\project thesis \startcomponent introduction \startchapter [title={Pakkende titel}, reference=chap:introduction] \startsection [title={Learning and Testing}] \todo{Gekopieerd van test methods paper.} 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]. \stopsection \startsection [title={Nominal Techniques}] In the second part of this thesis, I will present results related to nominal automata. Usually, nominal techniques are introduced in order to solve problems involving name binding in topics like lambda calculus. However, for our applications it makes more sense to focus more on the symmetries (of names) first and show how this is useful in automata theory. 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]. \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. 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}}. However, a trace $\kw{register}(\kw{hello})\, \kw{login}(\kw{bye})$ is different. This is an example of symmetry: the values \quotation{\kw{hello}} and \quotation{\kw{bye}} can be permuted, or interchanged. Before formalising the permutations, 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 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 \cite[]. As noted before, we want to let permutations act on objects constructed from names, such as lambda terms, tuples, and words. 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 $G$. \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. \item Another interesting non-trivial example is the set $\Pm$ itself. There are three different actions which are natural: \startformula\startalign \NC \pi \cdot_1 \sigma \NC = \pi \sigma \NR \NC \pi \cdot_2 \sigma \NC = \sigma \pi^{-1} \NR \NC \pi \cdot_3 \sigma \NC = \pi \sigma \pi^{-1} \NR \stopalign\stopformula Here the group multiplication is written by juxtaposition. The first two actions are left-multiplication and right-multiplication respectively. The latter is called conjugation. For each of them, one can verify the requirements. \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 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. \startexample We will describe the orbits for some non-trivial $\Pm$-sets. \startitemize \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. \item The set $\atoms^{\omega}$ has \emph{uncountably many orbits}. To see this we will order the elements of $\atoms = \{ a_0, a_1, a_2, \ldots \}$. 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_0 \NR \NC x^{\sigma}_{i+1} \NC = \startmathcases \NC a_i, \NC if $\sigma(i) = 0$ \NR \NC a_{i+1}, \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. 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. \stopitemize \stopsubsection \stopsection \referencesifcomponent \stopchapter \stopcomponent