# -*- coding: utf-8 -*-
#+TITLE: Concepts
#+SETUPFILE: 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 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 [[http://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 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 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.png :cmdline -Tpng :var txt=buchi-example1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-buchi1.png]]
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.png :cmdline -Tpng :var txt=buchi-example2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-buchi.png]]
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.png :cmdline -Tpng :var txt=buchi-example3 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-buchi3.png]]
* 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 < tmp.$$
ltl2tgba -s6 'p0 | GFp1' | pr -m -t tmp.$$ -
#+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.png :cmdline -Tpng :var txt=never-ex1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-never1.png]]
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.png]]
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.png :cmdline -Tpng :var txt=lbtt-ex2 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-lbtt2.png]]
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.a
#+END_SRC
#+BEGIN_SRC dot :file concept-dstar.png :cmdline -Tpng :var txt=dstar-example1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-dstar.png]]
* 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 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.a
#+END_SRC
#+BEGIN_SRC dot :file concept-hoa.png :cmdline -Tpng :var txt=hoa1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-hoa.png]]
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 (yet?) support all automata that can
be expressed using the HOA format. For instance it has no support for
alternating automata, or for the =Alphabet:=-based automata introduced
in v1.1 of HOA (only =AP:=-based automata are supported).
The present support for the HOA format in Spot, is discussed on [[file:hoa.org][a
separate page]].
* 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.png :cmdline -Tpng :var txt=ltl2tgba1 :exports results
$txt
#+END_SRC
#+RESULTS:
[[file:concept-ltl2tgba.png]]
[[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:
The Spot project can be broken down into several main parts:
- =libbddx=: a customized version of [[http://sourceforge.net/projects/buddy/][the BuDDy library]], for manipulating [[#bdd][BDDs]].
- =libspot=: the main library, containing a C++11 implementation of all the
data structures and algorithms. This depends on =libddx=.
- [[file:tools.org][command-line tools]]: built upon the =libspot= library, exporting some of its
features to shell users
- Python bindings for =libbddx= and =libspot=: those make it possible to write
python scripts for specific tasks, and allow interactive use of the library
via environments such a [[http://ipython.org][IPython/Jupyter]].