
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is generated for a subset of the atomic propositions and "ANDed" to
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the random formula. The =--tree-size= option has no influence on the
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weak-fairness formula appended.
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -n 5 a b c --weak-fairness
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#+END_SRC
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#+RESULTS:
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: G(Gb W (Gb M c)) & GFb
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: GFb & !(GFb -> Fa) & GFa
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: GFb & (((c xor Xc) R c) R Gc) & !c
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: GFb & GFa & (X(1 U Xb) M Fb)
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: GFb & (!XFb U !(Xa W 0)) & GFc
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Boolean formulas may be output with the =-B= option:
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -B -n 5 a b c
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#+END_SRC
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#+RESULTS:
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: !a -> !b
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: !(!(a -> (b xor c)) -> !a)
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: !c | (!c xor (c xor (c xor !c)))
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: !b
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: a xor !(!b <-> (!a -> c))
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In that case, priorities should be set with =--boolean-priorities=.
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Finally, PSL formulas may be output using the =-P= option. However
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keep in mind that since LTL formulas are PSL formulas, generating
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random PSL formula may produce many LTL formulas that do not use any
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PSL operator (this is even more so the case when simplifications are
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enabled with =-r=).
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -P -n 5 a b c
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#+END_SRC
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#+RESULTS:
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: G(Gb M ((c & Xb) | (!a M 1)))
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: !(Fa xor X({{c | {a xor !b}}[*]}[]-> Fb))
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: c & Gc
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: 0
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: 1
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As shown with the =--dump-priorities= output below, tweaking the
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priorities used to generated PSL formulas requires three different
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options:
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -P a b c --dump-priorities
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#+END_SRC
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#+RESULTS:
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#+begin_example
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Use --ltl-priorities to set the following LTL priorities:
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ap 3
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false 1
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true 1
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not 1
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F 1
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G 1
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X 1
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Closure 1
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equiv 1
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implies 1
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xor 1
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R 1
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U 1
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W 1
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M 1
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and 1
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or 1
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EConcat 1
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UConcat 1
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Use --sere-priorities to set the following SERE priorities:
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eword 1
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boolform 1
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star 1
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star_b 1
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and 1
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andNLM 1
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or 1
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concat 1
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fusion 1
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Use --boolean-priorities to set the following Boolean formula priorities:
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ap 3
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false 1
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true 1
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not 1
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equiv 1
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implies 1
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xor 1
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and 1
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or 1
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#+end_example
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The =--ltl-priorities= option we have seen previously now recognize
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some new PSL-specific operators: =Closure= is the ={sere}= operator,
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=EConcat= is the ={sere}<>->f= operator, and =UConcat= is the
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={sere}[]->f= operator. When these operator are selected, they
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require a SERE argument which is generated according to the priorities
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set by =--sere-priorities=: =eword= is the empty word, =boolform= is a
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Boolean formula (generated using the priorities set by
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=--boolean-priorities=), =star= is the unbounded Kleen star, while
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=star_b= is the bounded version, and =andNLM= is the
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non-length-matching variant of the =and= operator.
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# LocalWords: randltl num toc LTL PSL SRC Gb sed utf UTF lbt LBT's
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# LocalWords: Xc GFb Gc Xb Fb XFb Xa dups ltlfilt Fc XXc GX GFc XG
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# LocalWords: rewritings ltl ap GF ANDed GFa boolean EConcat eword
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# LocalWords: UConcat boolform andNLM concat Kleen eval setenv setq
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# LocalWords: getenv
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