Rewrote intro to nom sets

content/introduction.tex

@@ -21,41 +21,71 @@ learning embedded controller software \cite[DBLP:conf/icfem/SmeenkMVJ15], learni
 \startsection
   [title={Nominal Techniques}]
 
-The theory of nominal sets has been introduced to cope with name binding.
+In the second part of this thesis, I will present results related to nominal automata.
-Dealing with names is very common in computer science and several techniques exist to do this.
+Usually, nominal techniques are introduced in order to solve problems involving name binding in topics like lambda calculus.
-For example, one can use a meta language to talk about bound and free variables.
+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.
-This is easy enough to use in theoretical research, but can be a burden to implement (there is quite a bit of bookkeeping).
-Another way is to use some sort of normal form, for example, by naming things with numbers (sequentially).
-One example of such a numbering scheme is using de Bruijn indices.
 
+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].
 
-\citet[GabbayP02] introduced a third option, based on nominal techniques.
+\startplacefigure
-Instead of relying on the meta level, we declare names to really exist:
+  [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.},
-\startformula
+   list={A simple register automaton.},
-\atoms = \{ a, b, c, \ldots \}.
+   reference=fig:login-system]
-\stopformula
+\hbox{\starttikzpicture[node distance=4cm]
-Here we understand that $a$ and $b$ are different names and these names can be directly used in lambda terms or other objects.
+  \node[state,initial]    (0) {$q_0$};
-(These names do not stand for variables!
+  \node[state with output,right of=0] (1) {$q_1$ \nodepart{lower} $r$};
-The names exist in their own right.)
+  \node[state with output,right of=1] (2) {$q_2$ \nodepart{lower} $r$};
-As a consequence, we can describe lambda terms such as
+  
-\startformula
+  \path[->]
-\lambda x . a \quad \text{and} \quad \lambda x . b \, .
+  (0) edge [loop below] node [below] {$\ast$ / \checknok}
-\stopformula
+  (0) edge node [above, align=center] {$\kw{register}(p)$ / \checkok \\ set $r := p$} (1)
-These two lambda terms are distinct, since they are constructed with different names.
+  (1) edge [loop below] node [below] {$\ast$ / \checknok}
-In order to talk about equivalence, we introduce permutations.
+  (1) edge [bend left, align=center] node [above] {$\kw{login}(p)$ / \checkok \\ if $r = p$} (2)
-If we write $\swap{a}{b}$ for the permutation which swaps $a$ and $b$, then we can relate the two lambda terms as follows.
+  (2) edge [loop below] node [right, align=left] {$\kw{view}()$ / \checkok \\ $\ast$ / \checknok}
-\startformula
+  (2) edge [bend left] node [below] {$\kw{logout}()$ / \checkok} (1);
-\swap{a}{b} \lambda x . a = \lambda x . b
+\stoptikzpicture}
-\stopformula
+\stopplacefigure
-How the permutations act on objects other than names will be explained later.
+
-This way we can relate the two terms, without talking about free or bound variables.
+To model the behaviour nicely, we want the domain of passwords to be infinite.
-We only talk about the names and permutations we declared.
+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.
 
-The idea that names and permutations are sufficient was picked up by \citet[DBLP:journals/corr/BojanczykKL14].
-They show that besides abstract syntax, nominal techniques are also interesting when modelling automata over infinite alphabets.
-Here the infinite alphabet will be the set of names and the automaton should respect the permutations.
-We will see why this is a natural way to model such automata in the next section.
-\todo{\pi-calculus en named sets door mensen in Pisa.}
 
 \startsubsection
   [title={What is a nominal set?}]
@@ -193,9 +223,26 @@ 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

environment/font.tex

@@ -18,7 +18,8 @@
 % (Neo) Euler heeft niet alle symbolen, maar Pagella wel
 \definefallbackfamily[mainfamily] [mm] [TeX Gyre Pagella Math] [range=0x0266D-0x0266F,rscale=0.95] % sharp and flat
 \definefallbackfamily[mainfamily] [mm] [TeX Gyre Pagella Math] [range={0x025A1,0x025B3}] % triangle, square
-\definefallbackfamily[mainfamily] [mm] [TeX Gyre Pagella Math] [range={0x022EE}] % vdots
+\definefallbackfamily[mainfamily] [mm] [TeX Gyre Pagella Math] [range={0x022EE, 0x02254}] % vdots, assignment
+% \definefallbackfamily[mainfamily] [mm] [TeX Gyre Pagella Math] [range={0x02713,0x02717}] % checkmark
 % Neo Euler voor wiskunde. Merk op dat dit niet cursief is!
 \definefontfamily[mainfamily] [mm] [Neo Euler] [rscale=0.95]
 % \definefontfamily[mainfamily] [mm] [Latin Modern Math]

environment/notation.tex

@@ -1,9 +1,13 @@
 \startenvironment notation
 
+\usesymbols[fontawesome]
+
 % Pijlen met text erboven
 \definemathstackers [arrowstack] [voffset=-.7\mathexheight, topcommand=\m]
 \definemathextensible [arrowstack] [tot] ["27F6]
 
+\define\checkok{\scale[height=.5em]{\symbol[fontawesome][ok]}} % ✓
+\define\checknok{\scale[height=.5em]{\symbol[fontawesome][remove]}} % find a font with ✗
 
 \define[1]\Fam{\mathcal{#1}}
 \define\lang{\mathcal{L}}

environment/tikz.tex

@@ -8,6 +8,7 @@
 % defaults
 \tikzset{node distance=2cm} % 'shorten >=2pt' only for automata
 \tikzset{/pgfplots/table/search path={data,../data}}
+\tikzset{initial text=}
 
 % observation table
 \tikzset{obs table/.style={matrix of math nodes, ampersand replacement=\&, column 1/.style={anchor=base west}}}