# -*- coding: utf-8 -*-
#+TITLE: Concepts
#+DESCRIPTION: Informal explanation of various concepts used in Spot.
#+INCLUDE: setup.org
#+HTML_LINK_UP: index.html

This page documents some of the concepts used in Spot, and whose
knowledge is usually assumed throughout the documentation.  The
presentation is informal on purpose.

* Atomic proposition (AP)
   :PROPERTIES:
   :CUSTOM_ID: ap
   :END:

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][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][Boolean formulas]] that
label the edges of automata.

* Boolean formula
   :PROPERTIES:
   :CUSTOM_ID: boolean
   :END:

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

* Binary Decision Diagrams (BDD)
   :PROPERTIES:
   :CUSTOM_ID: bdd
   :END:

A Binary Decision Diagram is a data structure for efficient
manipulation of [[#boolean][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 [[http://configit.com/configit_wordpress/wp-content/uploads/2013/07/bdd-eap.pdf][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 [[#buchi][automata]]. Spot uses a
customized version of [[https://sourceforge.net/projects/buddy/][the BuDDy library]] for manipulating BDDs.

* ω-word
   :PROPERTIES:
   :CUSTOM_ID: word
   :END:

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 [[#ap][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 [[https://en.wikipedia.org/wiki/Canonical_normal_form#Minterms][minterms]] over $AP$:
\[(a\land b\land \bar c);(a\land \bar b\land \bar c);
  (a\land b\land \bar c);(a\land \bar b\land \bar c);
  (a\land b\land \bar c);(a\land \bar b\land \bar c);\ldots\]

* ω-Automaton
   :PROPERTIES:
   :CUSTOM_ID: automaton
   :END:

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

Those look like the classical [[https://en.wikipedia.org/wiki/Nondeterministic_finite_automaton][Nondeterministic Finite Automata]] in the
sense that they also have states and transitions.  However ω-automata
recognize [[#word][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][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][Boolean formulas]]
represented using [[#bdd][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][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 [[#buchi][Büchi
automaton]] described next.

* Büchi automaton
   :PROPERTIES:
   :CUSTOM_ID: buchi
   :END:

A Büchi automaton is a simple kind of [[#automaton][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.

#+NAME: buchi-example1
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'G(a) & GF(b)' -B -d
#+END_SRC

#+BEGIN_SRC dot :file concept-buchi1.svg :var txt=buchi-example1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-buchi1.svg]]


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


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

#+NAME: buchi-example2
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'G(door_open -> light_on)' -d -C
#+END_SRC

#+BEGIN_SRC dot :file concept-buchi2.svg :var txt=buchi-example2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-buchi.svg]]

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.

#+NAME: buchi-example3
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'G(a <-> Xb)' -B -d
#+END_SRC
#+BEGIN_SRC dot :file concept-buchi3.svg :var txt=buchi-example3 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-buchi3.svg]]

* Transitions vs. Edges
   :PROPERTIES:
   :CUSTOM_ID: trans-edge
   :END:

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:
#+NAME: te1
#+BEGIN_SRC sh :results verbatim :exports none
cat >concept-te.hoa <<EOF
HOA: v1
Acceptance: 0 t
Start: 0
States: 2
AP: 2 "a" "b"
--BODY--
State: 0 0 0 1 1
State: 1 1 0 0 0
--END--
EOF
autfilt --merge concept-te.hoa -d.A
#+END_SRC
#+BEGIN_SRC dot :file concept-te1.svg :var txt=te1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-te1.svg]]

#+NAME: size
#+BEGIN_SRC sh :exports none
autfilt --merge --stats=$arg concept-te.hoa
#+END_SRC

The above automaton has call_size(arg="%e")[:results raw] edges and
call_size(arg="%t")[:results raw] transitions.

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

#+NAME: te2
#+BEGIN_SRC sh :results verbatim :exports none
autfilt concept-te.hoa -d.A
#+END_SRC
#+BEGIN_SRC dot :file concept-te2.svg :var txt=te2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-te2.svg]]

The above is actually a different automaton from the point of view of
Spot: it is an automaton with call_size(arg="%t")[:results raw] 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
[[file:oaut.org::#stats][=--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][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.

* Acceptance sets & generalized Büchi acceptance
   :PROPERTIES:
   :CUSTOM_ID: acceptance-set
   :END:

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.

#+NAME: gen-buchi-example1
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'GFa & GFb' | autfilt -S --sat-minimize -d
#+END_SRC

#+BEGIN_SRC dot :file concept-gba1.svg :var txt=gen-buchi-example1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-gba1.svg]]

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).

#+NAME: gen-buchi-example2
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'GFa & GFb' -S -d
#+END_SRC

#+BEGIN_SRC dot :file concept-gba2.svg :var txt=gen-buchi-example2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-gba2.svg]]


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:

#+NAME: gen-buchi-example-ba
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'GFa & GFb' -B -d
#+END_SRC

#+BEGIN_SRC dot :file concept-gba-vs-ba.svg :var txt=gen-buchi-example-ba :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-gba-vs-ba.svg]]

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.

#+NAME: gen-buchi-example-ba2
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'GFa & GFb' -B -d.b
#+END_SRC

#+BEGIN_SRC dot :file concept-gba-vs-ba2.svg :var txt=gen-buchi-example-ba2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-gba-vs-ba2.svg]]

* Transition-based, vs. State-based acceptance
   :PROPERTIES:
   :CUSTOM_ID: trans-acc
   :END:

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 Generalized Büchi Automaton
(TGBA).

#+NAME: tgba-example1
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'GF(a & X(a U b))' -d
#+END_SRC
#+BEGIN_SRC dot :file concept-tgba1.svg :var txt=tgba-example1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-tgba1.svg]]

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 allows for more compact automata.
The typical example is the LTL formula =GFa= (infinitely often $a$)
that can be represented using a one-state transition-based Büchi
automaton:
#+NAME: tgba-example2
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'GFa' -d
#+END_SRC
#+BEGIN_SRC dot :file concept-tgba2.svg :var txt=tgba-example2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-tgba2.svg]]

While the same property require a 2-state Büchi automaton using
state-based acceptance:

#+NAME: tgba-example3
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba 'GFa' -B -d
#+END_SRC
#+BEGIN_SRC dot :file concept-tba-vs-ba.svg :var txt=tgba-example3 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-tba-vs-ba.svg]]


#+BEGIN_implem
Internally, instead of representing /acceptance sets/ as actual sets
of edges, Spot labels each edge of the automaton by a bit-vector that
lists the acceptance sets an edge belongs to.

There is [[#property-flags][a flag]] inside each automaton that tells Spot if an automaton
uses state-based or transition-based acceptance.  However, regardless
of the value of this flag, membership to acceptance sets is always
stored on transitions.  In the case of an automaton with state-based
acceptance, the convention is that all transition leaving a state will
carry the acceptance-set membership of that state.  Doing so allows us
to interpret an automaton state-based acceptance as if it was an
automaton with transition-based acceptance whenever needed.
#+END_implem

* Acceptance condition
   :PROPERTIES:
   :CUSTOM_ID: acceptance-condition
   :END:

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][HOA]] format and are also
recognized by Spot).  The second column give the encoding as a
formula.  Everything here is case-sensitive.

#+BEGIN_SRC python :results verbatim raw :exports results
  import spot
  # org-mode recognize | as a table delemiter even in ~|~ or =|= but the
  # documented workaround to use \vert{} does not work in ~\vert{}~ or =\vert{}=.
  # So until we have a better solution, let's leave the =...= mode to display
  # \vert{} characters.
  def line(arg):
    return ('| {} | {} |\n'
            .format(arg, '={}='.format(spot.acc_code(arg)).replace(' | ','|')
                                                          .replace('|','= \\vert{} =')))
  return "".join(map(line,
                     ["none", "all", "Buchi", "generalized-Buchi 2",
                      "generalized-Buchi 3", "co-Buchi",
                      "generalized-co-Buchi 2", "generalized-co-Buchi 3",
                      "Rabin 1", "Rabin 2", "Rabin 3", "Streett 1",
                      "Streett 2", "Streett 3",
                      "generalized-Rabin 3 1 0 2", "parity min odd 5",
                      "parity max even 5"]))
#+END_SRC

#+RESULTS:
| none                      | =f=                                                                                 |
| all                       | =t=                                                                                 |
| Buchi                     | =Inf(0)=                                                                            |
| generalized-Buchi 2       | =Inf(0)&Inf(1)=                                                                     |
| generalized-Buchi 3       | =Inf(0)&Inf(1)&Inf(2)=                                                              |
| co-Buchi                  | =Fin(0)=                                                                            |
| generalized-co-Buchi 2    | =Fin(0)= \vert{} =Fin(1)=                                                           |
| generalized-co-Buchi 3    | =Fin(0)= \vert{} =Fin(1)= \vert{} =Fin(2)=                                          |
| Rabin 1                   | =Fin(0) & Inf(1)=                                                                   |
| Rabin 2                   | =(Fin(0) & Inf(1))= \vert{} =(Fin(2) & Inf(3))=                                     |
| Rabin 3                   | =(Fin(0) & Inf(1))= \vert{} =(Fin(2) & Inf(3))= \vert{} =(Fin(4) & Inf(5))=         |
| Streett 1                 | =Fin(0)= \vert{} =Inf(1)=                                                           |
| Streett 2                 | =(Fin(0)= \vert{} =Inf(1)) & (Fin(2)= \vert{} =Inf(3))=                             |
| Streett 3                 | =(Fin(0)= \vert{} =Inf(1)) & (Fin(2)= \vert{} =Inf(3)) & (Fin(4)= \vert{} =Inf(5))= |
| generalized-Rabin 3 1 0 2 | =(Fin(0) & Inf(1))= \vert{} =Fin(2)= \vert{} =(Fin(3) & (Inf(4)&Inf(5)))=           |
| parity min odd 5          | =Fin(0) & (Inf(1)= \vert{} =(Fin(2) & (Inf(3)= \vert{} =Fin(4))))=                  |
| parity max even 5         | =Inf(4)= \vert{} =(Fin(3) & (Inf(2)= \vert{} =(Fin(1) & Inf(0))))=                  |

* ω-Automaton with generalized acceptance
   :PROPERTIES:
   :CUSTOM_ID: automaton-generalized
   :END:

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:

#+NAME: twa-example1
#+BEGIN_SRC sh :results verbatim :exports none
ltlfilt -l -f 'GFa | FGb' | ltl2dstar --output-format=hoa - - | autfilt --merge -d
#+END_SRC
#+BEGIN_SRC dot :file concept-twa1.svg :var txt=twa-example1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-twa1.svg]]

* Alternating ω-automata

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 [[http://adl.github.io/hoaf/][Hanoi Omega-Automa format]] for more precise semantics.)

For instance the following alternating co-Büchi ω-automaton was
generated by [[https://sourceforge.net/projects/ltl3ba/][=ltl3ba 1.1.3=]] for the LTL formula =GF(a <-> Xb)=.

#+NAME: concepts-alt
#+BEGIN_SRC sh :results verbatim :exports none
autfilt -d.ba <<EOF
HOA: v1
States: 5
Start: 0
acc-name: co-Buchi
Acceptance: 1 Fin(0)
AP: 2 "a" "b"
properties: trans-labels explicit-labels state-acc univ-branch very-weak
--BODY--
State: 0 "GF(a <-> Xb)"
 [(!0)] 3&0
 [(0)] 2&0
 [t] 1&0
State: 1 "F(a <-> Xb)" {0}
 [(!0)] 3
 [(0)] 2
 [t] 1
State: 2 "b"
 [(1)] 4
State: 3 "!b"
 [(!1)] 4
State: 4 "t"
 [t] 4
--END--
EOF
#+END_SRC

#+BEGIN_SRC dot :file concepts-alt.svg :var txt=concepts-alt :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concepts-alt.svg]]

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][HOA format, introduced below]].

* Never claims
   :PROPERTIES:
   :CUSTOM_ID: neverclaim
   :END:

   Never claims are used by [[http://spinroot.com/][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.

#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba -s 'p0 | GFp1' > tmp.$$
ltl2tgba -s6 'p0 | GFp1' | pr -w80 -m -t tmp.$$ -
rm tmp.$$
#+END_SRC

#+RESULTS:
#+begin_example
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
}					}
#+end_example

#+NAME: never-ex1
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba -Bd 'p0 | GFp1'
#+END_SRC
#+BEGIN_SRC dot :file concept-never1.svg :var txt=never-ex1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-never1.svg]]

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 claim that use other features (such as variables) of the
Promela language.

* LBTT's format
   :PROPERTIES:
   :CUSTOM_ID: lbtt
   :END:

This format was originally introduced by [[http://www.tcs.hut.fi/Software/maria/tools/lbt/][LBT]], a tool for translating
LTL to (state-based) generalized Büchi automata, and then used by
[[http://www.tcs.hut.fi/Software/lbtt/][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:

#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --ba --lbtt 'p0 | GFp1'
#+END_SRC

#+RESULTS:
#+begin_example
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
#+end_example

[[file:concept-never1.svg]]

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:

#+BEGIN_SRC sh :results verbatim :exports results
ltl2tgba --lbtt 'p0 | GFp1'
#+END_SRC

#+RESULTS:
#+begin_example
3 1t
0 1
1 -1 p0
2 -1 ! p0
-1
1 0
1 0 -1 t
-1
2 0
2 0 -1 p1
2 -1 ! p1
-1
#+end_example

#+NAME: lbtt-ex2
#+BEGIN_SRC sh :results verbatim :exports none
ltl2tgba -d 'p0 | GFp1'
#+END_SRC
#+BEGIN_SRC dot :file concept-lbtt2.svg :var txt=lbtt-ex2 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-lbtt2.svg]]

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

A second, but independent extension, was done in [[http://ltl2dstar.de/][=ltl2dstar=]], allowing
atomic propositions that are different from =p0=, =p1=, =p2=, etc.

Both extensions are supported by Spot.

* DSTAR format
   :PROPERTIES:
   :CUSTOM_ID: dstar
   :END:

The DSTAR format is the native format of [[http://ltl2dstar.de/][=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][HOA]] format should be
preferred.  =ltl2dstar= can now also output HOA directly.

Here is one Rabin automaton in the DSTAR format:

#+BEGIN_SRC sh :results verbatim :exports results
echo '| F G p0 G F p1' | ltl2dstar --output-format=native - -
#+END_SRC

#+RESULTS:
#+begin_example
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
#+end_example

#+NAME: dstar-example1
#+BEGIN_SRC sh :results verbatim :exports none
echo '| F G p0 G F p1' | ltl2dstar --output-format=native - - | autfilt -d
#+END_SRC
#+BEGIN_SRC dot :file concept-dstar.svg :var txt=dstar-example1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-dstar.svg]]

* Hanoi Omega-Automaton format (HOA)
   :PROPERTIES:
   :CUSTOM_ID: hoa
   :END:

The [[http://adl.github.io/hoaf/][HOA format]] inherits several features from the [[#dstar][DSTAR format]], but
extends it in many ways, including support for non-deterministic
automata, alternating automata, and for arbitrary acceptance conditions.

#+BEGIN_SRC sh :results verbatim :exports results
ltldo ltl2dstar -f 'FGp0 | GFp1' --name=%f
#+END_SRC

#+RESULTS:
#+begin_example
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--
#+end_example

#+NAME: hoa1
#+BEGIN_SRC sh :results verbatim :exports none
ltldo ltl2dstar -f 'FGp0 | GFp1' -d
#+END_SRC
#+BEGIN_SRC dot :file concept-hoa.svg :var txt=hoa1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-hoa.svg]]


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 [[file:hoa.org][a separate page]], with a section
dedicated to the [[file:hoa.org::#restrictions][restrictions]].

* Linear-time Temporal Logic (LTL)
   :PROPERTIES:
   :CUSTOM_ID: ltl
   :END:

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 [[#ap][atomic proposition]] can have a different value.

The following basic operators are supported:

| LTL formula | meaning                                                                                        |
|-------------+------------------------------------------------------------------------------------------------|
| =f=         | the formula =f= is true immediately                                                            |
| =X f=       | =f= will be true in the next step                                                              |
| =F f=       | =f= will become true eventually (it could be true immediately, or on the future)               |
| =G f=       | =f= is always true from now on                                                                 |
| =f U g=     | =f= has to be true until =g= becomes true (and =g= /will/ become true)                         |
| =f W g=     | =f= has to be true until =g= becomes true (=f= should stay true if =g= never becomes true)     |
| =f R g=     | =g= has to be true until =f&g= becomes true (=g= should stay true if =f&g= never becomes true) |
| =f M g=     | =g= has to be true until =f&g= becomes true (and =f&g= /will/ become true)                     |

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 [[file:ioltl.org][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 [[https://spot.lrde.epita.fr/tl.pdf][tl.pdf]] document.

* Property Specification Language (PSL)
   :PROPERTIES:
   :CUSTOM_ID: psl
   :END:

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.

| PSL formula  | meaning                                                                 |
|--------------+-------------------------------------------------------------------------|
| ={e}<>->f=   | =f= should hold on the last instant of some one prefix that matches =e= |
| ={e}[]->f=   | =f= should hold on the last instant of all prefixes that match =e=      |

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 [[#ap][atomic propositions]] and the following
operators:

| SERE                 | meaning                                                                           |
|----------------------+-----------------------------------------------------------------------------------|
| =e1;e2=              | =e1= followed by =e2= (concatenation)                                             |
| =e1:e2=              | =e1= fused with =e2=: =e2= has to start matching on the last letter matching =e1= |
| =e1= \vert\vert =e2= | =e1= or =e2= have to match (union)                                                |
| =e1 && e2=           | =e1= and =e2= have to match (intersection)                                        |
| =e1 & e2=            | =e2= should match a prefix of what =e1= matches, or vice-versa                    |
| =e[*]=               | =e= should be matched a finite number of times (Kleene star)                      |
| =e[*2..3]=           | same as =(e;e)= \vert\vert =(e;e;e)=                                              |
| =e[+]=               | =e= should be matched a finite number of times, and at least once                 |


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
[[https://spot.lrde.epita.fr/tl.pdf][tl.pdf]] document.

* Translation of temporal logic to automata
   :PROPERTIES:
   :CUSTOM_ID: ltl2tgba
   :END:

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

Internally the translator produces [[#trans-acc][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 [[#psl][above]].

#+NAME: ltl2tgba1
#+BEGIN_SRC sh :results verbatim :exports code
ltl2tgba '{(1;1)[*]}[]->a' -d
#+END_SRC
#+BEGIN_SRC dot :file concept-ltl2tgba.svg :var txt=ltl2tgba1 :exports results
$txt
#+END_SRC

#+RESULTS:
[[file:concept-ltl2tgba.svg]]


[[file:tut10.org][Another page shows how to translate an LTL formula into a never claim]]
from the command-line, Python, or C++.

* Architecture of Spot
   :PROPERTIES:
   :CUSTOM_ID: architecture
   :END:

[[file:arch.svg]]

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 Spot.

  - =libbddx= is a customized version of [[https://sourceforge.net/projects/buddy/][the BuDDy library]], for
    manipulating [[#bdd][BDDs]].
  - =libspot= is the main library, containing a C++14 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 [[file:tools.org][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 [[http://fmt.cs.utwente.nl/tools/ltsmin/][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 [[https://gitlab.lrde.epita.fr/spot/spot/blob/next/tests/ltsmin/README][=tests/ltsmin/README=]]
    for details.
  - In addition to the C++14 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 [[http://juptter.org][IPython/Jupyter]] notebook.

* Automaton property flags
   :PROPERTIES:
   :CUSTOM_ID: property-flags
   :END:

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 [[#trans-acc][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:

| flag name            | meaning when =true=                                                                          |
|----------------------+----------------------------------------------------------------------------------------------|
| =state_acc=          | automaton should be considered as having state-based acceptance                              |
| =inherently_weak=    | accepting and rejecting cycles cannot be mixed in the same SCC                               |
| =weak=               | transitions of an SCC all belong to the same acceptance sets                                 |
| =very_weak=          | weak automaton where all SCCs have size 1                                                    |
| =terminal=           | automaton is weak, accepting SCCs are complete, accepting edges may not go to rejecting SCCs |
| =complete=           | for any letter ℓ, each state has is at least one outgoing transition compatible with ℓ       |
| =deterministic=      | there is at most one run *recognizing* a word, but not necessarily accepting it              |
| =semi_deterministic= | any nondeterminism occurs before entering an accepting SCC                                   |
| =unambiguous=        | there is at most one run *accepting* a word (but it might be recognized several time)        |
| =stutter_invariant=  | the property recognized by the automaton is [[https://www.lrde.epita.fr/~adl/dl/adl/michaud.15.spin.pdf][stutter-invariant]]                                |

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 [[file:hoa.org::#property-bits][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.

* Named properties for automata
   :PROPERTIES:
   :CUSTOM_ID: named-properties
   :END:

In addition to [[#property-flags][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:

| key name            | (pointed) value type           | description                                                                                                                                           |
|---------------------+--------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------|
| ~automaton-name~    | ~std::string~                  | name for the automaton, for instance to display in the HOA format                                                                                     |
| ~product-states~    | ~const spot::product_states~   | vector of pairs of states giving the left and right operands of each state in a product automaton                                                     |
| ~original-states~   | ~std::vector<unsigned>~        | original state number before transformation (used by some algorithms like =degeneralize()=)                                                           |
| ~original-clauses~  | ~std::vector<unsigned>~        | original DNF clause associated to each state in automata created by =dnf_to_streett()=                                                                |
| ~state-names~       | ~std::vector<std::string>~     | vector naming each state of the automaton, for display purpose                                                                                        |
| ~highlight-edges~   | ~std::map<unsigned, unsigned>~ | map of (edge number, color number) for highlighting the output                                                                                        |
| ~highlight-states~  | ~std::map<unsigned, unsigned>~ | map of (state number, color number) for highlighting the output                                                                                       |
| ~incomplete-states~ | ~std::set<unsigned>~           | set of states numbers that should be displayed as incomplete  (used internally by ~print_dot()~ when truncating large automata)                       |
| ~degen-levels~      | ~std::vector<unsigned>~        | level associated to each state by the degeneralization algorithm                                                                                      |
| ~simulated-states~  | ~std::vector<unsigned>~        | map states of the original automaton to states if the current automaton in the result of simulation-based reductions                                  |
| ~synthesis-outputs~ | ~bdd~                          | conjunction of controllable atomic propositions (used by ~print_aiger()~ to determine which propositions should be encoded as outputs of the circuit) |
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 [[file:hoa.org::#named-properties][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.
