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Voorbeeld ADS in elkaar gepleurd

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Joshua Moerman 2018-11-06 21:53:55 +01:00
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@ -5,8 +5,8 @@
[title={FSM-based Test Methods}, [title={FSM-based Test Methods},
reference=chap:test-methods] reference=chap:test-methods]
In this chapter we will discuss some of the theory of test generation methods. In this chapter we will discuss some of the theory of test generation methods for black box conformance testing.
From this theoretical discussion we derive a new algorithm: the \emph{hybrid ADS methods}, which is applied on a case study in the next chapter (\in{Chapter}[chap:applying-automata-learning]). From a theoretical discussion we derive a new algorithm: the \emph{hybrid ADS methods}, which is applied on a case study in the next chapter (\in{Chapter}[chap:applying-automata-learning]).
A key aspect of such methods is the size of the obtained test suite. A key aspect of such methods is the size of the obtained test suite.
On one hand we want to cover as much as the specification as possible. On one hand we want to cover as much as the specification as possible.
On the other hand: testing takes time, so we want to minimise the size of a test suite. On the other hand: testing takes time, so we want to minimise the size of a test suite.
@ -175,6 +175,10 @@ We are often interested in the case of $m$-completeness, where $m = n + k$ for s
Here $k$ will stand for the number of \emph{extra states} we can test. Here $k$ will stand for the number of \emph{extra states} we can test.
The issue of an unknown bound is addressed later in the paper. The issue of an unknown bound is addressed later in the paper.
Note the order of the quantifiers in the above definition.
We ask for a single test suite which works for all implementations of bounded size.
This is crucial for black box testing, as we do not know the implementation, so the test suite has to work for all of them.
\stopsubsection \stopsubsection
\startsubsection \startsubsection
@ -231,25 +235,16 @@ We require that all sets are \emph{prefix-closed}, however, we only give the max
\item A \defn{state identifier} for a state $s \in M$ is a set $W_s$ which contains a separating sequence for every other state $t \in M$. \item A \defn{state identifier} for a state $s \in M$ is a set $W_s$ which contains a separating sequence for every other state $t \in M$.
\item A set of state identifiers $\{ W_s \}_{s}$ is \defn{harmonised} if a separating sequence $w$ for states $s$ and $t$ exists in both $W_s$ and $W_t$. \item A set of state identifiers $\{ W_s \}_{s}$ is \defn{harmonised} if a separating sequence $w$ for states $s$ and $t$ exists in both $W_s$ and $W_t$.
This is also called a \defn{separating family}. This is also called a \defn{separating family}.
\item Following the results of \citet[DBLP:journals/tc/LeeY94], a separating family where each set is the prefix-closure of a single word is called an \defn{adaptive distinguishing sequence} (ADS).
\stopitemize \stopitemize
\stopdefinition \stopdefinition
The property of being harmonised might seem a bit strange. The property of being harmonised might seem a bit strange.
This property ensure that the same tests are used for different states. This property ensure that the same tests are used for different states.
This extra consistency within a test suite is necessary for some test methods. This extra consistency within a test suite is necessary for some test methods.
A counterexample is given in \in{Example}[ex:uio-counterexample]. We return to this notion in more detail in \in{Example}[ex:uio-counterexample].
An ADS is of special interest since they can identify a state using a single word.
It is called an adaptive sequence, since it has a tree structure which depends on the output of the machine.
To see this, consider the first symbols of each of the sequences in the family.
Since the family is harmonised and each set is given by a single word, there is only one first symbol.
So we test each state with one symbol, observe the output, and then continue with the remainder of the sequence (which depends on the output).
\todo{plaatje?}
We may obtain a characterisation set by taking the union of state identifiers for each state. We may obtain a characterisation set by taking the union of state identifiers for each state.
For every machine we can construct a set of harmonised state identifiers as will be shown in \in{Chapter}[chap:separating-sequences] and hence every machine has a characterisation set. For every machine we can construct a set of harmonised state identifiers as will be shown in \in{Chapter}[chap:separating-sequences] and hence every machine has a characterisation set.
However, an adaptive distinguishing sequence may not exist, even if every state has an UIO.
\startexample \startexample
As mentioned before, state $s_2$ from \in{Figure}[fig:running-example] has a state identifier $\{aa, ac, b\}$. As mentioned before, state $s_2$ from \in{Figure}[fig:running-example] has a state identifier $\{aa, ac, b\}$.
@ -258,9 +253,73 @@ Since the other states have UIOs, we can pick singleton sets as state identifier
For example, state $s_0$ has the UIO $aa$, so a state identifier for $s_0$ is $W_0 = \{ aa \}$. For example, state $s_0$ has the UIO $aa$, so a state identifier for $s_0$ is $W_0 = \{ aa \}$.
Similarly, we can take $W_1 = \{ a \}$ and $W_3 = \{ c \}$. Similarly, we can take $W_1 = \{ a \}$ and $W_3 = \{ c \}$.
But note that such a family will \emph{not} be harmonised since the sets $\{ a \}$ and $\{ c \}$ have no common separating sequence. But note that such a family will \emph{not} be harmonised since the sets $\{ a \}$ and $\{ c \}$ have no common separating sequence.
There is no ADS for this machine.
\stopexample \stopexample
One more type of state identifier is of our interest: the \emph{adaptive distinguishing sequence}.
It it the strongest type of state identifier, and as a result not many machine have one.
\startdefinition
Following the results of \citet[DBLP:journals/tc/LeeY94], a separating family where each set is the prefix-closure of a single word is called an \defn{adaptive distinguishing sequence} (ADS).
\stopdefinition
An ADS is of special interest since they can identify a state using a single word.
It is called an adaptive sequence, since it has a tree structure which depends on the output of the machine.
To see this, consider the first symbols of each of the sequences in the family.
Since the family is harmonised and each set is given by a single word, there is only one first symbol.
So we test each state with one symbol, observe the output, and then continue with the remainder of the sequence (which depends on the output).
\todo{Verwijs naar plaatje}
Given an ADS, there exists an UIO for every state.
The converse -- if every state has an UIO, then the machine admits an ADS-- does not hold.
The machine in \in{Figure}[fig:running-example] admits no ADS, since $s_2$ has no UIO.
\startplacefigure
[title={(a): A Mealy machine with alphabets $I = \{a, b\}$ and $O = \{0, 1\}$ and (b): an ADS for (a). },
list={A Mealy machine and its ADS.}
reference=fig:ads-example]
\startcombination[nx=2, ny=1, align=center]
{\hbox{
\starttikzpicture
\node[initial, state] (0) {$s_0$};
\node[state] (1) [right=of 0] {$s_1$};
\node[state] (2) [below=of 1] {$s_2$};
\node[state] (3) [left =of 2] {$s_3$};
\path[->]
(0) edge node [above] {$a/0, b/0$} (1)
(1) edge node [right] {$a/1$} (2)
(2) edge node [above] {$a/0, b/0$} (3)
(3) edge node [left, align=left] {$a/1,$ \\ $b/0$} (0)
(1) edge [loop right] node {$b/0$} (1);
\stoptikzpicture}} {(a)}
{\hbox{
\starttikzpicture[node distance=.75cm]
\node (0) {$a$};
\node [below left=of 0] (1) {$b$};
\node [below=of 1] (3) {$a$};
\node [below left=of 3] (5) {$2$};
\node [below right=of 3] (6) {$0$};
\node [below right=of 0] (2) {$a$};
\node [below=of 2] (4) {$b$};
\node [below=of 4] (7) {$a$};
\node [below left=of 7] (8) {$1$};
\node [below right=of 7] (9) {$3$};
\path[->]
(0)+(0,.75) edge (0)
(0) edge node [above left] {$0$} (1)
(0) edge node [above right] {$1$} (2)
(1) edge node [left] {$0$} (3)
(3) edge node [above left] {$0$} (5)
(3) edge node [above right] {$1$} (6)
(2) edge node [right] {$0$} (4)
(4) edge node [right] {$0$} (7)
(7) edge node [above left] {$0$} (8)
(7) edge node [above right] {$1$} (9);
\stoptikzpicture}} {(b)}
\stopcombination
\stopplacefigure
\todo{Dit staat een beetje random}
Besides sequences which separate states, we also need sequences which brings a machine to specified states. Besides sequences which separate states, we also need sequences which brings a machine to specified states.
\startdefinition \startdefinition