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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:

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).

Date: 2013-07-09T14:04+0200

Author: Alexandre Duret-Lutz

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