bb2c697072
Part of #176. * doc/org/autfilt.org, doc/org/compile.org, doc/org/concepts.org, doc/org/csv.org, doc/org/dstar2tgba.org, doc/org/genltl.org, doc/org/hoa.org, doc/org/install.org, doc/org/ioltl.org, doc/org/ltl2tgba.org, doc/org/ltl2tgta.org, doc/org/ltlcross.org, doc/org/ltldo.org, doc/org/ltlfilt.org, doc/org/ltlgrind.org, doc/org/oaut.org, doc/org/randaut.org, doc/org/randltl.org, doc/org/satmin.org, doc/org/tools.org, doc/org/tut.org, doc/org/tut01.org, doc/org/tut02.org, doc/org/tut03.org, doc/org/tut10.org, doc/org/tut20.org, doc/org/tut21.org, doc/org/tut22.org, doc/org/tut30.org, doc/org/upgrade2.org: Here. * doc/org/index.org: Also add keywords in case it is useful, and use a more descripting title for search engines.
330 lines
10 KiB
Org Mode
330 lines
10 KiB
Org Mode
# -*- coding: utf-8 -*-
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#+TITLE: =randltl=
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#+DESCRIPTION: Spot command-line tool for generating random LTL formulas.
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#+SETUPFILE: setup.org
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#+HTML_LINK_UP: 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 fave 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 -n5 a b c
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#+END_SRC
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#+RESULTS:
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: 1
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: !(!((a U Gb) U b) U GFa)
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: GF((b R !a) U (Xc M 1))
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: (b <-> Xc) xor Fb
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: FXb R (a R (1 U b))
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Note that the result does not always use all atomic propositions.
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If you do not care about how the atomic propositions are named,
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you can give a nonnegative number instead:
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#+BEGIN_SRC sh :results verbatim :exports both
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randltl -n5 3
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#+END_SRC
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#+RESULTS:
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: 1
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: !(!((p0 U Gp1) U p1) U GFp0)
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: GF((p1 R !p0) U (Xp2 M 1))
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: (p1 <-> Xp2) xor Fp1
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: FXp1 R (p0 R (1 U p1))
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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 12
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randltl -8 12
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randltl -s 12
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randltl --wring 12
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#+END_SRC
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#+RESULTS:
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: V ! X ^ G p4 M p10 F ! p11 V G p3 p5
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: ¬○(□p4⊕(p10 M ◇¬p11)) R (□p3 R p5)
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: !X(([]p4 && !(<>!p11 U (p10 && <>!p11))) || (![]p4 && (<>!p11 U (p10 && <>!p11)))) V ([]p3 V p5)
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: (!(X((G(p4=1)) ^ ((F(p11=0)) U ((p10=1) * (F(p11=0))))))) R ((G(p3=1)) R (p5=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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initializes 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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#+NAME: result-seed
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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: result-seed
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: 1
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: b W 0
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: 1
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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 as seen in the very first example of this page.
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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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: (((b xor c) xor (a U (Gc M c))) | (!b xor (a M b))) W (b & Fc)
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: 0
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: !((c xor Gb) -> !Fb) xor !(Gc M 1)
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: 1
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: (a | ((c | G(!a W (!a W b))) W (b & (c M b)))) <-> (b R c)
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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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: 1
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: 0
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: ((G!b | (!c & F!b) | (c & Gb)) & GF!c) | (Fb & ((!c & Gb) | (c & F!b)) & FGc)
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: (b R c) | (!a & ((!c & F(a & !b)) M (!c W !b)))
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: XGa
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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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: 1
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: 1 U b
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: 0
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: F!((b U !a) U Gc)
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: !b U G(b R c)
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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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: GFb
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: ((b | XXa) M Ga) & GFa & GFc
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: GFb & GFc & !(!((a U Gb) U b) U GFa)
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: GFb & GFa & GFc & F(!X!b U (!c M 1))
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: GFb & GFa & GFc & Xc
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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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: !b
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: b & !b
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: b
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: !c xor (!a xor b)
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: !a -> (!c <-> (b xor !(a xor !b)))
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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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: 1
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: a R c
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: F({[*0] | {{[*0] | a[*]}}[*]}[]-> 0)
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: (a | ((a U c) M a)) M 1
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: 0
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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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