Commit 78fd7bea by Alexandre Duret-Lutz

### org: Add a Concepts page.

* doc/org/concepts.org: New file.
* NEWS: Mention the new page.
 # -*- 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