Exploring the temporal hierarchy of Manna & Pnueli

Table of Contents

A hierarchy of temporal properties was defined by Manna & Pnueli in their PODC'90 paper.

This hierarchy relates "properties" (i.e., omega-regular languages) to structural properties of the automata that can recognize them.

Description of the classes

The hierarchy is built from the classes pictured in the following diagram, where each class includes everything below it. For instance, the recurrence class includes the obligation class which also includes the safety and guarantee classes, as well as the unnamed intersection of safety and guarantee (B in the picture).

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Forget about the LTL properties and about the red letters displayed in this picture for a moment.

  • The reactivity class represents all possible omega-regular languages, i.e., all languages that can be recognized by a non-deterministic Büchi automaton.
  • The recurrence subclass contains all properties that can be recognized by a deterministic Büchi automaton.
  • The dual class, persistence properties, are those that can be recognized by a weak Büchi automaton (i.e., in each SCC either all states are accepting, or all states are rejecting).
  • The intersection of recurrence and persistence classes form the obligation properties: any of those can be recognized by a weak and deterministic Büchi automaton.
  • Guarantee properties are a subclass of obligation properties that can be recognized by terminal Büchi automata (i.e., upon reaching an accepting state, any suffix will be accepted).
  • Safety properties are the dual of Guarantee properties: they can be recognized by ω-automata that accept all their runs (i.e., the acceptance condition is "true"). Note that since these automata are not necessary complete, it is still possible for some words not to be accepted. If we interpret the ω-automata with "true" acceptance as finite automata with all states marked as final, we obtain monitors, i.e., finite automata that recognize all finite prefixes that can be extended into valid ω-words.
  • Finally, at the very bottom is an unnamed class that contains Safety properties that are also Guarantee properties: those are properties that can be represented by monitors in which the only cycles are self-loops labeled by true.

The "LTL normal forms" displayed in the above figure help to visualize the type of LTL formulas contained in each of these class. But note that (1) this hierarchy applies to all omega-regular properties, not just LTL-defined properties, and (2) the LTL expression displayed in the figure are actually normal forms in the sense that if an LTL-defined property belongs to the given class, then there exists an equivalent LTL property under the stated form, were \(p\), \(q\), \(p_i\) and \(q_i\) are subexpressions that may use only Boolean operators, the next operator (\(\mathsf{X}\)), and past-LTL operators (which are not supported by Spot). The combination of these allowed operators only makes it possible to express constraints on finite prefixes.

Obligations can be thought of as Boolean combinations of safety and guarantee properties, while reactivity properties are Boolean combinations of recurrence and persistence properties. The negation of a safety property is a guarantee property (and vice versa), and the same duality hold for persistence and recurrence properties.

The red letters in each of these seven classes are keys used in Spot to denote the classes.

Deciding class membership

The --format=%h option can be used to display the "class key" of the most precise class to which a formula belongs.

ltlfilt -f 'a U b' --format=%h

If you find hard to remember the class name corresponding to the class keys, you can request verbose output with %[v]h:

ltlfilt -f 'a U b' --format='%[v]h'

But actually any guarantee property is also an obligation, a recurrence, a persistence, and a reactivity property. You can get the complete list of classes using %[w]h or %[vw]h:

ltlfilt -f 'a U b' --format='%[w]h = %[vw]h'
GOPRT = guarantee obligation persistence recurrence reactivity

This --format option is also supported by randltl, genltl, and ltlgrind. So for instance if you want to classify the 55 LTL patterns of Dwyers et al. (FMSP'98) using this hierarchy, try:

genltl --dac-patterns --format='%[v]h' | sort | uniq -c
 1 guarantee
 2 obligation
 1 persistence
 2 reactivity
12 recurrence
37 safety

In this output, the most precise class is given for each formula, and the count of formulas for each subclass is given. We have to remember that the recurrence class also includes obligation, safety, and guarantee properties. In this list, there are no formulas that belong to the intersection of the guarantee and safety classes (it would have been listed as guarantee safety).

From this list, only 3 formulas are not recurrence properties (i.e., not recognized by deterministic Büchi automata): one of them is a persistence formula, the other two cannot be classified better than in the reactivity class. Let's pretend we are interested in those three non-recurrence formulas, we can use ltlfilt -v --recurrence to remove all recurrence properties from the genltl --dac-pattern output:

genltl --dac-patterns |
  ltlfilt -v --recurrence --format='%[v]h, %f'
persistence, G!p0 | F(p0 & (!p1 W p2))
reactivity, G(p0 -> ((p1 -> (!p2 U (!p2 & p3 & X(!p2 U p4)))) U (p2 | G(p1 -> (p3 & XFp4)))))
reactivity, G(p0 -> ((p1 -> (!p2 U (!p2 & p3 & !p4 & X((!p2 & !p4) U p5)))) U (p2 | G(p1 -> (p3 & !p4 & X(!p4 U p5))))))

Similar filtering options exist for other classes. Since these tests are automata-based, they work with PSL formulas as well. For instance, here is how to generate 10 random recurrence PSL formulas that are not LTL formulas, and that are not obligations:

randltl --psl -n -1 a b |
   ltlfilt -v --ltl |
   ltlfilt -v --obligation |
   ltlfilt --recurrence -n10
{[*]}<>-> (!a & XG!Ga)
F({[*2][*]}[]-> (b M 1))
{[*0] | b[*]}<>-> GFb
F{1:{{a | !b} | [*0]}} W (a & b)
X{b | [*0]} & GF!b
{a[*]}[]-> !Gb
!({{[*] & b[*]}[:*1..2]}<>-> F(Ga R b))
a -> G(G{{!{a xor b}}[*]} -> Gb)
GFb & ({[*]}[]-> a)
XXGF!XX({[*]}[]-> a)

Note that the order of the ltlfilt filters could be changed provided the -n10 stays at the end. For instance, we could first keep all recurrence before removing obligations and then removing LTL formulas. The order given above actually starts with the easier checks first and keep the most complex tests at the end of the pipeline, so they are applied to fewer formulas. Testing whether a formula is an LTL formula is very cheap, testing if a formula is an obligation is harder (it may involve a translation to automata and a powerset construction), and testing whether a formula is a recurrence is the most costly procedure (it involves a translation as well, plus conversion to deterministic Rabin automata, and an attempt to convert the automaton back to deterministic Büchi). As a rule of thumb, testing classes that are lower in the hierarchy is cheaper.

Since option -o (for specifying output file) also honors %-escape sequences, we can use it with %h to split a list of formulas in 7 possible files. Here is a generation of 200 random LTL formulas binned into aptly named files:

randltl -n 200 a b -o random-%h.ltl
wc -l random-?.ltl
 59 random-B.ltl
 46 random-G.ltl
 11 random-O.ltl
 19 random-P.ltl
 15 random-R.ltl
 44 random-S.ltl
  6 random-T.ltl
200 total

Deciding classes membership syntactically

LTL formulas can also be classified into related classes which we shall call syntactic-safety, syntactic-guarantee, etc. See tl.pdf for the grammar of each syntactic class. Any LTL-definable property of class C can be defined by an LTL formulas in class syntactic-C, but an LTL formula can describe a property of class C even if that formula is not in class syntactic-C (we just know that some equivalent formula is in class syntactic-C).

ltlfilt has options like --syntactic-guarantee, --syntactic-persistence, etc. to match formulas from these classes.

Here is how to generate 10 random LTL formulas that describe safety properties, but that are not in the syntactic-safety class:

randltl -n-1 a b |
  ltlfilt -v --syntactic-safety |
  ltlfilt --safety -n10
F!(!b <-> FGb)
!Fb xor G((b xor (Xa M b)) U b)
a W F(a -> b)
((0 R Xa) R a) -> Fa
X(Xb & (!Ga R Ga))
!Fb <-> (Fb | !X(0))
(1 U b) | F(Fb W (a <-> FXa))
(a M 1) | (!a W a)
(G!a W ((b M 1) -> Fa)) -> !a
!a -> ((a xor !GFa) W 0)

Since all those formulas describe safety properties, an exercise would be to suggest equivalent formulas that are in the syntactic-safety fragment. For instance b M Gb can be rewritten as just Gb, which belongs to this fragment. In this particular case, ltlfilt --simplify recognizes this:

ltlfilt --simplify -f 'b M Gb'

However, in the general case Spot is not able to provide the equivalent formula from the appropriate syntactic class.

What to do with each class?


Spot implements algorithms from Löding (Efficient minimization of deterministic weak ω-automata, IPL 2001) and Dax et al. (Mechanizing the powerset constructions for restricted classes of ω-automata, ATVA'07) in order to detect obligation properties, and produce minimal weak deterministic automata for them.

When running ltl2tgba -D on a formula that represents an obligation property, you are guaranteed to obtain a minimal (in the number of states) deterministic weak Büchi automaton that recognizes it. Note that since the obligation class includes the safety and guarantee classes, minimal deterministic automata will also be produced for those classes. Dax et al.'s determinization of obligation properties combined with Löding's minimization renders obsolete older algorithms (and tools) that produced minimal deterministic automata but only for the subclasses of safety or guarantee.

If ltl2tgba is run without -D (but still with the default --high optimization level), the minimal weak deterministic automaton will only be output if it is smaller than the non-deterministic automaton the translator could produce before determinization and minimization.

For instance Fa R b is an obligation:

ltlfilt -f 'Fa R b' --format='%[v]h'

If we translate it without -D we get a 3-state non-deterministic automaton (here we use autfilt --highlight-nondet to show where the non-determinism occurs):

ltl2tgba 'Fa R b' | autfilt --highlight-nondet -d

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Note that the default translation used by ltl2tgba will turn any syntactic persistence formulas (this includes obligations formulas) into a weak automaton. In a weak automaton, the acceptance condition could be defined in terms of SCCs, i.e., the cycles of some SCCs are either all accepting, or all rejecting. As a consequence, if there is no incentive to use transition-based acceptance, state-based acceptance is output by default.

With ltl2tgba -D we get a (minimal) deterministic weak Büchi automaton instead.

ltl2tgba -D 'Fa R b' -d

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When we called ltl2tgba, without the option -D, the two automata (non-deterministic and deterministic) were constructed, but the deterministic one was discarded because it was bigger. Using -D forces the deterministic automaton to be used regardless of its size.

The detection and minimization of obligation properties is also used by autfilt when simplifying deterministic automata (they need to be deterministic so that autfilt can easily compute their complement).

For instance, let us use ltl2dstar to construct a Streett automaton for the obligation property Ga | XFb.

ltldo 'ltl2dstar --automata=streett' -f 'Ga | XFb' -d

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We can now minimize this automaton with:

ltldo 'ltl2dstar --automata=streett' -f 'Ga | XFb' | autfilt -D -C -d

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Here we have used option -C to keep the automaton complete, so that the comparison with ltl2dstar is fair, since ltl2dstar always output complete automata.


Guarantee properties can be translated into terminal automata. There is nothing particular in Spot about guarantee properties, they are all handled like obligations.

Again, using -D will always produce (minimal) deterministic Büchi automata, even if they are larger than the non-deterministic version. The output should be a terminal automaton in either case,

An example is a U Xb:

ltlfilt -f 'a U Xb' --format='%[v]h'
ltl2tgba 'a U Xb' | autfilt --highlight-nondet -d

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ltl2tgba -D 'a U Xb' -d

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Safety properties also form a subclass of obligation properties, and again there is no code specific to them in the translation. However, the safety class corresponds to what can be represented faithfully by monitors, i.e., automata that accept all their infinite runs.

For most safety formulas, the acceptance output by ltl2tgba will already be t (meaning that all runs are accepting). However since the translator does not do anything particular about safety formulas, it is possible to find some pathological formulas for which the translator outputs a non-deterministic Büchi automaton where not all runs are accepting.

Here is an example:

ltlfilt -f '(a W Gb) M b' --format='%[v]h'
ltl2tgba '(a W Gb) M b' | autfilt --highlight-nondet -d

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Actually, marking all states of this automaton as accepting would not be wrong, the translator simply does not know it.

Using -D will fix that: it then produces a deterministic automaton that is guaranteed to be minimal, and where all runs are accepting.

ltl2tgba -D '(a W Gb) M b' -d

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If you are working with safety formula, and know you want to work with monitors, you can use the -M option of ltl2tgba. In this case this will output the same automaton, but using the universal acceptance (i.e. t). You can interpret this output as a monitor (i.e., a finite automaton that accepts all prefixes that can be extended into valid ω-words).

ltl2tgba -M '(a W Gb) M b' | autfilt --highlight-nondet -d

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ltl2tgba -M -D '(a W Gb) M b' -d

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Note that the -M option can be used with formulas that are not safety properties. In this case, the output monitor will recognize a language larger than that of the property.


Recurrence properties can be represented by deterministic Büchi automata.

For the subclass of obligation properties, using -D is a sure way to obtain a deterministic automaton (and even a minimal one), but for the recurrence properties that are not obligations the translator does not make too much effort to produce deterministic automata, even with -D (this might change in the future).

All properties that are not in the persistence class (this includes the recurrence properties that are not obligations) can benefit from transition-based acceptance. In other words using transition-based acceptance will often produce shorter automata.

The typical example is GFa, which can be translated into a 1-state transition-based Büchi automaton:

ltlfilt -f 'GFa' --format='%[v]h'
ltl2tgba 'GFa' -d

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Using state-based acceptance, at least two states are required. For instance:

ltl2tgba -S 'GFa' -d

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Here is an example of a formula for which ltl2tgba does not produce a deterministic automaton, even with -D.

ltlfilt -f 'G(Gb | Fa)' --format='%[v]h'
ltl2tgba -D 'G(Gb | Fa)' | autfilt --highlight-nondet -d

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One way to obtain a deterministic Büchi automaton (it has to exist, since this is a recurrence property), is to request a deterministic automaton with parity acceptance using -P. The number of color output with -P is always reduced to the minimal number possible, so for a recurrence property the output automaton can only have one of three possible acceptance: Inf(0), t, or f.

ltl2tgba -P -D 'G(Gb | Fa)' -d

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Note that if the acceptance is t, the property is a monitor, and if its f, the property is false. In any way, if you would like to obtain a DBA for any recurrent property, a sure way to avoid these difference is to pipe the result through autfilt -B

ltl2tgba -P -D 'G(Gb | Fa)' | autfilt -B -d

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It is likely that ltl2tgba -B -D will implement these steps in the future, but so originally -D was only expressing a preference not a requirement.


Since persistence properties are outside the recurrence class, they cannot be represented by deterministic Büchi automata. The typical persistence formula is FGa, and using -D on this is hopeless.

ltl2tgba -D FGa -d

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However since the negation of FGa is a recurrence, this negation can be represented by a deterministic Büchi automaton, which means that FGa could be represented by a deterministic co-Büchi automaton. ltl2tgba does not generate co-Büchi acceptance, but we can do the complementation ourselves:

ltl2tgba --negate -D FGa | autfilt --complement -d

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Note that in this example, we know that GFa is trivial enough that ltl2tgba -D GFa will generate a deterministic automaton. In the general case we might have to determinize the automaton using -P -D as we did in the previous section. For persistence properties, -P -D should return an automaton whose acceptance is one of Fin(0), t, or f.

Persistence properties can be represented by weak Büchi automata. The translator is aware of that, so when it detects that the input formula is a syntactic-persistence, it simplifies its translation slightly to ensure that the output will use at most one acceptance set. (It is possible to define a persistence property using an LTL formula that is not a syntactic-persistence, in that case this optimization is simply not applied.)

If the input is a weak property that is not syntactically weak, the output will not necessarily be weak. One costly way to obtain a weak automaton for a formula \(\varphi\) would be to first compute a deterministic co-Büchi automaton \(\varphi\) then transform that into a Büchi automaton.

Let's do that on the persistence formula F(G!a | G(b U a)), just for the fun of it.

ltlfilt -f 'F(G!a | G(b U a))' --format='%[v]h'

Unfortunately the default output of the translation is not weak:

ltl2tgba 'F(G!a | G(b U a))' -d

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So let's determinize using parity acceptance:

ltl2tgba -P -D 'F(G!a | G(b U a))' -d

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And finally we convert the result back to Büchi with autfilt -B.

ltl2tgba -P -D 'F(G!a | G(b U a))' | autfilt -B --highlight-nondet --small -d

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That is indeed, a weak non-deterministic automaton.