\project thesis
\startcomponent introduction

\startchapter
  [title={Introduction},
   reference=chap:introduction]

When I was younger, I often learned how to use new toys by playing around with them, i.e., by pressing button randomly, and observing its behaviour.
I would only resort to the manual -- or ask \quotation{experts} -- to confirm my hypotheses.
Now that I am older, I do mostly the same.
But now I know that this is an established computer science technique, called \emph{model learning}.

In short, model learning
\footnote{There are many names for the type of learning, such as \emph{active automata learning}.
The generic name \quotation{model learning} is chosen as a counterpoint to model checking.}
is a automated technique to construct a model -- often a type of \emph{automaton} -- from a black box system.
The aim of this is to reverse-engineer the system, to find bugs, to verify, or to understand the system in one way or another.
It is \emph{not just random testing}: the information learned during the interaction with the system is actively used to guide following interactions.

In this thesis, I report my contributions to the field of model learning.
In the first part, I give results concerning \emph{black box testing} of automata.
Testing is a crucial part in learning software behaviour and often remains a bottleneck in applications of model learning.
In the second part, I show how \emph{nominal techniques} can be used to learn automata over structured infinite alphabets.
This was directly motivated by work on learning networks protocols which rely on identifiers or sequence numbers.

But before we get ahead of ourselves, we should first understand what we mean by learning, as learning means very different things to different people.
For pedagogics scientist learning may involve concepts such as teachers, scholars, blended learning, and active learning.
Data scientist may think of data compression, feature extraction, and neural networks.
In this thesis we are mostly concerned with software verification.
But even in the field of verification several types of learning are relevant.


\startsection
  [title={Model Learning}]

In the context of software verification, we often look at stateful computation with inputs and outputs.
For this reason, it makes sense to look at \emph{words}, or \emph{traces}.
For an alphabet $\Sigma$, we denote the set of words by $\Sigma^{*}$.

The learning problem is defined as follows.
There is some fixed, but unknown to us, language $\lang \subseteq \Sigma^{*}$.
This language may define the behaviour of a software component, a property in model checking, a set of traces from a protocol, etc.
We wish to infer a description of $\lang$ by only using finitely many words.
Note that this is not an easy task, as $\lang$ is often infinite.

Such a learning problem can be stated and solved in a variety of ways.
In the applications we do in our research group, we often try to infer a model of a software component.
(\in{Chapter}[chap:applying-automata-learning] describes such an application.)
In such cases, a learning algorithm can interact with the software.
So it makes sense to study a learning paradigm which allows for \emph{queries}, and not just a data set of samples.
\footnote{Instead of query learning, people also use the term \emph{active learning}.}

A typical query learning framework was established by \citet[DBLP:journals/iandc/Angluin87].
In her framework, the learning algorithm may pose two types of queries to a \emph{teacher} (or \emph{oracle}):

\description{Membership queries (MQs)}
The learner poses such a query by providing a word $w \in \Sigma^{*}$ to the teacher.
The teacher will then reply whether $w \in \lang$ or not.
This type of query is often generalised to more general output, so the teacher replies with $\lang(w)$.
In some papers, such a query is then called an \emph{output query}.

\description{Equivalence queries (EQs)}
The learner can provide a hypothesised description of $\lang$.
If the hypothesis is correct, the teacher replies with \kw{yes}.
If, however, the hypothesis is incorrect, the teacher replies with \kw{no}, together with a counterexample (i.e., a word which is in $\lang$ but not in the hypothesis or vice versa).

With these queries, the learner algorithm is supposed to converge to a correct model.
This type of learning is hence called \emph{exact learning}.
\citet[DBLP:journals/iandc/Angluin87] showed that one can do this efficiently for deterministic finite automata DFAs (when $\lang$ is in the class of regular languages).

Another paradigm which is relevant for our type of applications is \emph{PAC-learning with membership queries}.
Here, the algorithm can again use MQs as before, but the EQs are replace by random sampling.
So the allowed query is:

\description{Random sample queries (EXs)}
If the learner poses this query (there are no parameters), the teacher responds with a random word $w$, together with its label $w \in \lang$.

Instead of requiring that the learner exactly learns the model, we only require the following.
The learner should \emph{probably} return a model which is \emph{approximate} to the target.
This gives the name probably approximately correct (PAC).
Note that there are two uncertainties: the probable and the approximate part.
Both part are bounded by parameters, so one can determine the confidence.

Of course, we are interested in \emph{efficient} learning algorithms.
This often means that we require a polynomial number of queries.
But polynomial in what?
The learner has no input, other than the access to a teacher.
We ask the algorithms to be polynomial in the \emph{size of the target} (i.e., the description which has yet to be learned).
In the case of PAC learning we also require it to be polynomial in the two parameters for confidence.

Deterministic automata can be efficiently learned in the PAC model.
In fact, any efficient exact learning algorithm with MQs and EQs can be transformed into an efficient PAC algorithm with MQs (see \citenp[KearnsV94], exercise 8.1).
For this reason, we mostly focus on the former type of learning in this thesis.
The transformation from exact learning to PAC learning is implemented by simply \emph{testing} the hypothesis with random samples.
This can be postponed until we actually implement a learning algorithm and apply it.

When using only EQs, only MQs, or only EXs, then there are hardness results for learning DFAs.
So the combinations MQs + EQs and MQs + EXs have been carefully been picked.
They provide a minimal basis for efficient learning.
See the book of \citet[KearnsV94] for such hardness results and more information on PAC learning.

So far, all the queries are assumed to be \emph{just there}.
Somehow, these are existing procedures which we can invoke with \kw{MQ}$(w)$, \kw{EQ}$(H)$, or \kw{EX}$()$.
This is a useful abstraction when designing a learning algorithm.
One can analyse the complexity (in terms of number of queries) independently of how these queries are resolved.
At some point in time, however, one has to implement them.
In our case of learning software behaviour, membership queries are easy:
Simply provide the word $w$ to a running instance of the software and observe the output.
\footnote{In reality, it is a bit harder than this.
There are plentiful of challenges to solve, such as timing, choosing your alphabet, choosing the kind of observations to make and being able to faithfully reset the software.}
Equivalence queries, however, are in general not doable.
Even if we have the (machine) code, it is often way too complicated to check equivalence.
That is why we resort to testing with EX queries.
The EX query from PAC learning normally assumes a fixed, unknown, distribution.
In our case, we implement a distribution to test against.
This cuts both ways:
On one hand, it allows us to only test behaviour we really case about, on the other hand the results are only as good as our choice of distribution.
We deviate even further from the PAC-model as we sometimes change our distribution while learning.
Yet, as applications show, this is a useful way of learning software behaviour.

\todo{Aannames: det. reset...}

\todo{Research questions}


\startsubsection
  [title={A few applications of learning}]

Since this thesis only contains one \quote{real-world} application on learning, I think it is good to mention a few others.
Although we remain in the context of learning software behaviour, the applications are quite different.

Finite state machine conformance testing is a core topic in testing literature that is relevant for communication protocols and other reactive systems.
The combination of conformance testing and automata learning is applied in practice and it proves to be an effective way of finding bugs \cite[Joeri], replacing legacy software \cite[Philips], or inferring models to be used in model checking \cite[MeinkeSindhu].

Several model learning applications:
learning embedded controller software \cite[DBLP:conf/icfem/SmeenkMVJ15], learning the TCP protocol \cite[DBLP:conf/cav/Fiterau-Brostean16] and learning the MQTT protocol \cite[DBLP:conf/icst/TapplerAB17].


\stopsubsection
\stopsection
\startsection
  [title={Testing Techniques}]

\todo{Even lang maken als Nominal Techniques.}

An important step in automata learning is equivalence checking.
Normally, this is abstracted away and done by an oracle, but we intend to implement such an oracle ourselves for our applications.
Concretely, the problem we need to solve is that of \emph{conformance checking}\footnote{It is also known as \emph{machine verification} or \emph{fault detection}.} and this was first described by \citet[Moore56].

The problem is as follows:
We are given the description of a finite state machine and a black box system, does the system behave exactly as the description?
We wish te determine this by running experiments on the system (as it is black box).
It should be clear that this is a hopelessly difficult task, as an error can be hidden arbitrarily deep in the system.
That is why we often assume some knowledge of the system.
In this thesis we often assume a \emph{bound on the number of states} of the system.
Under these conditions, \citet[Moore56] already solved the problem.
Unfortunately, his experiment is exponential in size, or in his own words: \quotation{fantastically large.}

Years later, \citet[DBLP:journals/tse/Chow78, Vasilevskii73] independently designed efficient experiments.
In particular, the set of experiments is polynomial in size.
These techniques will be discussed in detail in \in{Chapter}[chap:test-methods].
More background and other related problems, as well as their complexity results, are well exposed in a survey of \citet[DBLP:journals/tc/LeeY94].


\stopsection
\startsection
  [title={Nominal Techniques}]

In the second part of this thesis, I will present results related to \emph{nominal automata}.
Usually, nominal techniques are introduced in order to solve problems involving name binding in topics like lambda calculus.
However, we use them to deal with \emph{register automata}.
These are automata which have an infinite alphabet, often thought of as input actions with \emph{data}.
The control flow of the automaton may actually depend on the data.
However, the data cannot be used in an arbitrary way as this would lead to many problems being undecidable.
\footnote{The class of automata with arbitrary data operations is sometimes called \emph{extended finite state machines}.}
A crucial concept in nominal automata is that of \emph{symmetries}.

To motivate the use of symmetries, we will look at an example of a register auttomaton.
In the following automaton we model a (not-so-realistic) login system for a single person.
The alphabet consists of the following actions:
\startformula\startalign
\NC \kw{register}(p) \NR
\NC \kw{login}(p) \NR
\NC \kw{logout}() \NR
\NC \kw{view}() \NR
\stopalign\stopformula
The \kw{register} action allows one to set a password $p$.
This can only be done when the system is initialised.
The \kw{login} and \kw{logout} actions speak for themselves and the \kw{view} action allows one to see the secret data (we abstract away from what the user actually gets to see here).
A simple automaton with roughly this behaviour is given in \in{Figure}[fig:login-system].
We will only informally discuss its semantics for now.

\startplacefigure
  [title={A simple register automaton. The symbol $\ast$ denotes any input otherwise not specified. The $r$ in states $q_1$ and $q_2$ is a register.},
   list={A simple register automaton.},
   reference=fig:login-system]
\hbox{\starttikzpicture[node distance=4cm]
  \node[state,initial]    (0) {$q_0$};
  \node[state with output,right of=0] (1) {$q_1$ \nodepart{lower} $r$};
  \node[state with output,right of=1] (2) {$q_2$ \nodepart{lower} $r$};
  
  \path[->]
  (0) edge [loop below] node [below] {$\ast$ / \checknok}
  (0) edge node [above, align=center] {$\kw{register}(p)$ / \checkok \\ set $r := p$} (1)
  (1) edge [loop below] node [below] {$\ast$ / \checknok}
  (1) edge [bend left, align=center] node [above] {$\kw{login}(p)$ / \checkok \\ if $r = p$} (2)
  (2) edge [loop below] node [right, align=left] {$\kw{view}()$ / \checkok \\ $\ast$ / \checknok}
  (2) edge [bend left] node [below] {$\kw{logout}()$ / \checkok} (1);
\stoptikzpicture}
\stopplacefigure

To model the behaviour nicely, we want the domain of passwords to be infinite.
After all, one should allow arbitrarily long passwords to be secure.
This means that a register automaton is actually an automaton over an infinite alphabet.

Common algorithms for automata, such as learning, will not work with an infinite alphabet.
Any loop which iterates over the alphabet will diverge.
In order to cope with this, we will use the \emph{symmetries} present in the alphabet.

Let us continue with the example and look at its symmetries.
If a person registers with a password \quotation{\kw{hello}} and consequently logins with \quotation{\kw{hello}}, then this is not distinguishable from a person registering and logging in with \quotation{\kw{bye}}.
This is an example of symmetry:
the values \quotation{\kw{hello}} and \quotation{\kw{bye}} can be permuted, or interchanged.
Note, however, that the trace $\kw{register}(\kw{hello})\, \kw{login}(\kw{bye})$ is different from the two before.
So we see that we cannot simply collapse the whole alphabet into one equivalence class.
For this reason, we keep the alphabet infinite and explicitly mention its symmetries.

Before formalising symmetries, let me remark the the use of symmetries in automata theory is not new.
One of the first to use symmetries was \citet[DBLP:conf/alt/Sakamoto97].
He devised an \LStar{} learning algorithm for register automata, much like the one presented in \in{Chapter}[chap:learning-nominal-automata].
The symmetries are crucial to reduce the problem to a finite alphabet and use the regular \LStar{} algorithm.
(\in{Chapter}[chap:learning-nominal-automata] shows how to do it with more general symmetries.)
Around the same time \citet[FerrariMT05] worked on automata theoretic algorithms for the \pi-calculus.
Their approach was based on the same symmetries and they developed a theory of \emph{named sets} to implement their algorithms.
Named sets are equivalent to nominal sets.
However, nominal sets are defined in a more elementary way.
The nominal sets we will soon see are introduced by \citet[GabbayP02] to solve certain problems in name binding in abstract syntaxes.
Although this is not really related to automata theory, it was picked up by \citet[DBLP:journals/corr/BojanczykKL14].
They provide an equivalence between register automata and nominal automata.
Additionally, they generalise the work on nominal sets to other symmetries.


\startsubsection
  [title={What is a nominal set?}]

Before we dive into the relation with automata, we will define the notion of nominal sets.

\startdefinition
Fix a countable, infinite set $\atoms = \{ a, b, \ldots \}$ of \emph{names} (sometimes called \emph{atoms}).
The elements of $\atoms$ bare no relationship to natural numbers, or other standard mathematical entities.

Define $\Pm = \{ \pi \colon \atoms \to \atoms \mid \pi \text{ is bijective} \}$ to be the set of permutations of names.
Together with function composition, $\Pm$ forms a \emph{group}.
For two elements $a$ and $b$ we define a particular bijection $\swap{a}{b} \in \Pm$ which swaps $a$ and $b$ and leaves all other elements fixed.
\stopdefinition

It is good to stress that the set of names has no other structure defined on it.
The names are abstract entities which can be compared for equality, but nothing else.
\footnote{We can have more structure on the set of atoms, this is discussed in \in{Section}[].}
This also means that although $a$ and $b$ are distinct names, they are interchangeable.
If we write $a \in \atoms$, then $a$ can stand for any of the names.
So if we write $a, b \in \atoms$, then $a$ and $b$ can refer to the same name, i.e., $a = b$.
In other words, we do not adapt the permutative convention by \citet[?].

As alluded to before, we want to have permutations act on objects constructed from names, such as words, states in an automaton and languages.
The notion of a group action captures exactly this.
In most cases we are interested in the group $\Pm$.
However, in order to be general enough for the next chapters, we introduce group actions for an arbitrary group $G$.
\todo{Notatie $1$ is groepseenheid, ${\cdot}$ is vermenigvuldiging en werking.}

\startdefinition
Let $X$ be a set.
A (left)
\footnote{Many authors use left actions.
However, we note that \citet[DBLP:journals/corr/BojanczykKL14] use a right action.
For them to have a well-defined group action, their group multiplication has to be defined as $g \cdot f = f \circ g$ (i.e., reverse function composition).}
\emph{$G$-action} is a function ${\cdot} \colon G \times X \to X$ satisfying:
\startformula\startalign[n=3]
\NC 1 \cdot x            \NC = x                    \NC \quad \forall x \in X \NR
\NC (g \cdot h) \cdot x  \NC = g \cdot (h \cdot x)  \NC \quad \forall x \in X, \forall g,h \in G \NR
\stopalign\stopformula
A set together with a $G$-action, $(X, {\cdot})$, is called a \emph{$G$-set}.
\stopdefinition

It is worth noting that we generally fix $G$ but we consider many sets with a $G$-action.
In a way all these sets will have the same symmetries (namely $G$).
Instead of writing $g \cdot x$ we will often write the group action by juxtaposition $g x$.
We will often write $X$ instead of $(X, {\cdot})$ when the intended action is clear from the context.
\footnote{One should be cautious, as a set often allows for many different $G$-actions.}

\startexample
We list several examples of group actions.
Many of them will be used later in this thesis.
\startitemize
\item
The set $\atoms$ itself admits a natural $\Pm$-action, defined by
\startformula \pi \cdot a = \pi(a). \stopformula
The two requirements are easily verified by a routine calculation.
We will also omit this verification for the upcoming examples.
\item
The set of words $\atoms^{*}$ has a $\Pm$-action which is defined point-wise:
\startformula \pi \cdot a_1 a_2 \ldots a_k = \pi(a_1) \pi(a_2) \ldots \pi(a_k) \stopformula
\item
Similarly, the set of infinite words $\atoms^{\omega}$ has such a $\Pm$-action:
\startformula \pi \cdot a_1 a_2 \ldots = \pi(a_1) \pi(a_2) \ldots \stopformula
\item
The empty set always admits a unique $G$-action for any $G$.
(This is unique since the domain $G \times \emptyset = \emptyset$.)
\startformula {\cdot} \colon G \times \emptyset \to \emptyset \stopformula
\item
The singleton set always admits a unique $G$-action for any $G$.
(This is unique since the codomain only has just one element.)
\startformula {\cdot} \colon G \times \{*\} \to \{*\} \stopformula
\item
For any set $X$, we can define a $G$-action by defining
\startformula g \cdot x = x \stopformula
for all the elements $x \in X$.
Such an action is called \emph{trivial}.
Note that the action on $\emptyset$ and $\{*\}$ are trivial, but the $\Pm$-actions on $\atoms$, $\atoms^{*}$ and $\atoms^{\omega}$ are not trivial.
\stopitemize
\stopexample

In the above examples, the non-trivial $\Pm$-sets are all infinite.
Yet, in a sense, the set $\atoms^{*}$ is bigger than the set $\atoms$.
To be able to quantify this, we introduce the notion of an orbit.

\startdefinition
Given a $G$-set $(X, {\cdot})$ and an element $x \in X$, we define the \emph{orbit of $x$} as the set
\startformula \orb(x) = \{ g x \mid g \in G \}. \stopformula
\stopdefinition

If for two elements $x, y \in X$ we have $\orb(x) = \orb(y)$, then we say that $x$ and $y$ are in the same orbit.
This precisely happens if there exists a $g$ such that $g x = y$.
The relation of \quotation{being in the same orbit} is an equivalence relation (it is reflexive as a group has an identity element, symmetric because of the inverses and transitive because of composition).
This relation partitions the set $X$ in a collection of orbits:
\startformula X = \bigcup_{x \in X} \orb(x). \stopformula

We can picture orbits in the following way.
\todo{PLAATJE}
As we wish to represent such sets (in order to run algorithms on them), we are especially interested in orbit-finite sets.
For such sets, we can represent the whole set by a collection of its orbits.
What remains to be represented are the orbits themselves.
An easy way to do is, is to choose a representative of the orbit $x \in \orb(x)$. (Any element will do as the other elements can be constructed via the group action.)
\todo{PLAATJE}

\startexample
We will describe the orbits for some $\Pm$-sets.
\startitemize
\item
For a trivial $G$-set $X$, each element defines its own orbit, since $\orb(x) = \{ g x \mid g \in G \}$ is a singleton set.
\item
The $\Pm$-set $\atoms$ only has \emph{one orbit}.
To see this, take two (distinct) elements $a, b \in \atoms$ and consider the bijection $\pi = \swap{a}{b}$.
Then we see that $\pi \cdot a = b$, meaning that $a$ and $b$ are in the same orbit.
So $\atoms$ is a single-orbit set.
\item
Before we tackle $\atoms^{*}$, we will analyse $\atoms^{2}$.
The set consists of exactly \emph{two orbits}:
\startformula\startalign
\NC \{ (a, a) \NC \mid a \in \atoms \} \NR
\NC \{ (a, b) \NC \mid a, b \in \atoms, a \neq b \} \NR
\stopalign\stopformula
This is because a bijection $\pi \in \Pm$ can never send an element of the form $(a, b)$ to an element of the form $(a, a)$ or vice versa.
It can, however send any element $(a, b)$ to $(c, d)$ and so on.
\item
The set $\atoms^{*}$ has \emph{countably many orbits}.
Since the action preserves the length of a word, we will show that the set has finitely many orbits for each length.
So consider the set $\atoms^{k}$ with the point-wise action.
An orbit of $\atoms^{k}$ is precisely determined by specifying which of the $k$ elements are equal to each other.
This is a partition of $k$ elements, and there exactly $B_k$, the $k$th Bell number, such partitions.
(As we have seen for $k = 2$, the second Bell number is $B_2 = 2$.
This quantity grows exponential in $k$.)
This shows that the set $\atoms^{*} = \bigcup_k \atoms^{k}$ has countably many orbits.
\stopitemize
\stopexample

Having finitely many orbits is not enough for a finite representation which we can use algorithmically.
We need an additional finiteness on the elements of a $G$-set,
namely the existence of a \emph{finite support}.
In order to define this, we need the notion of a data symmetry.

\startdefinition
A \emph{data symmetry} is a pair $(\mathcal{D}, G)$, where $\mathcal{D}$ is a structure and $G \leq \Sym(\mathcal{D})$ is a subgroup of the automorphism group of $\mathcal{D}$.
\stopdefinition

\startdefinition
Let $X$ be a $G$-set and $x \in X$.
A set $C \subset \mathcal{D}$ \emph{supports} $x$ if for all $g \in G$ with $g|_C = \id|_C$ we have $g \cdot x = x$.

A $G$-set $X$ is called \emph{nominal} if every element has a finite support.
\stopdefinition

In a way, if an element is supported by a finite set $C$, it means that the element is somehow constructed from only the elements in $C$.
We can see this from the definition, as changing any element outside of $C$ will leave the element $x$ fixed.

\startexample
\startitemize
\item
The sets $\atoms$, $\atoms^{k}$, $\atoms^{*}$ are all nominal.
For an element $a_1 a_2 \ldots a_k \in \atoms^{*}$, its support is simply given by $\{a_1, a_2, \ldots, a_k\}$.
\stopitemize
\stopexample

These examples show that being orbit-finite and nominal are orthogonal properties.
\todo{Een voorbeeld is uitgesteld.}
There are $G$-sets which are orbit-finite, but non-nominal.
Conversely, there are nominal sets which are not orbit-finite.


\stopsubsection
\startsubsection
  [title={Nominal automata}]

\todo{Model the example above as nominal automata}


\stopsubsection
\startsubsection
  [title={More interesting examples of nominal sets}]

The set $\atoms^{\omega}$ has \emph{uncountably many orbits}.
To see this, fix two distinct elements $a, b \in \atoms$.
Now, let $\sigma \in 2^{\omega}$ be an element of the Cantor space.
We define the following sequence $x^{\sigma} \in \atoms^{\omega}$:
\startformula\startalign
\NC x^{\sigma}_0     \NC = a \NR
\NC x^{\sigma}_{i+1} \NC =
  \startmathcases
  \NC a, \NC if $\sigma(i) = 0$ \NR
  \NC b, \NC if $\sigma(i) = 1$ \NR
  \stopmathcases \NR
\stopalign\stopformula
Now for two distinct elements $\sigma, \tau \in 2^{\omega}$, the elements $x^{\sigma}$ and $x^{\tau}$ are different.
More importantly, their orbits $\orb(x^{\sigma})$ and $\orb(x^{\tau})$ are different too.
This shows that there is an injective map from $2^{\omega}$ to the orbits of $\atoms^{\omega}$.
This concludes that $\atoms^{\omega}$ has uncountably many orbits.

The set $\atoms^{\omega}$ is \emph{not} nominal.
To see this, let us order the elements of $\atoms$ as $\atoms = \{ a_1, a_2, a_3, \ldots \}$.
Now the element $a_1 a_2 a_3 \in \atoms^{\omega}$ is not finitely supported.
\todo{fs subset van $\atoms^{\omega}$?}

The set $\{ X \subset \atoms \mid X \text{ is not finite nor co-finite} \}$ (with the group action given by direct image) is a single orbit set, but it is not a nominal set.

The last example above needs a bit more clarification.
In the book of \citet[Pitts13], the group of permutations is defined to be
\startformula
G_{< \omega} = \{ \pi \in \Perm \mid \pi(x) \neq x \text{ for finitely many } x \}.
\stopformula
This is the subgroup of $\Pm$ of \emph{finite} permutation.
The set $\{ X \subset \atoms \mid X \text{ is not finite nor co-finite} \}$ has infinitely many orbits when considered as a $G_{< \omega}$-set, but only one orbit as a $\Pm$-set.
This poses the question which group we should consider (for example, \citet[DBLP:journals/corr/BojanczykKL14] use the whole group $\Pm$).
For nominal sets, there is no difference: nominal $G_{< \omega}$-sets and nominal $\Pm$-sets are equivalent, as shown by \citet[Pitts13].
It is only for non-nominal sets that we can distinguish them.
We will mostly work with the set of all permutations $\Pm$.

Another interesting non-trivial example is the set $\Pm$ itself.
There are three different interesting actions one can define:
\startformula\startalign
\NC \pi \cdot_{l} \sigma \NC = \pi \sigma \NR
\NC \pi \cdot_{r} \sigma \NC = \sigma \pi^{-1} \NR
\NC \pi \cdot_{c} \sigma \NC = \pi \sigma \pi^{-1} \NR
\stopalign\stopformula
Here the group multiplication is written by juxtaposition.
The first two actions are \emph{left-multiplication} and \emph{right-multiplication} respectively.
The latter is called \emph{conjugation}.
For each of them, one can verify the requirements.


\stopsubsection
\stopsection
\referencesifcomponent
\stopchapter
\stopcomponent