renaming core to chain and some other changes

Some changes:

- The core crate is renamed to "chain".

- The crate "viz" is added, which will provide layered graph drawing
  algorithms.

- A function is added to convert from a grammar to the regular
  language of its left-linear closures.

- A function is added to convert from a nondeterministic finite
  automaton to its "null" closure.  A null closure is the same
  automaton with edges added, as if some edges are "null".  Whether an
  edge is null is determined by a function.
  Combined with the previous change, we can convert a grammar to the
  regular language of the null closure of its left-linear closures.

---

Now it remains to test more grammars and add an Atom trait, before
finishing the part about compilations.

4 files changed, 1376 insertions, 0 deletions

diff --git a/chain/Cargo.toml b/chain/Cargo.toml
new file mode 100644
index 0000000..32eed8b
--- /dev/null
+++ b/chain/Cargo.toml
@@ -0,0 +1,10 @@
+[package]
+name = "chain"
+version = "0.1.0"
+edition = "2021"
+
+# See more keys and their definitions at https://doc.rust-lang.org/cargo/reference/manifest.html
+
+[dependencies]
+nfa = { path = "../nfa" }
+graph = { path = "../graph" }
\ No newline at end of file
diff --git a/chain/src/grammar.rs b/chain/src/grammar.rs
new file mode 100644
index 0000000..287fbca
--- /dev/null
+++ b/chain/src/grammar.rs
@@ -0,0 +1,1247 @@
+#![warn(missing_docs)]
+//! This file implements the extected behaviours of grammars.
+
+// NOTE: We shall first start with a parser that works at the level of
+// characters.  The purpose is to first experiment with the workings
+// and the performance of the algorithms, before optimising by using
+// regular expressions to classify inputs into tokens.  In other
+// words, the current focus is not on the optimisations, whereas
+// scanners are for optimisations only, so to speak.
+
+#![allow(unused_imports)]
+use nfa::{
+    default::{
+        nfa::DefaultNFA,
+        regex::{DefaultRegParser, DefaultRegex, ParseDirection, ParseError, RegexType},
+    },
+    DOption, DesRec, Nfa, Regex, SoC,
+};
+
+use graph::{adlist::ALGBuilder, builder::Builder, Graph};
+
+use std::{
+    collections::HashSet,
+    fmt::{Display, Write},
+};
+
+/// The type of a terminal.
+///
+/// For the time being this is a wrapper around a string, but in the
+/// future it may hold more information of scanners.
+#[derive(Debug, Clone, Eq, PartialEq)]
+pub struct Terminal {
+    // If we want to use scanners, per chance add them as a new field
+    // here.
+    name: String,
+}
+
+impl Terminal {
+    /// Create a terminal with the given name.
+    #[inline]
+    pub fn new(name: String) -> Self {
+        Self { name }
+    }
+
+    /// Return the name of the terminal.
+    #[inline]
+    pub fn name(&self) -> &str {
+        &self.name
+    }
+}
+
+/// The type of a non-terminal.
+///
+/// This is just a wrapper around a string.
+#[derive(Debug, Clone)]
+pub struct Nonterminal(String);
+
+impl Nonterminal {
+    /// Return the name of the nonterminal.
+    ///
+    /// Just to improve readability.
+    #[inline]
+    pub fn name(&self) -> &str {
+        &self.0
+    }
+}
+
+/// The type of a terminal or a non-terminal.
+///
+/// Only an index is stored here.  Actual data are stored in two other
+/// arrays.
+#[derive(Debug, Hash, Eq, PartialEq, Clone, Copy, Ord, PartialOrd)]
+pub enum TNT {
+    /// Terminal variant
+    Ter(usize),
+    /// Nonterminal variant
+    Non(usize),
+}
+
+impl Display for TNT {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        match self {
+            TNT::Ter(t) => write!(f, "T({t})"),
+            TNT::Non(n) => write!(f, "N({n})"),
+        }
+    }
+}
+
+/// Errors related to grammar operations.
+#[derive(Debug, Copy, Clone)]
+#[non_exhaustive]
+pub enum Error {
+    /// The first component is the index, and the second the bound.
+    IndexOutOfBounds(usize, usize),
+    /// Fail to build the N-th regular expression, due to the
+    /// ParseError.
+    BuildFail(usize, ParseError),
+    /// fail to build NFA
+    NFAFail(nfa::error::Error),
+}
+
+impl From<nfa::error::Error> for Error {
+    fn from(nfae: nfa::error::Error) -> Self {
+        Self::NFAFail(nfae)
+    }
+}
+
+impl Display for Error {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        match self {
+            Error::IndexOutOfBounds(i, b) => write!(f, "index {i} out of bound {b}"),
+            Error::BuildFail(n, pe) => write!(
+                f,
+                "Failed to build the {n}-th regular expression due to error: {pe}"
+            ),
+            Error::NFAFail(nfae) => write!(f, "failed to build NFA because of {nfae}"),
+        }
+    }
+}
+
+impl std::error::Error for Error {}
+
+/// A rule is a regular expression of terminals or non-terminals.
+#[derive(Debug, Clone)]
+pub struct Rule {
+    regex: DefaultRegex<TNT>,
+}
+
+impl Rule {
+    /// Return true if and only if the rule is empty.
+    #[inline]
+    pub fn is_empty(&self) -> bool {
+        self.regex.is_empty()
+    }
+
+    /// Return the length of the rule.
+    #[inline]
+    pub fn len(&self) -> usize {
+        self.regex.len()
+    }
+}
+
+/// The type of a grammar.
+#[derive(Debug, Clone, Default)]
+pub struct Grammar {
+    ter: Vec<Terminal>,
+    non: Vec<Nonterminal>,
+    rules: Vec<Rule>,
+    firsts: Vec<HashSet<Option<usize>>>,
+    first_nodes: Vec<Vec<usize>>,
+}
+
+/// A private type to aid the recursive looping of rergular
+/// expressions.
+#[derive(Copy, Clone)]
+enum StackElement {
+    Seen(usize),
+    Unseen(usize),
+}
+
+impl StackElement {
+    fn index(self) -> usize {
+        match self {
+            Self::Seen(index) => index,
+            Self::Unseen(index) => index,
+        }
+    }
+
+    fn is_seen(self) -> bool {
+        matches!(self, Self::Seen(_))
+    }
+}
+
+impl Grammar {
+    /// Construct a grammar from a vector of terminals, a vector of
+    /// non-terminals, and a vector of rules for the non-temrinals.
+    ///
+    /// # Panic
+    ///
+    /// If the length of `non` is not equal to that of `rules`, then
+    /// the function panics.
+    pub fn new(ter: Vec<Terminal>, non: Vec<Nonterminal>, rules: Vec<Rule>) -> Self {
+        assert_eq!(non.len(), rules.len());
+
+        // One more room is reserved for the `None` value.
+        let firsts = std::iter::repeat_with(|| HashSet::with_capacity(ter.len() + 1))
+            .take(non.len())
+            .collect();
+
+        let first_nodes = rules
+            .iter()
+            .map(|rule| Vec::with_capacity(rule.len()))
+            .collect();
+
+        Self {
+            ter,
+            non,
+            rules,
+            firsts,
+            first_nodes,
+        }
+    }
+
+    /// Return the name of a terminal or a non-terminal.
+    pub fn name_of_tnt(&self, tnt: TNT) -> Result<String, Error> {
+        match tnt {
+            TNT::Ter(t) => Ok(format!(
+                "T{}",
+                self.ter
+                    .get(t)
+                    .ok_or(Error::IndexOutOfBounds(t, self.ter.len()))?
+                    .name()
+            )),
+            TNT::Non(n) => Ok(format!(
+                "N{}",
+                self.non
+                    .get(n)
+                    .ok_or(Error::IndexOutOfBounds(n, self.non.len()))?
+                    .name()
+            )),
+        }
+    }
+
+    /// Return true if and only if there are no non-terminals in the
+    /// grammar.
+    #[inline]
+    pub fn is_empty(&self) -> bool {
+        self.non.is_empty()
+    }
+
+    /// Return the total length of all rules.
+    #[inline]
+    pub fn total(&self) -> usize {
+        self.rules.iter().map(Rule::len).sum()
+    }
+
+    /// Return the number of terminals.
+    #[inline]
+    pub fn ter_num(&self) -> usize {
+        self.ter.len()
+    }
+
+    /// Return the number of non-terminals.
+    #[inline]
+    pub fn non_num(&self) -> usize {
+        self.non.len()
+    }
+
+    /// Convert a non-terminal `N` to `N + TER_NUM`, so that we use a
+    /// single number to represent terminals and non-terminals.
+    ///
+    /// # Bounds
+    ///
+    /// If a terminal index is greater than or equal to the number of
+    /// terminals, then this signals an error; mutatis mutandis for
+    /// non-terminals.
+    ///
+    /// # Related
+    ///
+    /// The inverse function is [`unpack_tnt`][Grammar::unpack_tnt].
+    #[inline]
+    pub fn pack_tnt(&self, tnt: TNT) -> Result<usize, Error> {
+        let ter_num = self.ter.len();
+        let non_num = self.non.len();
+
+        match tnt {
+            TNT::Ter(t) => {
+                if t >= ter_num {
+                    Err(Error::IndexOutOfBounds(t, ter_num))
+                } else {
+                    Ok(t)
+                }
+            }
+            TNT::Non(n) => {
+                if n >= non_num {
+                    Err(Error::IndexOutOfBounds(n, non_num))
+                } else {
+                    Ok(n + ter_num)
+                }
+            }
+        }
+    }
+
+    /// Convert a single number to either a terminal or a
+    /// non-terminal.
+    ///
+    /// # Bounds
+    ///
+    /// If the number is greater than or equal to the sum of the
+    /// numbers of terminals and of non-terminals, then this signals
+    /// an error.
+    ///
+    /// # Related
+    ///
+    /// This is the inverse of [`pack_tnt`][Grammar::pack_tnt].
+    #[inline]
+    pub fn unpack_tnt(&self, flat: usize) -> Result<TNT, Error> {
+        let ter_num = self.ter.len();
+        let non_num = self.non.len();
+
+        if flat < ter_num {
+            Ok(TNT::Ter(flat))
+        } else if flat < ter_num + non_num {
+            Ok(TNT::Non(flat - ter_num))
+        } else {
+            Err(Error::IndexOutOfBounds(flat, ter_num + non_num))
+        }
+    }
+
+    /// Return true if and only if the non-terminal is nullable.
+    #[inline]
+    pub fn is_nullable(&self, non_terminal: usize) -> Result<bool, Error> {
+        Ok(self
+            .firsts
+            .get(non_terminal)
+            .ok_or(Error::IndexOutOfBounds(non_terminal, self.non.len()))?
+            .contains(&None))
+    }
+
+    /// For a NON_TERMINAL, return an iterator that goes over the
+    /// nodes that are reachable from the non-terminal through an
+    /// empty transition of the nondeterministic finite automaton.
+    #[inline]
+    pub fn first_nodes_of(&self, non_terminal: usize) -> Result<std::slice::Iter<usize>, Error> {
+        Ok(self
+            .first_nodes
+            .get(non_terminal)
+            .ok_or(Error::IndexOutOfBounds(non_terminal, self.non.len()))?
+            .iter())
+    }
+
+    /// Compute the set of terminals that can appear as the first
+    /// terminal in some left-linear derivation of a non-terminal, for
+    /// every non-terminal.
+    ///
+    /// This is an algorithm that computes the transitive closure,
+    /// which is a common approach for this task.  But perhaps there
+    /// are more efficient approaches?
+    ///
+    /// Also the function computes the set of "reachable nodes" in the
+    /// process, and records the information in the `first_nodes`
+    /// attribute.
+    pub fn compute_firsts(&mut self) -> Result<(), Error> {
+        let mut updated = true;
+
+        let non_len = self.non_num();
+
+        use StackElement::{Seen, Unseen};
+
+        while updated {
+            updated = false;
+
+            for (n, regex) in self.rules.iter().map(|rule| &rule.regex).enumerate() {
+                let root = if let Some(root) = regex.root() {
+                    root
+                } else {
+                    if !self.is_nullable(n)? {
+                        updated = true;
+
+                        self.firsts.get_mut(n).unwrap().insert(None);
+
+                        // The default construction of a grammar
+                        // reserves some space for each vector, so
+                        // explicitly setting this can reduce some
+                        // minor memory overhead.
+                        let pointer = self.first_nodes.get_mut(n).unwrap();
+
+                        pointer.clear();
+                        pointer.shrink_to_fit();
+                    }
+
+                    continue;
+                };
+
+                let regex_len = regex.len();
+
+                let mut stack: Vec<StackElement> = Vec::with_capacity(regex_len);
+
+                stack.push(Unseen(root));
+
+                let mut children_sets_stack: Vec<HashSet<Option<usize>>> =
+                    Vec::with_capacity(regex_len);
+
+                let mut children_nodes_stack: Vec<HashSet<usize>> = Vec::with_capacity(regex_len);
+
+                while let Some(top) = stack.pop() {
+                    let top_index = top.index();
+                    let is_seen = top.is_seen();
+
+                    match regex
+                        .vertex_label(top_index)
+                        .map_err(|_| Error::IndexOutOfBounds(top_index, regex_len))?
+                    {
+                        RegexType::Kleene => {
+                            if !is_seen {
+                                stack.push(Seen(top_index));
+
+                                for child in regex.children_of(top_index).unwrap() {
+                                    stack.push(Unseen(child));
+                                }
+                            } else {
+                                let degree = regex.degree(top_index).unwrap();
+                                let children_stack_len = children_sets_stack.len();
+                                let children_nodes_len = children_nodes_stack.len();
+
+                                assert!(
+                                    children_stack_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                assert!(
+                                    children_nodes_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                let mut this_set = HashSet::new();
+
+                                this_set.insert(None);
+
+                                let mut this_nodes = HashSet::new();
+
+                                this_nodes.insert(top_index);
+
+                                if degree == 0 {
+                                    children_sets_stack.push(this_set);
+                                    children_nodes_stack.push(this_nodes);
+
+                                    continue;
+                                }
+
+                                let mut stop = false;
+
+                                for (child_set, child_nodes) in children_sets_stack
+                                    .drain((children_stack_len - degree)..)
+                                    .zip(
+                                        children_nodes_stack.drain((children_nodes_len - degree)..),
+                                    )
+                                {
+                                    if stop {
+                                        break;
+                                    }
+
+                                    if !child_set.contains(&None) {
+                                        stop = true;
+                                    }
+
+                                    this_set.extend(child_set);
+                                    this_nodes.extend(child_nodes);
+                                }
+
+                                children_sets_stack.push(this_set);
+                                children_nodes_stack.push(this_nodes);
+                            }
+                        }
+                        RegexType::Plus => {
+                            if !is_seen {
+                                stack.push(Seen(top_index));
+
+                                for child in regex.children_of(top_index).unwrap() {
+                                    stack.push(Unseen(child));
+                                }
+                            } else {
+                                let degree = regex.degree(top_index).unwrap();
+                                let children_stack_len = children_sets_stack.len();
+                                let children_nodes_len = children_nodes_stack.len();
+
+                                assert!(
+                                    children_stack_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                assert!(
+                                    children_nodes_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                let mut this_set = HashSet::new();
+
+                                let mut this_nodes = HashSet::new();
+
+                                this_nodes.insert(top_index);
+
+                                if degree == 0 {
+                                    this_set.insert(None);
+
+                                    children_sets_stack.push(this_set);
+                                    children_nodes_stack.push(this_nodes);
+
+                                    continue;
+                                }
+
+                                let mut stop = false;
+
+                                for (child_set, child_nodes) in children_sets_stack
+                                    .drain((children_stack_len - degree)..)
+                                    .zip(
+                                        children_nodes_stack.drain((children_nodes_len - degree)..),
+                                    )
+                                {
+                                    if stop {
+                                        break;
+                                    }
+
+                                    if !child_set.contains(&None) {
+                                        stop = true;
+                                    }
+
+                                    this_set.extend(child_set);
+                                    this_nodes.extend(child_nodes);
+                                }
+
+                                if stop {
+                                    this_set.remove(&None);
+                                }
+
+                                children_sets_stack.push(this_set);
+                                children_nodes_stack.push(this_nodes);
+                            }
+                        }
+                        RegexType::Optional => {
+                            if !is_seen {
+                                stack.push(Seen(top_index));
+
+                                for child in regex.children_of(top_index).unwrap() {
+                                    stack.push(Unseen(child));
+                                }
+                            } else {
+                                let degree = regex.degree(top_index).unwrap();
+                                let children_stack_len = children_sets_stack.len();
+                                let children_nodes_len = children_nodes_stack.len();
+
+                                assert!(
+                                    children_stack_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                assert!(
+                                    children_nodes_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                let mut this_set = HashSet::new();
+
+                                this_set.insert(None);
+
+                                let mut this_nodes = HashSet::new();
+
+                                this_nodes.insert(top_index);
+
+                                if degree == 0 {
+                                    children_sets_stack.push(this_set);
+                                    children_nodes_stack.push(this_nodes);
+
+                                    continue;
+                                }
+
+                                let mut stop = false;
+
+                                for (child_set, child_nodes) in children_sets_stack
+                                    .drain((children_stack_len - degree)..)
+                                    .zip(
+                                        children_nodes_stack.drain((children_nodes_len - degree)..),
+                                    )
+                                {
+                                    if stop {
+                                        break;
+                                    }
+
+                                    if !child_set.contains(&None) {
+                                        stop = true;
+                                    }
+
+                                    this_set.extend(child_set.iter().copied());
+                                    this_nodes.extend(child_nodes.iter().copied());
+                                }
+
+                                children_sets_stack.push(this_set);
+                                children_nodes_stack.push(this_nodes);
+                            }
+                        }
+                        RegexType::Or => {
+                            if !is_seen {
+                                stack.push(Seen(top_index));
+
+                                for child in regex.children_of(top_index).unwrap() {
+                                    stack.push(Unseen(child));
+                                }
+                            } else {
+                                let degree = regex.degree(top_index).unwrap();
+                                let children_stack_len = children_sets_stack.len();
+                                let children_nodes_len = children_nodes_stack.len();
+
+                                assert!(
+                                    children_stack_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                assert!(
+                                    children_nodes_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                let mut this_set = HashSet::new();
+
+                                let mut this_nodes = HashSet::new();
+
+                                this_nodes.insert(top_index);
+
+                                if degree == 0 {
+                                    this_set.insert(None);
+
+                                    children_sets_stack.push(this_set);
+                                    children_nodes_stack.push(this_nodes);
+
+                                    continue;
+                                }
+
+                                for (child_set, child_nodes) in children_sets_stack
+                                    .drain((children_stack_len - degree)..)
+                                    .zip(
+                                        children_nodes_stack.drain((children_nodes_len - degree)..),
+                                    )
+                                {
+                                    this_set.extend(child_set.iter().copied());
+                                    this_nodes.extend(child_nodes.iter().copied());
+                                }
+
+                                children_sets_stack.push(this_set);
+                                children_nodes_stack.push(this_nodes);
+                            }
+                        }
+                        RegexType::Paren => {
+                            // Only for printing purposes
+                            let mut this_set = HashSet::new();
+
+                            this_set.insert(None);
+
+                            children_sets_stack.push(this_set);
+
+                            let mut this_nodes = HashSet::new();
+
+                            this_nodes.insert(top_index);
+
+                            children_nodes_stack.push(this_nodes);
+                        }
+                        RegexType::Empty => {
+                            if !is_seen {
+                                stack.push(Seen(top_index));
+
+                                for child in regex.children_of(top_index).unwrap().rev() {
+                                    stack.push(Unseen(child));
+                                }
+                            } else {
+                                let degree = regex.degree(top_index).unwrap();
+                                let children_stack_len = children_sets_stack.len();
+                                let children_nodes_len = children_nodes_stack.len();
+
+                                assert!(
+                                    children_stack_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                assert!(
+                                    children_nodes_len >= degree,
+                                    "not enough stack elements for {top_index}"
+                                );
+
+                                let mut this_set = HashSet::new();
+
+                                let mut this_nodes = HashSet::new();
+
+                                this_nodes.insert(top_index);
+
+                                if degree == 0 {
+                                    this_set.insert(None);
+
+                                    children_sets_stack.push(this_set);
+                                    children_nodes_stack.push(this_nodes);
+
+                                    continue;
+                                }
+
+                                let mut stop = false;
+
+                                for (child_set, child_nodes) in children_sets_stack
+                                    .drain((children_stack_len - degree)..)
+                                    .zip(
+                                        children_nodes_stack.drain((children_nodes_len - degree)..),
+                                    )
+                                {
+                                    if stop {
+                                        break;
+                                    }
+
+                                    if !child_set.contains(&None) {
+                                        stop = true;
+                                    }
+
+                                    this_set.extend(child_set.iter().copied());
+                                    this_nodes.extend(child_nodes.iter().copied());
+                                }
+
+                                if stop {
+                                    this_set.remove(&None);
+                                }
+
+                                children_sets_stack.push(this_set);
+                                children_nodes_stack.push(this_nodes);
+                            }
+                        }
+                        RegexType::Lit(tnt) => {
+                            match tnt {
+                                TNT::Ter(t) => {
+                                    let mut this_set = HashSet::with_capacity(1);
+
+                                    this_set.insert(Some(t));
+
+                                    children_sets_stack.push(this_set);
+                                }
+                                TNT::Non(non) => {
+                                    let this_set = self
+                                        .firsts
+                                        .get(non)
+                                        .ok_or(Error::IndexOutOfBounds(non, non_len))?
+                                        .clone();
+
+                                    children_sets_stack.push(this_set);
+                                }
+                            }
+
+                            let mut this_nodes = HashSet::with_capacity(1);
+                            this_nodes.insert(top_index);
+
+                            children_nodes_stack.push(this_nodes);
+                        }
+                    }
+                }
+
+                assert_eq!(
+                    children_sets_stack.len(),
+                    1,
+                    "Too many elements left at the end"
+                );
+
+                assert_eq!(
+                    children_nodes_stack.len(),
+                    1,
+                    "Too many elements left at the end"
+                );
+
+                for first in children_sets_stack.pop().unwrap().into_iter() {
+                    if !self.firsts.get(n).unwrap().contains(&first) {
+                        updated = true;
+
+                        self.firsts.get_mut(n).unwrap().insert(first);
+                    }
+                }
+
+                *self.first_nodes.get_mut(n).unwrap() =
+                    children_nodes_stack.pop().unwrap().into_iter().collect();
+            }
+        }
+
+        Ok(())
+    }
+
+    /// Return the regular language of the left-linear closures of
+    /// non-terminals in the grammar.
+    ///
+    /// The resulting vector is guaranteed to be of the same length as
+    /// the number of non-terminals.
+    ///
+    /// The resulting regular language is not "self-contained".  That
+    /// is to say, its terminals indices are packed indices and are
+    /// meaningless without the interpretation of the grammar.  They
+    /// should be converted to a nondeterministic finite automaton and
+    /// then to its null closure later on.
+    pub fn left_closure(&self) -> Result<Vec<DefaultRegex<TNT>>, Error> {
+        let non_len = self.non_num();
+
+        let mut result = Vec::with_capacity(non_len);
+
+        for (n, rule) in self.rules.iter().enumerate() {
+            let regex = &rule.regex;
+
+            let regex_root = if let Some(root) = regex.root() {
+                root
+            } else {
+                result.push(Default::default());
+
+                continue;
+            };
+
+            let regex_len = regex.len();
+
+            /// A convenient macro to retrieve the children from the
+            /// original regular expression with error propagation.
+            macro_rules! children {
+                ($n:expr) => {
+                    regex
+                        .children_of($n)
+                        .map_err(|_| Error::IndexOutOfBounds($n, regex_len))?
+                };
+            }
+
+            /// A convenient macro to retrieve the label from the
+            /// original regular expression with error propagation.
+            macro_rules! label {
+                ($n:expr) => {
+                    regex
+                        .vertex_label($n)
+                        .map_err(|_| Error::IndexOutOfBounds($n, regex_len))?
+                };
+            }
+
+            let parents = regex.parents_array().map_err(|e| match e {
+                nfa::error::Error::UnknownNode(n) => Error::IndexOutOfBounds(n, regex_len),
+                nfa::error::Error::Cycle => Error::BuildFail(n, ParseError::Cycle),
+                _ => unreachable!(),
+            })?;
+
+            use RegexType::*;
+            use TNT::*;
+
+            let mut local_result: Vec<RegexType<TNT>> = Vec::with_capacity(regex_len * 2);
+            let mut builder = ALGBuilder::default();
+
+            /// Perform a depth-first copy
+            macro_rules! df_copy {
+                ($parent:expr, $index:expr) => {
+                    match label!($index) {
+                        Kleene | Plus | Optional | Or | Paren | Empty => {
+                            let mut stack = vec![($parent, $index)];
+
+                            while let Some((top_parent, top_index)) = stack.pop() {
+                                let label = label!(top_index);
+                                let label = match label {
+                                    Lit(top_tnt) => Lit(Ter(self.pack_tnt(top_tnt).unwrap())),
+                                    _ => label,
+                                };
+
+                                local_result.push(label);
+
+                                let new = builder.add_vertex();
+
+                                builder.add_edge(top_parent, new, ()).unwrap();
+
+                                stack.extend(children!(top_index).map(|child| (new, child)));
+                            }
+                        }
+                        Lit(remain_tnt) => {
+                            local_result.push(Lit(Ter(self.pack_tnt(remain_tnt).unwrap())));
+                            let new = builder.add_vertex();
+                            builder.add_edge($parent, new, ()).unwrap();
+                        }
+                    }
+                };
+            }
+
+            local_result.push(Or);
+            builder.add_vertex();
+
+            local_result.push(Lit(Ter(self.pack_tnt(Non(n)).unwrap())));
+            let non_lit_index = builder.add_vertex();
+
+            builder.add_edge(0, non_lit_index, ()).unwrap();
+
+            for first_node in self.first_nodes_of(n)?.copied() {
+                assert!(first_node < parents.len());
+
+                let tnt = match label!(first_node) {
+                    Lit(tnt) => tnt,
+                    _ => continue,
+                };
+
+                let mut parents_chain = {
+                    let mut result = Vec::new();
+                    let mut stack = Vec::with_capacity(parents.len());
+
+                    stack.push(first_node);
+
+                    while let Some(top) = stack.pop() {
+                        assert!(top < parents.len());
+                        if let Some(parent) = parents.get(top).copied().unwrap() {
+                            result.push(parent);
+                            stack.push(parent.0);
+                        }
+                    }
+
+                    result.reverse();
+
+                    result
+                };
+
+                if parents_chain.is_empty() {
+                    local_result.push(Lit(tnt));
+                    let lit_index = builder.add_vertex();
+                    builder.add_edge(0, lit_index, ()).unwrap();
+
+                    continue;
+                }
+
+                assert!(parents_chain.first().unwrap().0 == regex_root);
+
+                // A different, "more local", root.
+                let mut root: usize;
+
+                // Handle the direct parent
+                let (parent_node, parent_edge_index) = parents_chain.pop().unwrap();
+
+                match label!(parent_node) {
+                    Kleene | Plus => {
+                        local_result.extend([Empty, Lit(tnt)]);
+
+                        root = builder.add_vertex();
+                        let lit_index = builder.add_vertex();
+                        builder.add_edge(root, lit_index, ()).unwrap();
+
+                        let iterator = children!(parent_node);
+
+                        for index in iterator.clone().skip(parent_edge_index + 1) {
+                            df_copy!(root, index);
+                        }
+
+                        local_result.push(Kleene);
+                        let new_parent = builder.add_vertex();
+                        builder.add_edge(root, new_parent, ()).unwrap();
+
+                        for index in iterator {
+                            df_copy!(new_parent, index);
+                        }
+                    }
+
+                    Or => {
+                        local_result.push(Lit(tnt));
+                        root = builder.add_vertex();
+                    }
+                    Optional | Empty => {
+                        // If this path is taken, it should not be
+                        // optional.
+                        local_result.extend([Empty, Lit(tnt)]);
+                        root = builder.add_vertex();
+                        let lit_index = builder.add_vertex();
+                        builder.add_edge(root, lit_index, ()).unwrap();
+
+                        for index in children!(parent_node).skip(parent_edge_index + 1) {
+                            df_copy!(root, index);
+                        }
+                    }
+                    Paren | Lit(_) => unreachable!(),
+                }
+
+                // Handle successive parents
+
+                for (node, edge_index) in parents_chain.into_iter() {
+                    let node_type = label!(node);
+
+                    match node_type {
+                        Kleene | Plus => {
+                            local_result.push(Empty);
+                            let new_index = builder.add_vertex();
+                            builder.add_edge(new_index, root, ()).unwrap();
+
+                            root = new_index;
+
+                            let iterator = children!(node);
+
+                            for index in iterator.clone().skip(edge_index + 1) {
+                                df_copy!(root, index);
+                            }
+
+                            local_result.push(Kleene);
+                            let new_parent = builder.add_vertex();
+                            builder.add_edge(root, new_parent, ()).unwrap();
+
+                            for index in iterator {
+                                df_copy!(new_parent, index);
+                            }
+                        }
+                        RegexType::Or => {}
+                        RegexType::Optional | RegexType::Empty => {
+                            local_result.push(Empty);
+                            let new_index = builder.add_vertex();
+                            builder.add_edge(new_index, root, ()).unwrap();
+                            root = new_index;
+
+                            for index in children!(node).skip(edge_index + 1) {
+                                df_copy!(root, index);
+                            }
+                        }
+                        RegexType::Paren | RegexType::Lit(_) => unreachable!(),
+                    }
+                }
+
+                builder.add_edge(0, root, ()).unwrap();
+            }
+
+            local_result.shrink_to_fit();
+
+            let graph = builder.build();
+
+            assert_eq!(graph.nodes_len(), local_result.len());
+
+            result.push(
+                DefaultRegex::new(graph, local_result)
+                    .map_err(|_| Error::BuildFail(n, ParseError::Cycle))?,
+            );
+        }
+
+        assert_eq!(result.len(), non_len);
+
+        Ok(result)
+    }
+
+    /// Convert the regular language of left-linear closures to its
+    /// equivalent nondeterministic finite automaton.
+    ///
+    /// In the generation of the left-linear closure, the terminals
+    /// and non-terminals are packed into an unsigned integer.  We
+    /// unpack them in converting to nondeterministic finite
+    /// automaton.
+    ///
+    /// The resulting nondeterministic finite automaton should be
+    /// transformed to its null-closure for use in our algorithm.
+    pub fn left_closure_to_nfa(
+        &self,
+        closure: &[DefaultRegex<TNT>],
+    ) -> Result<DefaultNFA<DOption<TNT>>, Error> {
+        let label_transform = |tnt| match tnt {
+            TNT::Ter(t) => {
+                let new_tnt = self.unpack_tnt(t).map_err(|e| match e {
+                    Error::IndexOutOfBounds(index, bound) => {
+                        graph::error::Error::IndexOutOfBounds(index, bound)
+                    }
+                    _ => unreachable!(),
+                })?;
+
+                Ok(SoC::Carry(new_tnt))
+            }
+            TNT::Non(n) => Ok(SoC::Sub(n)),
+        };
+
+        DefaultNFA::to_nfa(closure, label_transform).map_err(Into::into)
+    }
+}
+
+impl Display for Grammar {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        assert_eq!(self.non.len(), self.rules.len());
+
+        for (nt, rule) in self.non.iter().zip(self.rules.iter()) {
+            write!(f, "{}: ", nt.name())?;
+
+            writeln!(
+                f,
+                "{}",
+                rule.regex.to_string_with(|tnt| format!(
+                    "({})",
+                    self.name_of_tnt(tnt)
+                        .unwrap_or_else(|_| format!("Unknown {tnt:?}"))
+                ))?
+            )?;
+        }
+
+        Ok(())
+    }
+}
+
+#[cfg(test)]
+mod test_grammar_helper {
+    use super::*;
+    use nfa::default::regex::{DefaultRegParser, ParseDirection, ParseError, RegexType};
+    use std::fmt::Write;
+
+    // Construct a grammar to test
+    pub fn new_grammar() -> Result<Grammar, Box<dyn std::error::Error>> {
+        let ter = vec![Terminal::new("a".to_owned()), Terminal::new("b".to_owned())];
+        let non = vec![
+            Nonterminal("start".to_owned()),
+            Nonterminal("end".to_owned()),
+        ];
+
+        /// A function to scan the inputs.
+        fn scan_tnt(
+            parser: &DefaultRegParser<TNT>,
+            input: &str,
+        ) -> Result<Option<(usize, RegexType<TNT>, ParseDirection)>, ParseError> {
+            use ParseDirection::*;
+            use RegexType::*;
+            use TNT::*;
+
+            let mut chars = input.chars();
+
+            if let Some(first) = chars.next() {
+                match first {
+                    '*' => Ok(Some((1, Kleene, Right))),
+                    '+' => Ok(Some((1, Plus, Right))),
+                    '?' => Ok(Some((1, Optional, Right))),
+                    '|' => Ok(Some((1, Empty, Up))),
+                    '(' => Ok(Some((1, Or, Down))),
+                    ')' => Ok(Some((1, Paren, Up))),
+                    'T' => {
+                        let mut name = String::new();
+                        let mut len = 1;
+
+                        while let Some(c) = chars.next() {
+                            if ('a'..='z').contains(&c) {
+                                len += 1;
+                                write!(name, "{c}").map_err(|_| ParseError::InvalidCharacter(c))?;
+                            } else {
+                                break;
+                            }
+                        }
+
+                        if let Some(t) = parser.query(&name, true) {
+                            Ok(Some((len, Lit(Ter(t)), Right)))
+                        } else {
+                            Err(ParseError::InvalidCharacter(first))
+                        }
+                    }
+                    'N' => {
+                        let mut name = String::new();
+                        let mut len = 1;
+
+                        while let Some(c) = chars.next() {
+                            if ('a'..='z').contains(&c) {
+                                len += 1;
+                                write!(name, "{c}").map_err(|_| ParseError::InvalidCharacter(c))?;
+                            } else {
+                                break;
+                            }
+                        }
+
+                        if let Some(n) = parser.query(&name, false) {
+                            Ok(Some((len, Lit(Non(n)), Right)))
+                        } else {
+                            Err(ParseError::InvalidCharacter(first))
+                        }
+                    }
+                    _ => Err(ParseError::InvalidCharacter(first)),
+                }
+            } else {
+                Ok(None)
+            }
+        }
+
+        let mut regex_parser: DefaultRegParser<TNT> = Default::default();
+
+        regex_parser.add_tnt("a", true);
+        regex_parser.add_tnt("b", true);
+        regex_parser.add_tnt("start", false);
+        regex_parser.add_tnt("end", false);
+
+        let regex_parser = regex_parser;
+
+        let rule1 = Rule {
+            regex: regex_parser
+                .parse("Ta*Tb+Nend+", Box::new(scan_tnt), true)?
+                .ok_or(ParseError::Invalid)?
+                .0,
+        };
+
+        let rule2 = Rule {
+            regex: regex_parser
+                .parse("Nstart?Nend*", Box::new(scan_tnt), true)?
+                .ok_or(ParseError::Invalid)?
+                .0,
+        };
+
+        let rules = vec![rule1, rule2];
+
+        Ok(Grammar::new(ter, non, rules))
+    }
+}
+
+#[cfg(test)]
+mod test_grammar_display {
+    use super::test_grammar_helper::new_grammar;
+
+    #[test]
+    fn test_display() -> Result<(), Box<dyn std::error::Error>> {
+        println!("{}", new_grammar()?);
+
+        Ok(())
+    }
+}
+
+#[cfg(test)]
+mod test_grammar_firsts {
+    use super::test_grammar_helper::new_grammar;
+    use super::*;
+
+    #[test]
+    fn test_firsts() -> Result<(), Box<dyn std::error::Error>> {
+        let mut grammar = new_grammar()?;
+
+        grammar.compute_firsts()?;
+
+        println!("grammar firsts: {:?}", grammar.firsts);
+        println!("grammar first nodes: {:?}", grammar.first_nodes);
+
+        Ok(())
+    }
+}
+
+#[cfg(test)]
+mod test_grammar_left_closure {
+    use super::test_grammar_helper::new_grammar;
+    use super::*;
+
+    pub fn new_closure_regex(
+        grammar: &mut Grammar,
+    ) -> Result<Vec<DefaultRegex<TNT>>, Box<dyn std::error::Error>> {
+        grammar.compute_firsts()?;
+
+        println!("grammar firsts: {:?}", grammar.firsts);
+        println!("grammar first nodes: {:?}", grammar.first_nodes);
+        println!("grammar:");
+        println!("{grammar}");
+
+        grammar.left_closure().map_err(Into::into)
+    }
+
+    #[test]
+    fn test_regex() -> Result<(), Box<dyn std::error::Error>> {
+        let mut grammar = new_grammar()?;
+
+        let vec_of_regexps = new_closure_regex(&mut grammar)?;
+
+        for regex in &vec_of_regexps {
+            println!("regex: {}", regex.to_string_with(|tnt| format!("{tnt}"))?);
+            // println!("regex: {regex:?}",);
+            println!("regex len = {}", regex.nodes_len());
+        }
+
+        Ok(())
+    }
+
+    #[test]
+    fn test_nfa() -> Result<(), Box<dyn std::error::Error>> {
+        let mut grammar = new_grammar()?;
+        let closure = new_closure_regex(&mut grammar)?;
+        let nfa = grammar.left_closure_to_nfa(&closure)?;
+        // TODO: print the nfa out
+        Ok(())
+    }
+}
diff --git a/chain/src/lib.rs b/chain/src/lib.rs
new file mode 100644
index 0000000..0ec4d4c
--- /dev/null
+++ b/chain/src/lib.rs
@@ -0,0 +1,25 @@
+//! This package implements the core algorithm of the entire
+//! workspace: parsing with derivatives by means of chain rule and
+//! regular nulling languages.
+//!
+//! Since I shall not name my crate "core" to avoid collisions with
+//! the Rust's own core, I decided to name this crate after what I
+//! think is the essence of this algorithm, the chain-rule for
+//! derivatives of languages.
+
+pub mod grammar;
+
+pub fn add(left: usize, right: usize) -> usize {
+    left + right
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn it_works() {
+        let result = add(2, 2);
+        assert_eq!(result, 4);
+    }
+}
diff --git a/chain/src/plan.org b/chain/src/plan.org
new file mode 100644
index 0000000..8c63492
--- /dev/null
+++ b/chain/src/plan.org
@@ -0,0 +1,94 @@
+#+TITLE: Plan of the package
+#+AUTHOR: Durand
+#+DATE: <2022-11-18 Ven 19:57>
+
+* Things to do [2/7]
+
+- [X] Implement builders for graphs
+- [X] Find sets of the first terminals for each non-terminal, in the
+  grammar module.
+- [-] NFA [4/5]
+  + [X] Add regular expression into NFA.
+  + [X] Convert a set of regular expressions into a nondeterministic
+    finite automaton, where non-terminals denote the indices of
+    regular expressions in the set to substitute.
+  + [X] Convert a grammar into its grammar of left-linear closures of
+    non-temrinals and their derivatives.
+  + [X] Convert nondeterministic finite automata to their null
+    closures.
+  + [ ] Test more grammars to ensure it works correctly.
+- [ ] Define the Atom trait.
+- [ ] Implement virtual nodes for each derivative of each atom: The
+  lack of this step might be the root cause of the failure of the
+  previous version of this package.
+- [ ] Implement languages. [0/2]
+  + [ ] First implement them as doubly labelled (directed acyclic)
+    graphs.
+  + [ ] Implement finding derivatives.
+- [ ] Implement semiring. [0/5]
+  + [ ] Define the trait.
+  + [ ] Implement the boolean semiring.
+  + [ ] Implement the natural number semiring.
+  + [ ] Implement the free semiring.
+  + [ ] Compute and record a semiring value when computing
+    derivatives.
+
+* Atomic Languages
+
+This describes the behaviours of atomic languages.  The atomic
+language consists of the null closure of any non-terminal symbol in
+the grammar, and their deriavtives by terminals and non-terminal.
+
+* Script for creating GIF animation
+
+[[https://gist.github.com/maelvls/5379127][a gist]]
+
+* Derivative Languages
+
+This is the main driving device of the algorithm.  Basically, the
+algorithm works by taking successive derivatives, according to the
+input symbol.  At each step, we calculate the derivative language.  In
+this process, we also compute some semiring value and store in a
+carrier.  The end result of the algorithm is the final semiring
+value.
+
+If one wants simply to determine if the input string belongs to the
+grammar, one chooses the semiring to be the field with two elements,
+the boolean field.  If one wants to find how many ways are there to
+derive a given input string, then one uses the semiring of natural
+numbers instead.  If one wants, moreover, to find all the possible
+ways to derive a particular input string, then one can use the
+free semiring on the set of terminals and non-terminals of the
+grammar.  Here the free semiring is the left-adjoint functor to the
+forgetful functor from the category of semirings to the category of
+sets.
+
+To be more specific, the free semiring on a set is given by sets of
+sequences of elements in the set.  The addition of the semiring is the
+set union operation, and the multiplication is taking the respective
+concatenations.
+
+** Semirings
+
+So we need a module to define the behaviours of semirings, and provide
+some common semirings implementations.  Then in the main driving force
+we can freely substitute different semirings, according to the
+particular needs.
+
+** Languages
+
+Then the main part is to define the behaviour of languages.  This
+should be easy enough, since we already have the mechanism of graphs,
+nondeterministic automata, and semirings.  All we need to do is to
+combine them together.
+
+* Testing ground
+
+I am in a strong need to test things out.  The most important one is
+to visualize each step of the derivation, in a human-friendly manner.
+I need this to examine whether my atomic languages are wrongly
+implemented, or my atomic languages are wrongly derived, or my
+understanding of the main algorithm is plain wrong.
+
+This is the main reason I started this rewrite of the package.
+