Concepts

An atomic proposition is a named Boolean variable that represents a simple property that must be true or false. It usually represents some property of a system. For instance light_on and door_open could be the names of two atomic propositions that are respectively true if the light is on and the door open, and false otherwise.

Atomic propositions are used to construct temporal logic formulas (see below) to specify properties of the system: for instance we might want to state that whenever the door is open, the light should be on. We could write that as the LTL formula G(door_open -> light_on) in which G is a temporal operator that means always.

Atomic propositions are also used to form the Boolean formulas that label the edges of automata.

A Boolean formula is formed from atomic propositions, the Boolean constants true and false, and standard Boolean operators like and, or, implies, xor, etc.

A Binary Decision Diagram is a data structure for efficient manipulation of Boolean formulas.

BDDs correspond to a kind of if-then-else normal form for Boolean formulas. If we fix the order in which the atomic propositions will be tested, that normal form is unique. BDDs are stored as directed acyclic graphs with sharing of subformulas.

For further information about BDDs, read for instance Henrik Reif Andersen's lecture notes.

In Spot, BDDs are one way to represent Boolean formulas, and in particular, they are used to labels the edges of automata. Spot uses a customized version of the BuDDy library for manipulating BDDs.

An ω-word (omega-word) is a word of infinite length. In our context, each letter is used to describe the state of a system at a given time, and the sequence of letters shows the evolution of the system as the (discrete) time is incremented.

If the set \(AP\) of atomic propositions is fixed, an ω-word over \(AP\) is an infinite sequence of subsets of \(AP\). In other words, there are \(2^{|AP|}\) possible letters to choose from, and these letters denote the set of atomic propositions that are true at a given instant.

For instance if \(AP=\{a,b,c\}\), the infinite sequence \[\{a,b\};\{a\};\{a,b\};\{a\};\{a,b\};\{a\};\ldots\] is an example of ω-word over \(AP\). This particular ω-word can be interpreted as the following scenario: atomic proposition \(a\) is always true, \(b\) is true at each other instant, and \(c\) is always false.

Note that instead of using sets of atomic propositions, it is equivalent to write that word using minterms over \(AP\): \[(a\land b\land \bar c);(a\land \bar b\land \bar c); (a\land b\land \bar c);(a\land \bar b\land \bar c); (a\land b\land \bar c);(a\land \bar b\land \bar c);\ldots\]

An ω-automaton is used to represent sets of ω-word.

Those look like the classical Nondeterministic Finite Automata in the sense that they also have states and transitions. However ω-automata recognize ω-words instead of finite words. In this context, the notion of final state makes no sense, and is replaced by the notion of acceptance condition: a run of the automaton (i.e., an infinite sequence alternating states and edges in a way that is compatible with the structure of the automaton) is accepting if it satisfies the constraint given by the acceptance condition.

In Spot, ω-automata have their edges labeled by Boolean formulas represented using BDDs. An ω-word is accepted by an ω-automaton if there exists an accepting run whose labels (those Boolean formulas) are compatible with the minterms used as letters in the word.

The language of an ω-automaton is the set of ω-words it accepts.

There are many kinds of ω-Automata and they mostly differ by their acceptance condition. The different types of acceptance condition, and whether the automata are deterministic or not can affect their expressive power.

One of the simplest and most common type of ω-Automata is the Büchi automaton described next.

A Büchi automaton is a simple kind of ω-Automaton in which a run is accepting iff it visits some accepting state infinitely often. Those accepting states are often denoted using a double circle.

For instance here is a Büchi automaton that accepts only words in which \(a\) is always true, and \(b\) is true infinitely often.

The above automaton would accept the ω-word we used previously as an example.

As a more concrete example, here is a (complete) Büchi automaton for the LTL formula G(door_open -> light_on) that specifies that light_on should be true whenever door_open is true.

The 1 displayed on the edge that loops on state 1 should be read as true, i.e., the Boolean formula that accepts any valuation of the atomic propositions.

The above automaton is complete: any possible ω-word over \(AP=\{\mathit{door\_open}, \mathit{light\_on}\}\) is recognized by some run. But not all those runs are accepting. In fact, there is only one run that is accepting: the one that loops continuously on state 0.

All the remaining runs eventually reach state 1 and stay there. Those runs recognize scenarios where at some point the door is open and the light is off. There is an infinite number of those runs: they differ by the number of times they loop on state 0. But since those runs reach state 1, it means they visited state 0 only a finite number of times, so they do not validate the acceptance condition.

There can be multiple accepting states, but it is enough to visit one infinitely often. For instance the following Büchi automaton accept all runs in which at all point \(a\) is true iff \(b\) is true at the next instant.

Since automata are labeled by Boolean formulas instead of letters it is sometimes useful to think of the formula-labeled edges of an automaton as a way to aggregate several letter-labeled transitions.

Whenever the distinction is important, for instance when giving the size of an automaton, we use the terms edge and transition to distinguish whether we are looking at the automaton as a graph, or whether we are actually considering all possible letters that may have been aggregated in an edge.

Here is a simple example:

The above automaton has edges and transitions.

This is because those formula-labeled edges actually simplify the following transition structure:

The above is actually a different automaton from the point of view of Spot: it is an automaton with edges and as many transitions.

Spot has some function to merge those "parallel transitions" into larger edges. Limiting the number of edges helps most of the algorithms that have to explore automata, since they have less successors to consider.

The distinction between edge and transition is something we try maintain in the various interfaces of Spot. For instance the --stats option has %e or %t to count either edges or transitions. The method used to add new edge into an automaton is called new_edge(...), not new_transition(...), because it takes a BDD (representing a Boolean formula) as label. However that naming convention is recent in the history of Spot. Spot versions up to 1.2.6 used to call everything transition (and what we now call transition was sometime called sub-transition), and traces of this history may still be present: do not hesitate to file bug reports if you uncover some confusing use of these terms.

As a rather straightforward generalization of the Büchi acceptance, let us consider that instead of one set of accepting states, we might have multiple sets of states. We call these sets acceptance sets. The generalized Büchi acceptance condition states that a run is accepting iff it visits at least one state of each acceptance set.

The Büchi convention of representing accepting states using a double circle is not going to work in the generalized Büchi case. So instead we label each state with the numbers of each acceptance set it belongs to.

In the automaton below, there are two acceptance sets denoted with ⓿ and ❶: all states labeled with ⓿ belong to acceptance set 0, and all states labeled with ❶ belong to set 1. Here each acceptance set contains a single state.

The accepting runs are only those that visit infinitely often both states, so that means this automaton accepts all words in which \(a\) and \(b\) are infinitely often true (not necessarily at the same time).

A state can of course belong to multiple acceptance sets, and an acceptance set may contain multiple states. For instance the following automaton has the same language as the previous one (despite its more complex look).

Speaking of size… Let us note that using a generalized Büchi acceptance condition makes it possible to build smaller automata than what we can do with Büchi acceptance. We have seen that the above language (infinitely often \(a\) and infinitely often \(b\)) can be built with a 2-state generalized-Büchi automaton, but the smallest equivalent Büchi automaton has three state:

Finally, let us point the obvious fact that a Büchi automaton is a particular case of generalized-Büchi acceptance with a single acceptance set. Depending on the context, it might be useful to represent Büchi automaton using double circles (as above), or numbered acceptance sets (as below). Spot's output routines have options for both.

So far we have discussed examples of state-based acceptance: acceptance sets are sets of states, runs are accepting if these visit infinitely often some state in each acceptance set, etc.

When transition-based acceptance is used, acceptance sets are now sets of edges (or set of transitions if you prefer), and runs are accepting if the edges they visit satisfy the acceptance condition.

Here is an example of Transition-based Büchi Automaton (TBA).

This automaton accept all ω-words that infinitely often match the pattern \(a^+;b\) (that is: a positive number of letters where \(a\) is true are followed by one letter where \(b\) is true).

Using transition-based acceptance often allows for more compact automata. For instance the above automaton would need at least 3 states with state-based acceptance:

Older versions of Spot (up to 1.2.6), used to support only Transition-based Generalized Büchi Automata (TGBA). This of course included support for non-generalized or state-based Büchi.

Today, Spot can work with more general forms of acceptance condition. An acceptance condition actually consists of two pieces: some acceptance sets, and a formula that tells how to use these acceptance sets.

Acceptance formulas are positive Boolean formula over atoms of the form t, f, Inf(n), or Fin(n), where n is a non-negative integer denoting an acceptance set.

• t denotes the true acceptance condition: any run is accepting
• f denotes the false acceptance condition: no run is accepting
• Inf(n) means that a run is accepting if it visits infinitely often the acceptance set n
• Fin(n) means that a run is accepting if it visits finitely often the acceptance set n

The above atoms can be combined using only the operator & and | (with obvious semantics), and parentheses for grouping. Note that there is no negation, but an acceptance condition can be negated swapping t and f, & and |, and Fin(n) and Inf(n).

For instance the formula Inf(0)&Inf(1) specifies that accepting runs should visit infinitely often the acceptance 0, and infinitely often the acceptance set 1. This corresponds the generalized Büchi acceptance with two sets.

The opposite acceptance condition Fin(0)|Fin(1) is known as generalized co-Büchi acceptance (with two sets). Accepting runs have to visit finitely often set 0 or finitely often set 1.

A Rabin acceptance condition with 3 pairs corresponds to the following formula: (Fin(0)&Inf(1)) | (Fin(2)&Inf(3)) | (Fin(4)&Inf(5))

The following table gives an overview of how some classical acceptance condition are encoded. The first column gives a name that is more human readable (those names are defined in the HOA format and are also recognized by Spot). The second column give the encoding as a formula. Everything here is case-sensitive.

Spot's automata support arbitrary acceptance conditions as discussed above. When displaying automata, it is convenient to display the acceptance condition as well. For instance here is a Rabin automaton produced by ltl2dstar for the LTL formula GFa | FGb, but displayed by Spot:

Alternating ω-automata are ω-automata in which the destination of an edge can be a group of states. If an edge has more than one destination, it is called a universal edge, and its destinations are referred to as its universal destinations.

When an alternating automaton evaluates a word, following a universal edge will have the same effect as forking the automaton to evaluate the rest of the word simultaneously from each universal destination. A run of an alternating automaton can therefore be pictured as a tree. The tree is accepting if all its branches satisfy the acceptance condition. (See the Hanoi Omega-Automata format for more precise semantics.)

For instance the following alternating co-Büchi ω-automaton was generated by ltl3ba 1.1.3 for the LTL formula GF(a <-> Xb).

In this picture, the universal edges appear as arrows with a white tip going to a small dot, from which additional arrows connect to the universal destinations. Here the three universal edges all leave the initial state, and connect to two universal destinations. Note that non-determinism is allowed between universal edges, for instance upon reading a word starting with "a", this automaton should non-deterministically decide to read the rest of the word from states GF(a<->Xb) and F(a<->Xb) (when taking the universal transition labeled by 1) or from states GF(a<->Xb) and b (when taking the universal transition labeled by a).

Alternation support in Spot is currently experimental, please report any issue. The only supported file format able to represent alternating automata is the HOA format, introduced below.

Never claims are used by Spin to represent Büchi automata; they are part of the Promela language.

Here are two never claims using different syntaxes to represent a Büchi automaton for the LTL formula p0 | GFp1 (that is: \(p_0\) or infinitely often \(p_1\)). The graphical representation of that automaton follows.

never { /* p0 | GFp1 */			never { /* p0 | GFp1 */
T0_init:				T0_init:
  if					  do
  :: (p0) -> goto accept_all		  :: atomic { (p0) -> assert(!(p0)) }
  :: (!(p0)) -> goto accept_S2		  :: (!(p0)) -> goto accept_S2
  fi;					  od;
accept_S2:				accept_S2:
  if					  do
  :: (p1) -> goto accept_S2		  :: (p1) -> goto accept_S2
  :: (!(p1)) -> goto T0_S3		  :: (!(p1)) -> goto T0_S3
  fi;					  od;
T0_S3:					T0_S3:
  if					  do
  :: (p1) -> goto accept_S2		  :: (p1) -> goto accept_S2
  :: (!(p1)) -> goto T0_S3		  :: (!(p1)) -> goto T0_S3
  fi;					  od;
accept_all:				accept_all:
  skip					  skip
}					}

The two different types of never claims differ only in a few syntactic elements: do..od instead of if..fi, assert instead of goto accept_all, etc. Older Spin releases used to output the first one, while newer Spin releases (starting with Spin 6.2.4) use the second syntax as they help Spin to produce more precise counterexamples.

Spot can read and write never claims in both syntaxes, but it cannot parse never claims that use other features (such as variables) of the Promela language.

This format was originally introduced by LBT, a tool for translating LTL to (state-based) generalized Büchi automata, and then used by LBTT, a tool for testing LTL-to-Büchi translators.

For instance the Büchi automaton we used as an example for never claims can be encoded as follows:

4 1
0 1 -1
1 p0
2 ! p0
-1
1 0 0 -1
1 t
-1
2 0 0 -1
2 p1
3 ! p1
-1
3 0 -1
2 p1
3 ! p1
-1

The format has been extended in two ways. First, LBTT extended it to support transition-based acceptance. This is indicated by a t on the first line:

3 1t
0 1
1 -1 p0
2 -1 ! p0
-1
1 0
1 0 -1 t
-1
2 0
2 -1 ! p1
2 0 -1 p1
-1

We call this format the LBTT format because of this extension.

A second, but independent extension, was done in ltl2dstar, allowing atomic propositions that are different from p0, p1, p2, etc.

Both extensions are supported by Spot.

The DSTAR format is the native format of ltl2dstar. It allows representing Deterministic Streett And Rabin automata, hence the name.

Spot can read the DSTAR format, but it does not output it. Adding output for this format would not be difficult, but it would also not be very useful: for all intents and purposes, the HOA format should be preferred. ltl2dstar can now also output HOA directly.

Here is one Rabin automaton in the DSTAR format:

DRA v2 explicit
Comment: "Union{Safra[NBA=2],Safra[NBA=2]}"
States: 4
Acceptance-Pairs: 2
Start: 0
AP: 2 "p0" "p1"
---
State: 0
Acc-Sig: -0
0
1
2
3
State: 1
Acc-Sig: +0
0
1
2
3
State: 2
Acc-Sig: -0 +1
0
1
2
3
State: 3
Acc-Sig: +0 +1
0
1
2
3

The HOA format inherits several features from the DSTAR format, but extends it in many ways, including support for non-deterministic automata, alternating automata, and for arbitrary acceptance conditions.

HOA: v1
name: "FGp0 | GFp1"
States: 4
Start: 0
AP: 2 "p0" "p1"
acc-name: Rabin 2
Acceptance: 4 (Fin(0) & Inf(1)) | (Fin(2) & Inf(3))
properties: trans-labels explicit-labels state-acc complete
properties: deterministic
--BODY--
State: 0 {0}
[!0&!1] 0
[0&!1] 1
[!0&1] 2
[0&1] 3
State: 1 {1}
[!0&!1] 0
[0&!1] 1
[!0&1] 2
[0&1] 3
State: 2 {0 3}
[!0&!1] 0
[0&!1] 1
[!0&1] 2
[0&1] 3
State: 3 {1 3}
[!0&!1] 0
[0&!1] 1
[!0&1] 2
[0&1] 3
--END--

Since this file format is the only one able to represent the range of ω-automata supported by Spot, and it its default output format.

However note that Spot does not support all automata that can be expressed using the HOA format. The present support for the HOA format in Spot, is discussed on a separate page, with a section dedicated to the restrictions.

The Linear-time Temporal Logic (LTL) extends propositional logic with operators that refer to the future. Some definitions of LTL also include past operators, but Spot only supports future operators. The view of the time is discrete: a scenario can be seen as a succession of steps in which each atomic proposition can have a different value.

The following basic operators are supported:

For instance the LTL formula G(request -> F(response)) specifies that whenever request atomic proposition is true, there exists a later instant (possibly the same) where response is true.

Spot supports several syntaxes for writing LTL formulas. For example some people prefer to write <> and [] instead of F and G, R is written V in some tools, etc.

For more discussion about the temporal operators and their semantics, see the tl.pdf document.

Spot supports the linear fragment of PSL, this basically extends LTL with semi-extended regular expressions. Those regular expressions can express finite languages and PSL introduces operators to use these finite languages as a prefix of a PSL formula.

In the above table e is a semi-extended expression, and f is a PSL (or LTL) formula.

Semi-extended regular expressions can be formed using Boolean expressions over atomic propositions and the following operators:

For example the formula {(1;1)[*]}[]->a can be interpreted as follows:

• the SERE (1;1)[*] matches all prefixes of even length (here 1 stands for the true formula, so it matches anything)
• the part ...[]->a requests that a should be true at the end of each matched prefix.

Therefore this formula ensures that a is true at every even instant (if we consider the first instant to be odd). This is the canonical example of formula that can be expressed in PSL but not in LTL.

A few other operators and syntactic sugar are supported. For more discussion about the temporal operators and their semantics, see the tl.pdf document.

Spot can translate any LTL or PSL formula into Büchi automata, or generalized Büchi automata.

Internally the translator produces Transition-based Generalized Büchi Automata (TGBA) but that automaton can then be simplified using several algorithms depending on what options were given.

Here is for instance a translation of {(1;1)[*]}[]->a discussed above.

ltl2tgba '{(1;1)[*]}[]->a' -d

Another page shows how to translate an LTL formula into a never claim from the command-line, Python, or C++.

The Spot project can be broken down into several parts, as shown above. Orange boxes are C/C++ libraries. Red boxes are command-line programs. Blue boxes are Python-related. The gray outline shows the components that are distributed and installed by the Spot tarball. Green boxes represent online services that build upon the Python layers.

• libbddx is a customized version of the BuDDy library, for manipulating BDDs.
• libspot is the main library, containing a C++17 implementation of all the data structures and algorithms. This depends on libddx.
• libspotgen is an auxiliary library that contains functions to generate families of automata, useful for benchmarking and testing
• all the supplied command-line tools distributed with Spot are built upon the libspot or libspotgen libraries
• libspotltsmin is a library that helps interfacing Spot with dynamic libraries that LTSmin uses to represent state-spaces. It currently supports libraries generated from Promela models using SpinS or a patched version of DiVinE, but you have to install those third-party tools first. See tests/ltsmin/README for details.
• In addition to the C++17 API, we also provide Python bindings for libspotgen, libspotltsmin, libbddx, and most of libspot. These are available by importing spot.gen, spot.ltsmin, bdd, and spot. Those Python bindings also includes some additional code to make them more usable in interactive environments such as the IPython/Jupyter notebook.
• While the online services described pictured in green are not distributed with the rest of Spot, their source-code is publicly available (in case you want to contribute or run a local version). The spot-sandbox website runs from a Docker container whose configuration can be found in this repository. The client and server parts of the online LTL translator can be found in this repository.

The automaton class used by Spot to represent ω-Automata is called twa (because we use TωA as a short for Transition-based ω-Automaton). As its names implies, the twa class supports only transition-based acceptance, but as discussed previously we can emulate state-based acceptance using transition-based acceptance by ensuring that all transitions leaving a state have the same acceptance set membership. In addition, there is a bit in the twa class that we can set to indicate that the automaton is meant to be considered with state-based acceptance: this allows some algorithms to make better choices.

There are actually several property flags that are stored into each automaton, and that can be queried or set by algorithms:

For each flag flagname, the twa class has a method prop_flagname() that returns the value of the flag as an instance of trival, and there is a method prop_flagname(trival newval) that sets that value.

trival instances can take three values: false, true, or trival::maybe. The idea is that algorithms should update flags as a side effect of their execution, but only if that does not induce some extra cost. For instance when translating an LTL formula into an automaton, we can set the stutter_invariant properties to true if the input formula does not use the X operator, but we would leave the flag to trival::maybe if X is used: the presence of such an operator X does not prevent the formula from being stutter-invariant, but it would require additional work to check.

As another example, if you write an algorithm that must check whether an automaton is universal, do not call the twa::prop_universal() method, because that might return trival::maybe. Instead, call spot::is_universal(...): that will respond in constant time if the universal property flag was either true or false, otherwise it will actually explore the automaton to decide its determinism. Note that there is also a spot::is_deterministic(...) function, which is equivalent to testing that the automaton is both universal and existential.

These automata properties are encoded into the HOA format, so they can be preserved when building a processing pipeline using the shell. However the HOA format has support for more properties that do not correspond to any twa flag.

In addition to property flags, automata in Spot can be tied to an arbitrary number of objects via a system of named properties that is implemented mostly as an std::map between std::string and void*.

A property can be used to store additional information about the automaton, that is not usually available via the automaton interface. The property can be set via the twa::set_named_prop(key, value) method, and queried with the twa::get_named_prop<type>(key) template method.

Here is a list of named properties currently used inside Spot:

Objects referenced via named properties are automatically destroyed when the automaton is destroyed, but this can be altered by passing a custom destructor as a third parameter to twa::set_named_prop().

These properties should be considered short-lived. They are usually not propagated to new automata that are created via transformation, unless the algorithm has been explicitly implemented to preserve that property. Algorithms that update the automaton in place should probably call release_named_properties() to ensure they do not inadvertently keep a stale property.

Most of the above properties are related to the graphical display of automata, or to their output in the HOA format. So they are usually set right before the automaton is output. The notable exception is product-states, which is a property present in automata returned by spot::product() function in case it is necessary to know the origins of each state.