`ltlfilt`

## Table of Contents

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 common input and output options.

Additionally, if no `-f`

or `-F`

option is specified, and `ltlfilt`

will read formulas from the standard input if it is not connected to a
terminal.

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.

ltlfilt -l -f 'p1 U (p2 & GFp3)' -f 'X<>[]p4'

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)) p3 | Xp7 | Fp6 ((Xp0 & Xp4) U Fp1) & XX(XFp5 U (p0 U XXp3)) p0 U (p1 & (p0 | p5))

## Altering the formula

The following options can be used to modify the formulas that have been read.

--boolean-to-isop rewrite Boolean subformulas as irredundant sum of products (implies at least -r1) --define[=FILENAME] when used with --relabel or --relabel-bool, output the relabeling map using #define statements --exclusive-ap=AP,AP,... if any of those APs occur in the formula, add a term ensuring two of them may not be true at the same time. Use this option multiple times to declare independent groups of exclusive propositions. --from-ltlf[=alive] transform LTLf (finite LTL) to LTL by introducing some 'alive' proposition --negate negate each formula --nnf rewrite formulas in negative normal form --relabel[=abc|pnn] relabel all atomic propositions, alphabetically unless specified otherwise --relabel-bool[=abc|pnn] relabel Boolean subexpressions, alphabetically unless specified otherwise --remove-wm rewrite operators W and M using U and R (this is an alias for --unabbreviate=WM) --remove-x remove X operators (valid only for stutter-insensitive properties) -r, --simplify[=LEVEL] simplify formulas according to LEVEL (see below); LEVEL is set to 3 if omitted --unabbreviate[=STR] remove all occurrences of the operators specified by STR, which must be a substring of "eFGiMRW^", where 'e', 'i', and '^' stand respectively for <->, ->, and xor. If no argument is passed, the subset "eFGiMW^" is used.

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

Finally, there is a second variant of the relabeling procedure that is
enabled by `--relabel-bool=abc`

or `--relabel-book=pnn`

. With this
option, Boolean subformulas that do not interfere with other
subformulas will be changed into atomic propositions. For instance:

ltlfilt -f '(a & !b) & GF(a & !b) & FG(!c)' --relabel-bool=pnn ltlfilt -f '(a & !b) & GF(a & !b) & FG(!c & a)' --relabel-bool=pnn

p0 & GFp0 & FGp1 p0 & p1 & GF(p0 & p1) & FG(p0 & p2)

In the first formula, the independent `a & !b`

and `!c`

subformulae
were respectively renamed `p0`

and `p1`

. In the second formula, ```
a &
!b
```

and `!c & a`

are dependent so they could not be renamed; instead
`a`

, `!b`

and `c`

were renamed as `p0`

, `p1`

and `p2`

.

This option was originally developed to remove superfluous formulas
from benchmarks of LTL translators. For instance the automata
generated for `GF(a|b)`

and `GF(p0)`

should be structurally
equivalent: replacing `p0`

by `a|b`

in the second automaton should
turn in into the first automaton, and vice-versa. (However algorithms
dealing with `GF(a|b)`

might be slower because they have to deal with
more atomic propositions.) So given a long list of LTL formulas, we
can combine `--relabel-bool`

and `-u`

to keep only one instance of
formulas that are equivalent after such relabeling. We also suggest
to use `--nnf`

so that `!FG(a -> b)`

would become `GF(p0)`

as well. For instance here are some LTL formulas extracted from an
industrial project:

ltlfilt --nnf -u --relabel-bool <<EOF G (hfe_rdy -> F !hfe_req) G (lup_sr_valid -> F lup_sr_clean ) G F (hfe_req) reset && X G (!reset) G ( (F hfe_clk) && (F ! hfe_clk) ) G ( (F lup_clk) && (F ! lup_clk) ) G F (lup_sr_clean) G ( ( !(lup_addr_5_ <-> (X lup_addr_5_)) || !(lup_addr_6_ <-> (X lup_addr_6_)) || !(lup_addr_7_ <-> (X lup_addr_7_)) || !(lup_addr_8_ <-> (X lup_addr_8_)) ) -> ( (X !lup_sr_clean) && X ( (!( !(lup_addr_5_ <-> (X lup_addr_5_)) || !(lup_addr_6_ <-> (X lup_addr_6_)) || !(lup_addr_7_ <-> (X lup_addr_7_)) || !(lup_addr_8_ <-> (X lup_addr_8_)) )) U lup_sr_clean ) ) ) G F ( !(lup_addr_5_ <-> (X lup_addr_5_)) || !(lup_addr_6_ <-> (X lup_addr_6_)) || !(lup_addr_7_ <-> (X lup_addr_7_)) || !(lup_addr_8_ <-> (X lup_addr_8_)) ) (lup_addr_8__5__eq_0) ((hfe_block_0__eq_0)&&(hfe_block_1__eq_0)&&(hfe_block_2__eq_0)&&(hfe_block_3__eq_0)) G ((lup_addr_8__5__eq_0) -> X( (lup_addr_8__5__eq_0) || (lup_addr_8__5__eq_1) ) ) G ((lup_addr_8__5__eq_1) -> X( (lup_addr_8__5__eq_1) || (lup_addr_8__5__eq_2) ) ) G ((lup_addr_8__5__eq_2) -> X( (lup_addr_8__5__eq_2) || (lup_addr_8__5__eq_3) ) ) G ((lup_addr_8__5__eq_3) -> X( (lup_addr_8__5__eq_3) || (lup_addr_8__5__eq_4) ) ) G ((lup_addr_8__5__eq_4) -> X( (lup_addr_8__5__eq_4) || (lup_addr_8__5__eq_5) ) ) G ((lup_addr_8__5__eq_5) -> X( (lup_addr_8__5__eq_5) || (lup_addr_8__5__eq_6) ) ) G ((lup_addr_8__5__eq_6) -> X( (lup_addr_8__5__eq_6) || (lup_addr_8__5__eq_7) ) ) G ((lup_addr_8__5__eq_7) -> X( (lup_addr_8__5__eq_7) || (lup_addr_8__5__eq_8) ) ) G ((lup_addr_8__5__eq_8) -> X( (lup_addr_8__5__eq_8) || (lup_addr_8__5__eq_9) ) ) G ((lup_addr_8__5__eq_9) -> X( (lup_addr_8__5__eq_9) || (lup_addr_8__5__eq_10) ) ) G ((lup_addr_8__5__eq_10) -> X( (lup_addr_8__5__eq_10) || (lup_addr_8__5__eq_11) ) ) G ((lup_addr_8__5__eq_11) -> X( (lup_addr_8__5__eq_11) || (lup_addr_8__5__eq_12) ) ) G ((lup_addr_8__5__eq_12) -> X( (lup_addr_8__5__eq_12) || (lup_addr_8__5__eq_13) ) ) G ((lup_addr_8__5__eq_13) -> X( (lup_addr_8__5__eq_13) || (lup_addr_8__5__eq_14) ) ) G ((lup_addr_8__5__eq_14) -> X( (lup_addr_8__5__eq_14) || (lup_addr_8__5__eq_15) ) ) G ((lup_addr_8__5__eq_15) -> X( (lup_addr_8__5__eq_15) || (lup_addr_8__5__eq_0) ) ) G (((X hfe_clk) -> hfe_clk)->((hfe_req->X hfe_req)&&((!hfe_req) -> (X !hfe_req)))) G (((X lup_clk) -> lup_clk)->((lup_sr_clean->X lup_sr_clean)&&((!lup_sr_clean) -> (X !lup_sr_clean)))) EOF

G(a | Fb) GFa a & XG!a G(Fa & F!a) G((((a & Xa) | (!a & X!a)) & ((b & Xb) | (!b & X!b)) & ((c & Xc) | (!c & X!c)) & ((d & Xd) | (!d & X!d))) | (X!e & X((((a & Xa) | (!a & X!a)) & ((b & Xb) | (!b & X!b)) & ((c & Xc) | (!c & X!c)) & ((d & Xd) | (!d & X!d))) U e))) GF((a & X!a) | (!a & Xa) | (b & X!b) | (!b & Xb) | (c & X!c) | (!c & Xc) | (d & X!d) | (!d & Xd)) a G(!a | X(a | b)) G((!b & Xb) | ((!a | Xa) & (a | X!a)))

Here 29 formulas were reduced into 9 formulas after relabeling of Boolean subexpression and removing of duplicate formulas. In other words the original set of formulas contains 9 different patterns.

An option that can be used in combination with `--relabel`

or
`--relabel-bool`

is `--define`

. This causes the correspondence
between old and new names to be printed as a set of `#define`

statements.

```
ltlfilt -f '(a & !b) & GF(a & !b) & FG(!c)' --relabel-bool=pnn --define --spin
```

#define p0 (a && !b) #define p1 (!c) p0 && []<>p0 && <>[]p1

This can be used for instance if you want to use some complex atomic
propositions with third-party translators that do not understand them.
For instance the following sequence show how to use `ltl3ba`

to create
a neverclaim for an LTL formula containing atomic propositions that
`ltl3ba`

cannot parse:

```
ltlfilt -f '"proc@loc1" U "proc@loc2"' --relabel=pnn --define=ltlex.def --spin |
ltl3ba -F - >ltlex.never
cat ltlex.def ltlex.never
```

#define p0 ((proc@loc1)) #define p1 ((proc@loc2)) never { /* p0 U p1 */ T0_init: if :: (!p1 && p0) -> goto T0_init :: (p1) -> goto accept_all fi; accept_all: skip }

As a side note, the tool `ltldo`

might be a simpler answer to this syntactic problem:

```
ltldo ltl3ba -f '"proc@loc1" U "proc@loc2"' --spin
```

never { T0_init: if :: ((proc@loc1) && (!(proc@loc2))) -> goto T0_init :: (proc@loc2) -> goto accept_all fi; accept_all: skip }

This case also relabels the formula before calling `ltl3ba`

, and it
then rename all the atomic propositions in the output.

## Filtering

`ltlfilt`

supports many ways to filter formulas:

--accept-word=WORD keep formulas that accept WORD --ap=RANGE match formulas with a number of atomic propositions in RANGE --boolean match Boolean formulas --bsize=RANGE match formulas with Boolean size in RANGE --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) --obligation match obligation formulas (even pathological) --persistence match persistence formulas (even pathological) --recurrence match recurrence formulas (even pathological) --reject-word=WORD keep formulas that reject WORD --safety match safety formulas (even pathological) --size=RANGE match formulas with size in RANGE --stutter-insensitive, --stutter-invariant match stutter-insensitive LTL formulas --suspendable synonym for --universal --eventual --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 --syntactic-stutter-invariant, --nox match stutter-invariant formulas syntactically (LTL-X or siPSL) --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' -n 10
```

b W (a U b) a U b !(!a R !b) b | (a U b) (a xor (a & b)) U b a U ((a | !a) R b) (b <-> !b) U (a U b) (a | b) U b (a U b) | Gb b M (a U 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`

is limited to 10 formulas using `-n 10`

. (As would using
`| head -n 10`

.) 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 -n 10

F((!b & GF!b) | (b & FGb)) a | G((a & GFa) | (!a & FG!a)) XXG(!a & (Fa W Gb)) (!a & (XX!a | (!a W F!b))) R !b G(Ga | (F!a & X!b)) b W ((!b & (a W XG!b)) | (b & (!a M XFb))) Xa W (b | ((!b M F!a) R !a)) Xa | (a & ((!a & F!b) | (a & Gb))) | (!a & ((a & F!b) | (!a & Gb))) (b M a) & XGb Xb | (!a U !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 & !b & Gb)) & ((!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 & !b & Gb)) & ((!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 (otherwise the output would have been empty).

The difference between `--size`

and `--bsize`

lies in the way Boolean
subformula are counted. With `--size`

the size of the formula is
exactly the number of atomic propositions and operators used. For
instance the following command generates 10 random formulas with size
5 (the reason `randltl`

uses `--tree-size=8`

is because the "tree" of
the formula generated randomly can be reduced by trivial
simplifications such as `!!f`

being rewritten to `f`

, yielding
formulas of smaller sizes).

randltl -n -1 --tree-size=8 a b | ltlfilt --size=5 -n 10

!F!Ga X!(a U b) !G(a & b) (b W a) W 0 b R X!b GF!Xa Xb & Ga a xor !Fb a xor FXb !(0 R Fb)

With `--bsize`

, any Boolean subformula is counted as "1" in the total
size. So `F(a & b & c)`

would have Boolean-size 2. This type of size
is probably a better way to classify formulas that are going to be
translated as automata, since transitions are labeled by Boolean
formulas: the complexity of the Boolean subformulas has little
influence on the overall translation. Here are 10 random formula with
Boolean-size 5:

randltl -n -1 --tree-size=12 a b | ltlfilt --bsize=5 -n 10

Gb xor Fa FX!Fa !(Fb U b) (a -> !b) & XFb X(b & Xb) 0 R (a U !b) XXa R !b (!a & !(!a xor b)) xor (0 R b) GF(1 U b) (a U b) R b

## Using `--format`

and `--output`

The `--format`

option can be used the alter the way formulas are output.
The list of supported `%`

-escape sequences are recalled in the `--help`

output:

%< the part of the line before the formula if it comes from a column extracted from a CSV file %> the part of the line after the formula if it comes from a column extracted from a CSV file %% a single % %b the Boolean-length of the formula (i.e., all Boolean subformulas count as 1) %f the formula (in the selected syntax) %F the name of the input file %h, %[vw]h the class of the formula is the Manna-Pnueli hierarchy ([v] replaces abbreviations by class names, [w] for all compatible classes) %L the original line number in the input file %[OP]n the nesting depth of operator OP. OP should be a single letter denoting the operator to count, or multiple letters to fuse several operators during depth evaluation. Add '~' to rewrite the formula in negative normal form before counting. %r wall-clock time elapsed in seconds (excluding parsing) %R, %[LETTERS]R CPU time (excluding parsing), in seconds; Add LETTERS to restrict to(u) user time, (s) system time, (p) parent process, or (c) children processes. %s the length (or size) of the formula %x, %[LETTERS]X, %[LETTERS]x number of atomic propositions used in the formula; add LETTERS to list atomic propositions with (n) no quoting, (s) occasional double-quotes with C-style escape, (d) double-quotes with C-style escape, (c) double-quotes with CSV-style escape, (p) between parentheses, any extra non-alphanumeric character will be used to separate propositions

As a trivial example, use

`--latex --format='$%f$'`

to enclose formula in LaTeX format with `$...$`

.

But `--format`

can be useful in more complex scenarios. For instance
you could print only the line numbers containing formulas matching
some criterion. In the following, we print only the numbers of the
lines of `scheck.ltl`

that contain guarantee formulas:

ltlfilt --lbt-input -F scheck.ltl --guarantee --format=%L

2 3 4

We could also prefix each formula by its size, in order to sort the file by formula size:

```
ltlfilt --lbt-input scheck.ltl --format='%s,%f' | sort -n
```

7,p0 U (p1 & (p0 | p5)) 7,p3 | Xp7 | Fp6 9,!(Gp0 | (Gp1 & Fp3)) 20,((Xp0 & Xp4) U Fp1) & XX(XFp5 U (p0 U XXp3))

More examples of how to use `--format`

to create CSV files are on a
separate page

The `--output`

option interprets its argument as an output filename,
but after evaluating the `%`

-escape sequence for each formula. This
makes it very easy to partition a list of formulas in different files.
For instance here is how to split `scheck.ltl`

according to formula
sizes.

```
ltlfilt --lbt-input scheck.ltl --output='scheck-%s.ltl'
wc -l scheck*.ltl
```

1 scheck-20.ltl 2 scheck-7.ltl 1 scheck-9.ltl 4 scheck.ltl 8 total

rm -f ltlex.def ltlex.never scheck.ltl