More intro nom sets
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Joshua Moerman
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@ -55,7 +55,7 @@ The idea that names and permutations are sufficient was picked up by \citet[DBLP
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They show that besides abstract syntax, nominal techniques are also interesting when modelling automata over infinite alphabets.
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Here the infinite alphabet will be the set of names and the automaton should respect the permutations.
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We will see why this is a natural way to model such automata in the next section.
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\todo{\pi-calculus en named sets door mensen in Pisa.}
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\startsubsection
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[title={What is a nominal set?}]
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@ -66,19 +66,30 @@ Before we dive into the relation with automata, we will define the notion of nom
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Fix a countable, infinite set $\atoms = \{ a, b, \ldots \}$ of \emph{names} (sometimes called \emph{atoms}).
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The elements of $\atoms$ bare no relationship to natural numbers, or other standard mathematical entities.
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Define $\Perm(\atoms) = \{ \pi \colon \atoms \to \atoms \mid \pi \text{ is bijective} \}$ to be the set of permutations of atoms.
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Together with function composition, $\Perm(\atoms)$ forms a group.
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Define $\Pm = \{ \pi \colon \atoms \to \atoms \mid \pi \text{ is bijective} \}$ to be the set of permutations of names.
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Together with function composition, $\Pm$ forms a group.
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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.
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\stopdefinition
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As noted before, we want to let permutations act on objects constructed from names (such as lambda terms).
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It is good to stress that the set of names has no other structure defined on it.
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The names are abstract entities which can be compared for equality, but nothing else.
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\footnote{We can have more structure on the set of atoms, this is discussed in \in{Section}[].}
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This also means that although $a$ and $b$ are distinct names, they are interchangeable.
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If we write $a \in \atoms$, then $a$ can stand for any of the names.
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So if we write $a, b \in \atoms$, then $a$ and $b$ can refer to the same name, i.e., $a = b$.
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In other words, we do not adapt the permutative convention by \cite[].
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As noted before, we want to let permutations act on objects constructed from names, such as lambda terms, tuples, and words.
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The notion of a group action captures exactly this.
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In most cases we are interested in the group $\Perm(\atoms)$.
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In most cases we are interested in the group $\Pm$.
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However, in order to be general enough for the next chapters, we introduce group actions for an arbitrary group $G$.
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\startdefinition
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Let $X$ be a set.
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A (left)
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\footnote{Most 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 composition has to be defined as $g \cdot f = f \circ g$ (i.e., reverse function composition).}
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\footnote{Many authors use left actions.
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However, we note that \citet[DBLP:journals/corr/BojanczykKL14] use a right action.
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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).}
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\emph{$G$-action} is a function ${\cdot} \colon G \times X \to X$ satisfying:
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\startformula\startalign[n=3]
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\NC 1 \cdot x \NC = x \NC \quad \forall x \in X \NR
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@ -89,6 +100,100 @@ A set together with a $G$-action, $(X, {\cdot})$, is called a \emph{$G$-set}.
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It is worth noting that we generally fix $G$ but we consider many sets with a $G$-action.
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In a way all these sets will have the same symmetries (namely $G$).
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Instead of writing $g \cdot x$ we will often write the group action by juxtaposition $g x$.
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We will often write $X$ instead of $(X, {\cdot})$ when the intended action is clear from the context.
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\footnote{One should be cautious, as a set often allows for many different $G$-actions.}
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\startexample
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We list several examples of group actions.
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Many of them will be used later in this thesis.
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\startitemize
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\item
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The set $\atoms$ itself admits a natural $\Pm$-action, defined by
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\startformula \pi \cdot a = \pi(a). \stopformula
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The two requirements are easily verified by a routine calculation.
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We will also omit this verification for the upcoming examples.
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\item
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The set of words $\atoms^{*}$ has a $\Pm$-action which is defined point-wise:
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\startformula \pi \cdot a_1 a_2 \ldots a_k = \pi(a_1) \pi(a_2) \ldots \pi(a_k) \stopformula
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\item
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Similarly, the set of infinite words $\atoms^{\omega}$ has such a $\Pm$-action:
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\startformula \pi \cdot a_1 a_2 \ldots = \pi(a_1) \pi(a_2) \ldots \stopformula
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\item
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The empty set always admits a unique $G$-action for any $G$.
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(This is unique since the domain $G \times \emptyset = \emptyset$.)
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\startformula {\cdot} \colon G \times \emptyset \to \emptyset \stopformula
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\item
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The singleton set always admits a unique $G$-action for any $G$.
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(This is unique since the codomain only has just one element.)
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\startformula {\cdot} \colon G \times \{*\} \to \{*\} \stopformula
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\item
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For any set $X$, we can define a $G$-action by defining
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\startformula g \cdot x = x \stopformula
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for all the elements $x \in X$.
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Such an action is called \emph{trivial}.
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Note that the action on $\emptyset$ and $\{*\}$ are trivial, but the $\Pm$-actions on $\atoms$, $\atoms^{*}$ and $\atoms^{\omega}$ are not trivial.
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\item
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Another interesting non-trivial example is the set $\Pm$ itself.
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There are three different actions which are natural:
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\startformula\startalign
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\NC \pi \cdot_1 \sigma \NC = \pi \sigma \NR
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\NC \pi \cdot_2 \sigma \NC = \sigma \pi^{-1} \NR
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\NC \pi \cdot_3 \sigma \NC = \pi \sigma \pi^{-1} \NR
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\stopalign\stopformula
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Here the group multiplication is written by juxtaposition.
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The first two actions are left-multiplication and right-multiplication respectively.
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The latter is called conjugation.
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For each of them, one can verify the requirements.
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\stopitemize
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\stopexample
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In the above examples, the non trivial $\Pm$-sets are all infinite.
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Yet, in a sense, the set $\atoms^{*}$ is bigger than the set $\atoms$.
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To be able to quantify this, we introduce the notion of an orbit.
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\startdefinition
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Given a $G$-set $(X, {\cdot})$ and an element $x \in X$, we define the \emph{orbit of $x$} as the set
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\startformula \orb(x) = \{ g x \mid g \in G \}. \stopformula
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\stopdefinition
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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.
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This precisely happens if there exists a $g$ such that $g x = y$.
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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).
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This relation partitions the set $X$ in a collection of orbits:
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\startformula X = \bigcup_{x \in X} \orb(x). \stopformula
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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.
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\startexample
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We will describe the orbits for some non-trivial $\Pm$-sets.
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\startitemize
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\item
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The $\Pm$-set $\atoms$ only has \emph{one orbit}.
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To see this, take two (distinct) elements $a, b \in \atoms$ and consider the bijection $\pi = \swap{a}{b}$.
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Then we see that $\pi \cdot a = b$, meaning that $a$ and $b$ are in the same orbit.
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So $\atoms$ is a single-orbit set.
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\item
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Before we tackle $\atoms^{*}$, we will analyse $\atoms^{2}$.
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The set consists of exactly \emph{two orbits}:
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\startformula\startalign
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\NC \{ (a, a) \NC \mid a \in \atoms \} \NR
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\NC \{ (a, b) \NC \mid a, b \in \atoms, a \neq b \} \NR
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\stopalign\stopformula
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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.
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It can, however send any element $(a, b)$ to $(c, d)$ and so on.
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\item
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The set $\atoms^{*}$ has \emph{countably many orbits}.
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Since the action preserves the length of a word, we will show that the set has finitely many orbits for each length.
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So consider the set $\atoms^{k}$ with the point-wise action.
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An orbit of $\atoms^{k}$ is precisely determined by specifying which of the $k$ elements are equal to each other.
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This is a partition of $k$ elements, and there exactly $B_k$, the $k$th Bell number, such partitions.
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(As we have seen for $k = 2$, the second Bell number is $B_2 = 2$.
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This quantity grows exponential in $k$.)
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This shows that the set $\atoms^{*} = \bigcup_k \atoms^{k}$ has countably many orbits.
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\item
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The set $\atoms^{\omega}$ has \emph{uncountably many orbits}.
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\stopitemize
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\stopsubsection
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@ -54,6 +54,7 @@
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\define\Nsize{N}
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\define\Sym{Sym}
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\define[2]\swap{(#1 \, #2)}
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\define\Pm{\Perm(\atoms)}
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\define\Lint{\mathcal{L}_{\text{int}}}
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\define\Lmax{\mathcal{L}_{\text{max}}}
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