#+TITLE: =ltlfilt=
#+EMAIL spot@lrde.epita.fr
#+OPTIONS: H:2 num:nil toc:t
#+LINK_UP: file:tools.html

This tool is a filter for LTL formulas.  (It will also work with PSL
formulas.)  It can be used to perform a number of tasks.  Essentially:
- converting formulas from one syntax to another,
- transforming formulas,
- selecting formulas matching some criterion.

* Changing syntaxes

Because it read and write formulas, =ltlfilt= accepts
all the [[file:ioltl.org][common input and output options]].

Additionally, if no =-f= or =-F= option is specified, =ltlfilt=
will read formulas from the standard input.

For instance the following will convert two LTL formulas expressed
using infix notation (with different names supported for the same
operators) and convert it into LBT's syntax.

#+BEGIN_SRC sh :results verbatim :exports results
ltlfilt -l -f 'p1 U (p2 & GFp3)' -f 'X<>[]p4'
#+END_SRC
#+RESULTS:
: U p1 & p2 G F p3
: X F G p4

Conversely, here is how to rewrite formulas expressed using the
LBT's Polish notation.  Let's take the following four formulas
taken from examples distributed with =scheck=:
#+BEGIN_SRC sh :results verbatim :exports both
cat >scheck.ltl<<EOF
! | G p0 & G p1 F p3
| | X p7 F p6 & | | t p3 p7 U | f p3 p3
& U & X p0 X p4 F p1 X X U X F p5 U p0 X X p3
U p0 & | p0 p5 p1
EOF
#+END_SRC
#+RESULTS:

These can be turned into something easier to read (to the human) with:
#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --lbt-input -F scheck.ltl
#+END_SRC
#+RESULTS:
: !(Gp0 | (Gp1 & Fp3))
: Xp7 | Fp6 | p3
: ((Xp0 & Xp4) U Fp1) & XX(XFp5 U (p0 U XXp3))
: p0 U ((p0 | p5) & p1)

* Altering the formula

As with [[file:randltl.org][=randltl=]], the =-r= option can be used to simplify formulas.

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --lbt-input -F scheck.ltl -r
#+END_SRC
#+RESULTS:
: F!p0 & (F!p1 | G!p3)
: p3 | Xp7 | Fp6
: Fp1 & XX(XFp5 U (p0 U XXp3))
: p0 U (p1 & (p0 | p5))

You may notice that operands of n-ary operators such as =&= or =|= can
be reordered by =ltlfilt= even when the formula is not changed
otherwise.  This is because Spot internally order all operands of
commutative and associative operators, and that this order depends on
the order in which the subformulas are first encountered.  Adding
transformation options such as =-r= may alter this order.  However
this difference is semantically insignificant.

Formulas can be easily negated using the =-n= option, rewritten into
negative normal form using the =--nnf= option, and the =W= and =M=
operators can be rewritten using =U= and =R= using the =--remove-wm=
option (note that this is already done when a formula is output in
Spin's syntax).

Another way to alter formula is to rename the atomic propositions it
uses.  The =--relabel=abc= will relabel all atomic propositions using
letters of the alphabet, while =--relabel=pnn= will use =p0=, =p1=,
etc. as in LBT's syntax.

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --lbt-input -F scheck.ltl -r --relabel=abc
#+END_SRC
#+RESULTS:
: F!a & (F!b | G!c)
: a | Xb | Fc
: Fa & XX(XFb U (c U XXd))
: a U (b & (a | c))

Note that the relabeling is reset between each formula: =p3= became
=c= in the first formula, but it became =d= in the third.

Another use of relabeling is to get rid of complex atomic propositions
such as the one shown when [[file:ioltl.org][presenting lenient mode]]:

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --lenient --relabel=pnn -f '(a < b) U (process[2]@ok)'
#+END_SRC
#+RESULTS:
: p0 U p1

* Filtering

=ltlfilt= supports many ways to filter formulas:

#+BEGIN_SRC sh :results verbatim :exports results
ltlfilt --help | sed -n '/Filtering options.*:/,/^$/p' | sed '1d;$d'
#+END_SRC
#+RESULTS:
#+begin_example
      --boolean              match Boolean formulas
      --bsize-max=INT        match formulas with Boolean size <= INT
      --bsize-min=INT        match formulas with Boolean size >= INT
      --equivalent-to=FORMULA   match formulas equivalent to FORMULA
      --eventual             match pure eventualities
      --guarantee            match guarantee formulas (even pathological)
      --implied-by=FORMULA   match formulas implied by FORMULA
      --imply=FORMULA        match formulas implying FORMULA
      --ltl                  match only LTL formulas (no PSL operator)
      --nox                  match X-free formulas
      --obligation           match obligation formulas (even pathological)
      --safety               match safety formulas (even pathological)
      --size-max=INT         match formulas with size <= INT
      --size-min=INT         match formulas with size >= INT
      --stutter-insensitive, --stutter-invariant
                             match stutter-insensitive LTL formulas
      --syntactic-guarantee  match syntactic-guarantee formulas
      --syntactic-obligation match syntactic-obligation formulas
      --syntactic-persistence   match syntactic-persistence formulas
      --syntactic-recurrence match syntactic-recurrence formulas
      --syntactic-safety     match syntactic-safety formulas
      --universal            match purely universal formulas
  -u, --unique               drop formulas that have already been output (not
                             affected by -v)
  -v, --invert-match         select non-matching formulas
#+end_example

Most of the above options should be self-explanatory.  For instance
the following command will extract all formulas from =scheck.ltl=
which do not represent guarantee properties.

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt --lbt-input -F scheck.ltl -v --guarantee
#+END_SRC
#+RESULTS:
: !(Gp0 | (Gp1 & Fp3))

Combining =ltlfilt= with [[file:randltl.org][=randltl=]] makes it easier to generate random
formulas that respect certain constraints.  For instance let us
generate 10 formulas that are equivalent to =a U b=:

#+BEGIN_SRC sh :results verbatim :exports both
randltl -n -1 a b | ltlfilt --equivalent-to 'a U b' | head -n 10
#+END_SRC
#+RESULTS:
#+begin_example
!(!a R !b)
(!Gb -> a) U b
a U b
Fb & (a W b)
((a <-> !(a | b)) W a) U ((!b M b) U b)
(b <-> (Xb M a)) -> b
(a | b) U b
((!b U b) -> (a W b)) U b
(a xor b) U b
b R (Fb & (a U (a W b)))
#+end_example

The =-n -1= option to =randltl= will cause it to output an infinite
stream of random formulas.  =ltlfilt=, which reads its standard input
by default, will select only those equivalent to =a U b=.  The output
of =ltlfilt= would still be an infinite stream of random formulas, so
we display only the first 10 using the standard =head= utility.  Less
trivial formulas could be obtained by adding the =-r= option to
=randltl= (or equivalently adding the =-r= and =-u= option to
=ltlfilt=).


Another similar example, that requires two calls to =ltlfilt=, is the
generation of random pathological safety formulas.  Pathological
safety formulas are safety formulas that do not /look/ so
syntactically.  We can generate some starting again with =randltl=,
then ignoring all syntactic safety formulas, and keeping only the
safety formulas in the remaining list.

#+BEGIN_SRC sh :results verbatim :exports both
randltl -r -n -1 a b | ltlfilt -v --syntactic-safety | ltlfilt --safety | head -n 10
#+END_SRC
#+RESULTS:
#+begin_example
(!a & Fa) R Xa
!a | (a & b) | (((!a & b) | (a & !b)) M (!a M X!a))
G(!a M Xa)
G((G!b & !a) | (a & Fb)) R a
G!a M !a
G(!a M ((!b & XGb) | (b & XF!b)))
F(b | G!b)
F(Xa | G!a)
G(XXa | (b & F!a))
G((!a & (!a M !b)) | (a & (a W b)))
#+end_example


=ltlfilt='s filtering ability can also be used to answer questions
about a single formula.  For instance is =a U (b U a)= equivalent to
=b U a=?

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt -f 'a U (b U a)' --equivalent-to 'b U a'
#+END_SRC
#+RESULTS:
: a U (b U a)

The command prints the formula and returns an exit status of 0 if the
two formulas are equivalent.  It would print nothing and set the exit
status to 1, were the two formulas not equivalent.


If the formula =F(a & X(!a & Gb))= stutter-invariant?

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt -f 'F(a & X(!a & Gb))' --stutter-invariant
#+END_SRC
#+RESULTS:
: F(a & X(!a & Gb))

Yes it is.  And since it is stutter-invariant, there exist some
equivalent formulas that do not use =X= operator.  The =--remove-x=
option gives one:

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt -f 'F(a & X(!a & Gb))' --remove-x
#+END_SRC
#+RESULTS:
: F(a & ((a & (a U (!a & Gb)) & ((!b U !a) | (b U !a))) | (!a & (!a U (a & !a & Gb)) & ((!b U a) | (b U a))) | (b & (b U (!a & Gb & !b)) & ((!a U !b) | (a U !b))) | (!b & (!b U (!a & b & Gb)) & ((!a U b) | (a U b))) | (!a & Gb & (G!a | Ga) & (Gb | G!b))))

We could even verify that the resulting horrible formula is equivalent
to the original one:

#+BEGIN_SRC sh :results verbatim :exports both
ltlfilt -f 'F(a & X(!a & Gb))' --remove-x | ltlfilt --equivalent-to 'F(a & X(!a & Gb))'
#+END_SRC
#+RESULTS:
: F(a & ((a & (a U (!a & Gb)) & ((!b U !a) | (b U !a))) | (!a & (!a U (a & !a & Gb)) & ((!b U a) | (b U a))) | (b & (b U (!a & Gb & !b)) & ((!a U !b) | (a U !b))) | (!b & (!b U (!a & b & Gb)) & ((!a U b) | (a U b))) | (!a & Gb & (G!a | Ga) & (Gb | G!b))))

It is therefore equivalent, but that is not a surprise since the
=--stutter-invariant= filter is actually implemented using exactly
[[http://homepages.inf.ed.ac.uk/kousha/note_on_stut_tl_lpi.ps][this procedure]] (calling the =remove_x()= function, and checking the
equivalence of the resulting formula with the original one).


#  LocalWords:  ltlfilt num toc LTL PSL syntaxes LBT's SRC GFp scheck
#  LocalWords:  ltl EOF lbt Gp Fp Xp XFp XXp randltl ary nnf wm abc
#  LocalWords:  pnn Xb Fc XFb XXd sed boolean bsize nox Gb Fb Xa XGb
#  LocalWords:  XF XXa
