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writing/2024-02-17-formal-languages.md

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+---
+layout: post
+title: Formal languages
+---
+
+[formal grammar in nLab](https://ncatlab.org/nlab/show/formal+grammar)
+
+[Introduction to Language Theory \| Ray Toal, Loyola Marymount University](https://cs.lmu.edu/~ray/notes/languagetheory/)
+
+[Formal languages and automata. *Encyclopedia of Mathematics*](https://encyclopediaofmath.org/wiki/Formal_languages_and_automata)
+
+<https://opendsa-server.cs.vt.edu/OpenDSA/Books/PIFLAS22/html/index.html>
+
+# Languages over an alphabet 
+
+References: [^language] [^Toal]
+
+[^language]: [Formal languages and automata. *Encyclopedia of Mathematics*](https://encyclopediaofmath.org/wiki/Formal_languages_and_automata)
+
+[^Toal]: [Introduction to Language Theory \| Ray Toal, Loyola Marymount University](https://cs.lmu.edu/~ray/notes/languagetheory/)
+
+The empty set $\emptyset$ is an **initial object** in the category of sets **Set**:
+for any set $Y$ there is a unique function $\emptyset_Y:\emptyset \to Y$. [^empty]
+
+[^empty]: [empty function in nLab](https://ncatlab.org/nlab/show/empty+function)
+
+For sets $X$ and $Y$, let
+
+```math
+\mathrm{hom}_{\mathbf{Set}}(X,Y)=Y^X
+```
+
+be the set of functions with domain $X$ and image contained in $Y$.
+In particular, $Y^\emptyset = \{\emptyset_Y\}$.
+
+Let $\mathbb{N}=\{0,1,2,\ldots\}$. For $n \in \mathbb{N}$, define
+
+$[n] = \{i \in \mathbb{N} : i < n\}.$
+
+Thus, $[0]=\{\}=\emptyset$, $[1]=\{0\}$, $[2]=\{0,1\}$, $[3]=\{0,1,2\}$, etc.
+
+For $m,n \in \mathbb{N}$, define
+
+$[m,n] = \{i \in \mathbb{N}: m \leq i < n\}.$
+
+If $m \geq n$, then $[m,n]=\emptyset$.
+
+Let $A$ be a finite set that we shall call **an alphabet**.
+
+Write $\epsilon_A=\emptyset_A$, the unique function
+$\emptyset \to A$. Thus $A^\emptyset = \{\epsilon_A\}$.
+
+A **word over the alphabet** $A$ **of length** $n$, $n \in \mathbb{N}$, is an element
+of $A^{[n]}$. [^monoid]
+
+[^monoid]: [free monoid in nLab](https://ncatlab.org/nlab/show/free+monoid)
+
+A word over $A$ of length 0 is an element of $A^{[0]}$, and
+
+$A^{[0]} = A^\emptyset = \{\epsilon_A\},$
+
+so the unique word over $A$ of length 0 is $\epsilon_A$.
+We call $\epsilon_A$ the **empty word** over the alphabet $A$.
+
+If $N$ is an empty set, then as we have stated above, $\emptyset^N = \{\emptyset\}$.
+If $N$ is a nonempty set, then $\emptyset^N = \emptyset$. Thus, if the alphabet $A$ is empty, then
+$A^{[n]}=\emptyset$ for $n > 0$.
+
+If $x \in A^{[m]}$ and $y \in A^{[n]}$, define the **concatenation** $x*y \in A^{[m+n]}$ by
+
+```math
+(x*y)(i) = \begin{cases}
+x(i)&i \in [m]\\
+y(i-m)&i \in [m,m+n]
+\end{cases},
+\quad i \in [m+n].
+```
+
+## Free monoid on a set
+
+Define
+
+```math
+A^* = \bigcup_{n \in \mathbb{N}} A^{[n]}.
+```
+
+The sets in this union are pairwise disjoint; this is true whether or not the alphabet $A$ is empty,
+for different reasons.
+If the alphabet $A$ is empty,
+then $A^{[0]}=\{\epsilon_A\}$, and $A^{[n]}=\emptyset$ for $n>0$; and in this case
+the sets are indeed pairwise disjoint. If the alphabet $A$ is nonempty, then
+a word of some length is not a word of any other length, so in this case the sets are
+pairwise disjoint.
+
+$(A^*,*,\epsilon_A)$ is the **free monoid on** $A$. [^monoid] ($A^*$ is also called the **Kleene star of** $A$.[^Kleene])
+
+[^Kleene]: [Kleene star \| Wikipedia](https://en.wikipedia.org/wiki/Kleene_star)
+
+[^DM4CS]: [Discrete Mathematics for Computer Science \| OpenDSA Project](https://icsatkcc.github.io/Discrete-Math-for-Computer-Science/s-Monoids.html)
+
+For $x \in A^*$, define $\mathrm{len}(x)$ to be the unique $n \in \mathbb{N}$ such that $x \in A^{[n]}$,
+the **length** of a word. [^list]
+
+[^list]: [list in nLab](https://ncatlab.org/nlab/show/list)
+
+If $A$ is an empty set, then $A^*=\{\epsilon_A\}$.
+
+If $A$ is a set with one element, say $A=\{a\}$, then for each $n>0$, the unique
+function $[n] \to A$ is $i \mapsto a$ for $i \in [n]$. Thus when $A=\{a\}$,
+it is the case that for $n>0$ the set $A^{[n]}$ has exactly one element, function
+
+$i \mapsto a, \quad i \in [n],$
+
+that is, the unique word of length $n$ over $A=\{a\}$,
+
+$\underbrace{a \cdots a}_{n}.$
+
+If $A=\{a\}$ then $x \mapsto \mathrm{len}(x)$ is an isomorphism of monoids
+$A^* \to \mathbb{N}$.
+
+A **language over the alphabet** $A$ is any subset of $A^*$,
+the free monoid on $A$.
+
+## Free semigroup on a set
+
+Define
+
+$A^+ = \bigcup_{n \in \mathbb{N} \setminus\{0\}} A^{[n]}.$
+
+$(A^+,*)$ is the **free semigroup on** $A$. [^semigroup]
+
+$A^+ = A^* \setminus \{\epsilon_A\}.$
+
+[^semigroup]: [Free semi-group. *Encyclopedia of Mathematics*](https://encyclopediaofmath.org/wiki/Free_semi-group)
+
+# Operations on languages 
+
+We remind ourselves that a language over an alphabet $A$ is any subset of the free monoid $A^*$. In other words,
+the power set $\mathscr{P}(A^*)$ is the set of all languages over an alphabet $A$.
+
+**Union.** For $L_1, L_2 \subset A^*$, the union $L_1 \cup L_2 \subset A^*$.
+
+**Concatenation.** For $L_1, L_2 \subset A^*$,
+
+$L_1 L_2 = \{x*y : x \in L_1, y \in L_2\} \subset A^*.$
+
+**Kleene star.** For $L \subset A^*$, the Kleene star of $L$ is [^Kleene]
+
+```math
+L^* = \bigcup_{n \in \mathbb{N}} L^{[n]} \subset A^*.
+```
+
+# Free group on a set
+
+## Words over a set
+
+Let $X$ be a set.
+
+If $X \neq \emptyset$, for $x \in X$ define $I(x)=x$.
+
+Let $Y$ be a set disjoint from $X$ for which there is a bijection $J:X \to Y$.
+
+For $x \in X$, write $x^{-1} = J(x)$.
+
+Let
+
+$$
+\begin{align*}
+A &= X \cup Y\\
+&= \left( \bigcup_{x \in X} \lbrace I(x) \rbrace \right) \cup \left( \bigcup_{x \in X} \lbrace J(x) \rbrace \right)\\
+&= \bigcup_{x \in X} \lbrace I(x), J(x) \rbrace,
+\end{align*}
+$$
+
+elements of which we call **characters**.
+
+The **Kleene star** of $A$ is
+
+$$
+A^* = \bigcup_{n \geq 0} A^n.
+$$
+
+Elements of $W(X)=A^*$ are called **words over the set** $X$.
+
+$$
+A^0 = \lbrace \epsilon \rbrace
+$$
+
+We call $\epsilon$ the **empty word**. [^1]
+
+[^1]: <https://arbital.com/p/5zr/empty_set_universal_property/>
+
+For $n \geq 1$ and for  $a \in A^n$, we have
+
+$$
+a = a_1 \cdots a_n, \qquad a_i \in A, 1 \leq i \leq n.
+$$
+
+## Reduction
+
+If $a_i a_{i+1} = I(x)J(x)$ for some $x \in X$
+or $a_i a_{i+1} = J(x)I(x)$ for some $x \in X$,
+we call
+
+$$
+a_1 \cdots a_i a_{i+1} \cdots a_n \longrightarrow a_1 \cdots a_{i-1} \epsilon a_{i+2} \cdots a_n 
+$$
+
+a **reduction**.
+
+If a word $a$ has no reductions, it is called a **reduced word**. [^2]
+
+[^2]: <https://arbital.com/p/freely_reduced_word/>
+
+Let $W_R(X)$ be the set of reduced words over $X$.
+
+## Symmetric group on a set
+Let $S(X)$ be the set of bijections $X \to X$.
+
+### $X = \emptyset$
+
+$$S(\emptyset) = \lbrace \emptyset \rbrace$$
+
+[^5]
+
+[^5]: <https://arbital.com/p/empty_set_universal_property/>
+
+### $X \neq \emptyset$
+
+For $\sigma, \tau \in S(X)$, define
+
+$$
+(\sigma \tau)(x) = (\sigma \circ \tau)(x) = \sigma(\tau(x)),\qquad x \in X.
+$$
+
+$\mathrm{id}_X(x)=x$ is the identity element of $S(X)$.
+
+We call $S(X)$ the **symmetric group on the set** $X$.
+
+## Free group on a set
+
+
+
+[^3]
+
+[^3]: [Formal definition of the free group \| Arbital](https://arbital.com/p/free_group/?l=5s1)
+
+
+
+## Universal property
+
+
+[^4]
+
+[^4]: [Free group universal property \| Arbital](https://arbital.com/p/free_group/?l=6gd)
+
+