ltlfilt
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.
Table of Contents
Changing syntaxes
Because it read and write formulas, ltlfilt
accepts
all the 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.
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
:
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
These can be turned into something easier to read (to the human) with:
ltlfilt --lbt-input -F scheck.ltl
!(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 randltl
, the -r
option can be used to simplify formulas.
ltlfilt --lbt-input -F scheck.ltl -r
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.
ltlfilt --lbt-input -F scheck.ltl -r --relabel=abc
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 presenting lenient mode:
ltlfilt --lenient --relabel=pnn -f '(a < b) U (process[2]@ok)'
p0 U p1
Filtering
ltlfilt
supports many ways to filter formulas:
--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
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.
ltlfilt --lbt-input -F scheck.ltl -v --guarantee
!(Gp0 | (Gp1 & Fp3))
Combining ltlfilt
with 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
:
randltl -n -1 a b | ltlfilt --equivalent-to 'a U b' | head -n 10
!(!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)))
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.
randltl -r -n -1 a b | ltlfilt -v --syntactic-safety | ltlfilt --safety | head -n 10
(!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)))
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
?
ltlfilt -f 'a U (b U a)' --equivalent-to 'b U a'
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.
Is the formula F(a & X(!a & Gb))
stutter-invariant?
ltlfilt -f 'F(a & X(!a & Gb))' --stutter-invariant
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:
ltlfilt -f 'F(a & X(!a & Gb))' --remove-x
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:
ltlfilt -f 'F(a & X(!a & Gb))' --remove-x | ltlfilt --equivalent-to 'F(a & X(!a & Gb))'
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
this procedure (calling the remove_x()
function, and building automata
to check the equivalence of the resulting formula with the original one).