a prototype of an item derivation forest

It seems to be complete now, but still awaits more tests to see where
the errors are, which should be plenty, haha.

7 files changed, 1180 insertions, 49 deletions

diff --git a/chain/Cargo.toml b/chain/Cargo.toml
index c55586d..496f5dd 100644
--- a/chain/Cargo.toml
+++ b/chain/Cargo.toml
@@ -13,7 +13,6 @@ test-print-viz = []
 nfa = { path = "../nfa" }
 graph = { path = "../graph" }
 grammar = { path = "../grammar" }
-forest = { path = "../forest" }
 
 [dev-dependencies]
 grammar = { path = "../grammar", features = ["test-helper"] }
diff --git a/chain/src/default.rs b/chain/src/default.rs
index 22befff..77a64ca 100644
--- a/chain/src/default.rs
+++ b/chain/src/default.rs
@@ -7,8 +7,10 @@
 
 use super::*;
 use crate::atom::{Atom, DefaultAtom};
+use crate::item::{
+    default::DefaultForest, generate_fragment, Forest, ForestLabel, ForestLabelError,
+};
 use core::fmt::Display;
-use forest::{default::DefaultForest, Forest};
 use grammar::{Error as GrammarError, GrammarLabel, GrammarLabelType, TNT};
 use graph::{labelled::DLGBuilder, Builder, DLGraph, Graph, LabelExtGraph, LabelGraph};
 
@@ -29,6 +31,10 @@ pub enum Error {
     NodeNoLabel(usize),
     /// Reserving memory fails.
     CannotReserve(TryReserveError),
+    /// Cannot create label for cloning nodes.
+    CannotClone(ForestLabelError),
+    /// Cannot find a suitable node to plant the new forest fragment.
+    CannotPlant,
     /// An invalid situation happens.
     Invalid,
 }
@@ -44,6 +50,8 @@ impl Display for Error {
             ),
             Self::NodeNoLabel(n) => write!(f, "node {n} has no labels while it should have one"),
             Self::CannotReserve(e) => write!(f, "cannot reserve memory: {e}"),
+            Self::CannotClone(fe) => write!(f, "cannot clone due to {fe}"),
+            Self::CannotPlant => write!(f, "cannot plant a new forest fragment"),
             Self::Invalid => write!(f, "invalid"),
         }
     }
@@ -69,6 +77,7 @@ impl From<ForestError> for Error {
             ForestError::DuplicatedNode(n) => Error::DuplicateNode(n),
             ForestError::InvalidGraphError(ge) => ge.into(),
             ForestError::NodeNoLabel(n) => Error::NodeNoLabel(n),
+            ForestError::LabelConversion(ce) => Error::CannotClone(ce),
         }
     }
 }
@@ -93,7 +102,7 @@ impl Default for DerIterIndex {
 }
 
 /// A complex type used for storing values of edges with two layers.
-type SecondTypeValue = (Parent, bool, Vec<(Edge, usize)>);
+type SecondTypeValue = (PaSe, bool, Vec<(Edge, usize)>);
 
 /// An iterator of TwoLayers.
 #[derive(Debug, Default)]
@@ -120,7 +129,7 @@ impl DerIter {
     fn add_second_layer(
         &mut self,
         label: usize,
-        forest_source: Parent,
+        forest_source: PaSe,
         accepting: bool,
         edges: Vec<(Edge, usize)>,
     ) {
@@ -186,7 +195,7 @@ pub struct DefaultChain {
     atom: DefaultAtom,
     current: usize,
     history: Vec<usize>,
-    forest: DefaultForest<GrammarLabel>,
+    forest: DefaultForest<ForestLabel<GrammarLabel>>,
     accepting_vec: Vec<bool>,
 }
 
@@ -205,7 +214,7 @@ impl DefaultChain {
 
     /// Return a reference to the associated forest.
     #[inline]
-    pub fn forest(&self) -> &DefaultForest<GrammarLabel> {
+    pub fn forest(&self) -> &DefaultForest<ForestLabel<GrammarLabel>> {
         &self.forest
     }
 
@@ -316,6 +325,7 @@ impl LabelExtGraph<Edge> for DefaultChain {
         let new = self.graph.extend(edges)?;
         let accepting_len = self.accepting_vec.len();
 
+        // Update the acceptace of the new node, if any.
         if self.accepting_vec.get(new).is_none() {
             // assert it can only grow by one node at a time.
             #[cfg(debug_assertions)]
@@ -377,14 +387,17 @@ impl Chain for DefaultChain {
 
         let empty_state = atom.empty();
 
+        // The zero-th non-terminal is assumed to be the starting
+        // non-terminal, by convention.
         let initial_nullable = atom
             .is_nullable(0)
             .map_err(|_| Error::IndexOutOfBounds(0, atom.non_num()))?;
 
+        // NOTE: Is it really correct to use `Parent::new(0, 0)` here?
         builder.add_edge(
             first,
             root,
-            Edge::new(empty_state, Parent::new(0, 0), initial_nullable),
+            Edge::new(empty_state, Default::default(), initial_nullable),
         )?;
 
         let graph = builder.build();
@@ -431,7 +444,7 @@ impl Chain for DefaultChain {
 
     type DerIter = DerIter;
 
-    fn derive(&mut self, t: usize, _pos: usize) -> Result<Self::DerIter, Self::Error> {
+    fn derive(&mut self, t: usize, pos: usize) -> Result<Self::DerIter, Self::Error> {
         use TNT::*;
 
         /// Convert an error telling us that an index is out of bounds.
@@ -445,7 +458,7 @@ impl Chain for DefaultChain {
                 GrammarError::IndexOutOfBounds(index, bound) => {
                     Error::IndexOutOfBounds(index, bound)
                 }
-                _ => panic!("wrong error kind"),
+                _ => Error::Invalid,
             }
         }
 
@@ -459,11 +472,11 @@ impl Chain for DefaultChain {
         ///
         /// The generated edges will be pushed to `output` directly,
         /// to save some allocations.
-        // TODO: Handle forests as well.
         fn generate_edges(
             chain: &DefaultChain,
             child_iter: impl Iterator<Item = usize> + ExactSizeIterator + Clone,
             atom_child_iter: impl Iterator<Item = usize> + Clone,
+            pase: PaSe,
             mut output: impl AsMut<Vec<(Edge, usize)>>,
         ) -> Result<(), Error> {
             // First check the values from iterators are all valid.
@@ -515,13 +528,11 @@ impl Chain for DefaultChain {
 
             // now push into output
 
-            let parent = Parent::new(0, 0);
-
             for atom_child in atom_child_iter {
                 let atom_child_accepting = chain.atom.is_accepting(atom_child).unwrap();
                 let atom_child_empty_node = chain.atom.is_empty_node(atom_child).unwrap();
 
-                let edge = Edge::new(atom_child, parent, atom_child_accepting);
+                let edge = Edge::new(atom_child, pase, atom_child_accepting);
 
                 if !atom_child_empty_node {
                     output.extend(child_iter.clone().map(|child| (edge, child)));
@@ -549,16 +560,41 @@ impl Chain for DefaultChain {
                     continue;
                 }
 
+                let atom_moved = atom_label.get_moved();
+
                 match *atom_label.get_value() {
                     Some(Ter(ter)) if ter == t => {
+                        // prepare forest fragment
+
+                        let fragment =
+                            generate_fragment([atom_moved.into(), Ter(ter).into()], pos)?;
+
+                        let new_pase = self.forest.insert_item(
+                            *label,
+                            fragment,
+                            atom_child_iter.clone(),
+                            &self.atom,
+                        )?;
+
                         generate_edges(
                             self,
                             child_iter.clone(),
                             atom_child_iter.clone(),
+                            new_pase,
                             &mut der_iter.singles,
                         )?;
                     }
                     Some(Non(non)) => {
+                        let first_fragment =
+                            generate_fragment([atom_moved.into(), Non(non).into()], pos)?;
+
+                        let first_segment_pase = self.forest.insert_item(
+                            *label,
+                            first_fragment,
+                            atom_child_iter.clone(),
+                            &self.atom,
+                        )?;
+
                         let virtual_node = self
                             .atom
                             .atom(non, t)
@@ -570,12 +606,66 @@ impl Chain for DefaultChain {
                                 .is_accepting(virtual_node)
                                 .map_err(index_out_of_bounds_conversion)?;
 
+                            let virtual_fragment = {
+                                // `non` is valid, as checked above
+                                let non_start = self.atom.nth_accumulator(non).unwrap() * 2;
+
+                                let mut result = DefaultForest::default();
+
+                                for (label, child_iter) in self.atom.labels_of(non_start)? {
+                                    if matches!(*label.get_value(),
+                                                Some(Ter(ter)) if ter == t)
+                                    {
+                                        for child in child_iter {
+                                            let mut line: Vec<_> = self
+                                                .atom
+                                                .query_expansion(non_start, child)
+                                                .map_err(index_out_of_bounds_conversion)?
+                                                .iter()
+                                                .copied()
+                                                .flatten()
+                                                .map(|(nt, rule)| [(*rule).into(), Non(*nt).into()])
+                                                .collect();
+
+                                            line.push([label.get_moved().into(), Ter(t).into()]);
+
+                                            let line: Vec<GrammarLabelType> =
+                                                line.into_iter().flatten().collect();
+
+                                            if result.is_empty() {
+                                                result = generate_fragment(line, pos)?;
+                                            } else {
+                                                result.plant(
+                                                    0,
+                                                    generate_fragment(line, pos)?,
+                                                    false,
+                                                )?;
+                                            }
+                                        }
+                                    }
+                                }
+
+                                result
+                            };
+
+                            // NOTE: We only need the PaSe from the
+                            // first segment, so we pass an empty
+                            // iterator, in which case the passed
+                            // label is only used for the PaSe.
+                            let virtual_pase = self.forest.insert_item(
+                                Edge::new(0, first_segment_pase, true),
+                                virtual_fragment,
+                                std::iter::empty(),
+                                &self.atom,
+                            )?;
+
                             let mut new_edges = Vec::new();
 
                             generate_edges(
                                 self,
                                 child_iter.clone(),
                                 atom_child_iter.clone(),
+                                first_segment_pase,
                                 &mut new_edges,
                             )?;
 
@@ -583,12 +673,10 @@ impl Chain for DefaultChain {
                                 der_iter.singles.extend(new_edges.clone());
                             }
 
-                            let parent = Parent::new(0, 0);
-
                             if !self.atom.is_empty_node(virtual_node).unwrap() {
                                 der_iter.add_second_layer(
                                     virtual_node,
-                                    parent,
+                                    virtual_pase,
                                     accepting,
                                     new_edges,
                                 );
@@ -601,7 +689,10 @@ impl Chain for DefaultChain {
                                     // `generate_edges`
                                     if self.atom.is_empty_node(atom_child).unwrap() {
                                         der_iter.singles.extend(child_iter.clone().map(|child| {
-                                            (Edge::new(virtual_node, parent, accepting), child)
+                                            (
+                                                Edge::new(virtual_node, virtual_pase, accepting),
+                                                child,
+                                            )
                                         }));
                                     }
                                 }
@@ -610,7 +701,8 @@ impl Chain for DefaultChain {
                                     // this has been checked in
                                     // `generate_edges`
                                     if self.atom.is_empty_node(atom_child).unwrap() {
-                                        // flat flat map, hmm...
+                                        // REVIEW: flat flat map,
+                                        // hmm...
                                         der_iter.singles.extend(child_iter.clone().flat_map(
                                             |child| {
                                                 self.graph.labels_of(child).unwrap().flat_map(
@@ -646,7 +738,7 @@ mod test_der_iter {
     fn test() -> Result<(), Box<dyn std::error::Error>> {
         let mut der_iter = DerIter::default();
 
-        let parent = Parent::new(0, 0);
+        let parent = Default::default();
 
         der_iter.singles.push((Edge::new(0, parent, true), 0));
 
diff --git a/chain/src/item/default.rs b/chain/src/item/default.rs
new file mode 100644
index 0000000..b71f940
--- /dev/null
+++ b/chain/src/item/default.rs
@@ -0,0 +1,547 @@
+//! This module provides a default implementation of iten derivation
+//! forest.
+
+use super::*;
+use graph::{
+    builder::BuilderMut, labelled::binary::PLGBuilderMut, Graph, LabelGraph, PLGraph, RedirectGraph,
+};
+
+use core::fmt::Display;
+
+/// The type of errors for forest operations.
+#[derive(Debug, Copy, Clone, PartialEq, Eq, Ord, PartialOrd)]
+pub enum Error {
+    /// An index is out of bounds.
+    ///
+    /// The first component is the index that is out of bounds, and
+    /// the second component is the current length of nodes.
+    IndexOutOfBounds(usize, usize),
+    /// The forest does not permit duplicate nodes but encounters a
+    /// repeated node.
+    DuplicatedNode(usize),
+    /// A node has no labels while it is required to have one.
+    NodeNoLabel(usize),
+    /// Encounter an invalid error in converting from an error of
+    /// graphs.
+    InvalidGraphError(GError),
+    /// Encounter an error when converting forest labels.
+    LabelConversion(ForestLabelError),
+}
+
+impl Display for Error {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        match self {
+            Error::IndexOutOfBounds(index, bound) => {
+                write!(f, "index {index} is out of bound {bound}")
+            }
+            Error::DuplicatedNode(n) => write!(f, "node {n} is duplicated"),
+            Error::InvalidGraphError(ge) => write!(f, "invalid error: {ge}"),
+            Error::NodeNoLabel(n) => write!(f, "node {n} has no labels, but it should have one"),
+            Error::LabelConversion(le) => write!(f, "fail to convert labels: {le}"),
+        }
+    }
+}
+
+impl std::error::Error for Error {}
+
+impl From<GError> for Error {
+    fn from(ge: GError) -> Self {
+        match ge {
+            GError::IndexOutOfBounds(index, bound) => Self::IndexOutOfBounds(index, bound),
+            GError::DuplicatedNode(n) => Self::DuplicatedNode(n),
+            _ => Self::InvalidGraphError(ge),
+        }
+    }
+}
+
+impl From<ForestLabelError> for Error {
+    fn from(le: ForestLabelError) -> Self {
+        Self::LabelConversion(le)
+    }
+}
+
+/// A default implementation of forest.
+#[derive(Debug, Clone)]
+pub struct DefaultForest<T: GraphLabel> {
+    graph: PLGraph<T>,
+    root: Option<usize>,
+}
+
+impl<T: GraphLabel> Default for DefaultForest<T> {
+    fn default() -> Self {
+        let graph = Default::default();
+        let root = None;
+
+        Self { graph, root }
+    }
+}
+
+impl<T: GraphLabel> AsRef<DefaultForest<T>> for DefaultForest<T> {
+    fn as_ref(&self) -> &DefaultForest<T> {
+        self
+    }
+}
+
+impl<T: GraphLabel> Graph for DefaultForest<T> {
+    type Iter<'a> = <PLGraph<T> as Graph>::Iter<'a>
+    where
+        Self: 'a;
+
+    #[inline]
+    fn is_empty(&self) -> bool {
+        self.graph.is_empty()
+    }
+
+    #[inline]
+    fn nodes_len(&self) -> usize {
+        self.graph.nodes_len()
+    }
+
+    #[inline]
+    fn edges_len(&self) -> Option<usize> {
+        self.graph.edges_len()
+    }
+
+    #[inline]
+    fn children_of(&self, node_id: usize) -> Result<Self::Iter<'_>, GError> {
+        self.graph.children_of(node_id)
+    }
+
+    #[inline]
+    fn degree(&self, node_id: usize) -> Result<usize, GError> {
+        self.graph.degree(node_id)
+    }
+
+    #[inline]
+    fn is_empty_node(&self, node_id: usize) -> Result<bool, GError> {
+        self.graph.is_empty_node(node_id)
+    }
+
+    #[inline]
+    fn has_edge(&self, source: usize, target: usize) -> Result<bool, GError> {
+        self.graph.has_edge(source, target)
+    }
+
+    fn replace_by_builder(&mut self, _builder: impl graph::Builder<Result = Self>) {
+        unimplemented!()
+    }
+}
+
+impl<T: GraphLabel> ParentsGraph for DefaultForest<T> {
+    type Iter<'a>= <PLGraph<T> as ParentsGraph>::Iter<'a>
+    where
+        Self:'a;
+
+    #[inline]
+    fn parents_of(&self, node_id: usize) -> Result<<Self as ParentsGraph>::Iter<'_>, GError> {
+        self.graph.parents_of(node_id)
+    }
+
+    #[inline]
+    fn nth_child(&self, node_id: usize, n: usize) -> Result<usize, GError> {
+        self.graph.nth_child(node_id, n)
+    }
+
+    #[inline]
+    fn edge_to_parent(
+        &self,
+        source: usize,
+        target: usize,
+    ) -> Result<Option<graph::Parent>, GError> {
+        self.graph.edge_to_parent(source, target)
+    }
+}
+
+impl<T: GraphLabel> LabelGraph<T> for DefaultForest<T> {
+    type Iter<'a> = std::iter::Empty<usize>
+    where
+        Self: 'a;
+
+    type LabelIter<'a> = std::iter::Empty<(&'a T, <Self as LabelGraph<T>>::Iter<'a>)>
+    where
+        Self: 'a,
+        T: 'a;
+
+    type EdgeLabelIter<'a> = std::iter::Empty<T>
+    where
+        Self: 'a,
+        T: 'a;
+
+    #[inline]
+    fn query_label(&self, label: T) -> Option<usize> {
+        self.graph.query_label(label)
+    }
+
+    #[inline]
+    fn vertex_label(&self, node_id: usize) -> Result<Option<T>, GError> {
+        self.graph.vertex_label(node_id)
+    }
+
+    fn edge_label(
+        &self,
+        _source: usize,
+        _target: usize,
+    ) -> Result<Self::EdgeLabelIter<'_>, GError> {
+        unimplemented!("edges have no labels")
+    }
+
+    fn find_children_with_label(
+        &self,
+        _node_id: usize,
+        _label: &T,
+    ) -> Result<<Self as LabelGraph<T>>::Iter<'_>, GError> {
+        unimplemented!("edges have no labels")
+    }
+
+    fn labels_of(&self, _node_id: usize) -> Result<Self::LabelIter<'_>, GError> {
+        unimplemented!("edges have no labels")
+    }
+
+    fn has_edge_label(&self, _node_id: usize, _label: &T, _target: usize) -> Result<bool, GError> {
+        unimplemented!("edges have no labels")
+    }
+}
+
+impl<T: GraphLabel> Forest<T> for DefaultForest<ForestLabel<T>> {
+    type Error = Error;
+
+    fn root(&self) -> Option<usize> {
+        self.root
+    }
+
+    fn new_leaf(label: T) -> Self {
+        let mut graph = PLGraph::default();
+
+        let mut builder = PLGBuilderMut::from_graph_mut(&mut graph);
+
+        let root = Some(builder.add_vertex(ForestLabel::from(label)));
+
+        Self { graph, root }
+    }
+
+    fn is_prefix<F>(&self, node_id: usize, fragment: F) -> Result<bool, Self::Error>
+    where
+        F: Borrow<Self>,
+    {
+        if !self.has_node(node_id) {
+            return Err(Error::IndexOutOfBounds(node_id, self.nodes_len()));
+        }
+
+        // We do a depth-first traversal to determine if every node
+        // encountered has the same set of children (labels taken into
+        // the consideration).
+
+        let fragment = fragment.borrow();
+
+        let mut frag_stack = Vec::with_capacity(fragment.nodes_len());
+
+        let mut self_stack = Vec::with_capacity(fragment.nodes_len());
+
+        let frag_root = if let Some(root) = fragment.root() {
+            root
+        } else {
+            // an empty forest is a prefix of any forest.
+            return Ok(true);
+        };
+
+        frag_stack.push(frag_root);
+        self_stack.push(node_id);
+
+        // defer popping
+        while let (Some(frag_top), Some(self_top)) =
+            (frag_stack.last().copied(), self_stack.last().copied())
+        {
+            frag_stack.pop();
+            self_stack.pop();
+
+            if fragment.vertex_label(frag_top)? != self.vertex_label(self_top)? {
+                // not a prefix
+                return Ok(false);
+            }
+
+            let mut self_children = self.children_of(self_top)?;
+
+            for child in fragment.children_of(frag_top)? {
+                if let Some(self_child) = self_children.next() {
+                    frag_stack.push(child);
+                    self_stack.push(self_child);
+                } else {
+                    // too few children
+                    return Ok(false);
+                }
+            }
+        }
+
+        // Check both stacks are empty at the end.
+        Ok(frag_stack.is_empty() && self_stack.is_empty())
+    }
+
+    fn plant<F>(&mut self, node_id: usize, fragment: F, planted: bool) -> Result<(), Self::Error>
+    where
+        F: Borrow<Self>,
+    {
+        // Convert self to a builder_mut, and traverse fragment in a
+        // depth-first manner and adjoin corresponding nodes along the
+        // way.
+
+        if !self.has_node(node_id) {
+            return Err(Error::IndexOutOfBounds(node_id, self.nodes_len()));
+        }
+
+        let fragment = fragment.borrow();
+
+        let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+        let root = if let Some(root) = fragment.root() {
+            root
+        } else {
+            // Nothing to do to plant an empty forest.
+            return Ok(());
+        };
+
+        // Just a dummy label for use in adding edges.
+        //
+        // REVIEW: I probably should refactor the API for builder_mut.
+        let root_label = fragment
+            .vertex_label(root)?
+            .ok_or(Error::NodeNoLabel(root))?;
+
+        let nodes_len = fragment.nodes_len();
+
+        /// If the fragment root has a duplicate label, the forest
+        /// will not grow, so we use the label to find the adjoined
+        /// node index.
+        ///
+        /// The nodes hava already been added to the forest, so it is
+        /// safe to call unwrap.
+        macro_rules! conversion (
+            ($node:expr) => {
+                {
+                    builder
+                        .query_label(
+                            fragment
+                                .vertex_label($node)?
+                                .ok_or(Error::NodeNoLabel($node))?
+                        ).unwrap()
+                }
+            }
+        );
+
+        // If the fragment has been planted before, we just add an
+        // edge.
+
+        if planted {
+            builder.add_edge(node_id, conversion!(root), root_label)?;
+
+            return Ok(());
+        }
+
+        // First adjoin those nodes and join the edges later.
+
+        for node in 0..nodes_len {
+            let label = fragment
+                .vertex_label(node)?
+                .ok_or(Error::NodeNoLabel(node))?;
+
+            builder.add_vertex(label);
+        }
+
+        // Don't forget to join the new sub-forest to the original
+        // forest, at the specified position.
+
+        builder.add_edge(node_id, conversion!(root), root_label)?;
+
+        // We can try to calculate the depth of fragments, if we need
+        // to lower the memory usage.  But in our use cases, we
+        // usually deal with fragments where each node has at most one
+        // child, so the depth is supposed to be equal to the length
+        // in this case.
+        let mut stack = Vec::with_capacity(fragment.nodes_len());
+
+        stack.push(root);
+
+        while let Some(top) = stack.pop() {
+            for child in fragment.children_of(top)? {
+                builder.add_edge(conversion!(top), conversion!(child), root_label)?;
+            }
+        }
+
+        Ok(())
+    }
+
+    fn clone_node(&mut self, node_id: usize) -> Result<usize, Self::Error> {
+        let builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+        let old_label = builder
+            .vertex_label(node_id)?
+            .ok_or(Error::NodeNoLabel(node_id))?;
+
+        if old_label.is_packed() {
+            return Err(ForestLabelError::ClonePack.into());
+        }
+
+        // We are sure old_label is not packed here, so it is safe to
+        // unwrap.
+        let new_label = old_label.clone(self).unwrap();
+
+        if old_label.clone_index().is_some() {
+            let mut parents = self.parents_of(node_id)?;
+
+            // Make sure it only has one parent, which is the
+            // representative node.
+            assert_eq!(parents.len(), 1);
+
+            let rep_node = parents.next().unwrap().node();
+
+            let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+            let new_clone = builder.add_vertex(new_label);
+
+            builder.add_edge(rep_node, new_clone, new_label)?;
+
+            // We checked its length is 1, so it is safe to unwrap
+            // here.
+            return Ok(new_clone);
+        }
+
+        let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+        // We are sure old_label is not a clone here, so it is safe to
+        // unwrap.
+        let rep_label = old_label.pack().unwrap();
+
+        // Make a new node
+        let new_index = builder.add_vertex(rep_label);
+
+        // Re-direct parents to the new node.
+        //
+        // This must be done before pointing the new node to the old
+        // node, otherwise that edge will be redirected as well.
+
+        // Unfortunately I cannot loop through parents and mutate them
+        // at the same time, so I first collect them into a vector.
+        let parents: Vec<_> = builder.parents_of(node_id)?.collect();
+
+        for parent in parents.into_iter() {
+            builder.redirect(parent.node(), parent.edge(), new_index)?;
+        }
+
+        // Point the new node to the old node.  OLD_LABEL is just a
+        // place holder.
+
+        builder.add_edge(new_index, node_id, old_label)?;
+
+        // Modify the label of the old node.
+
+        builder.set_label(node_id, new_label)?;
+
+        // Make another clone
+
+        // new_label is cloned, so is guaranteed not to be packed, and
+        // we are safe to unwrap.
+        let new_label = new_label.clone(self).unwrap();
+
+        let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+        let new_clone = builder.add_vertex(new_label);
+
+        builder.add_edge(new_index, new_clone, new_label)?;
+
+        Ok(new_clone)
+    }
+}
+
+#[cfg(test)]
+mod item_test {
+    use super::*;
+
+    macro_rules! leaf (
+        ($label:expr, $type:tt) =>{
+            DefaultForest::<ForestLabel<$type>>::new_leaf($label)
+        };
+        ($label:expr) => {
+            DefaultForest::<ForestLabel<usize>>::new_leaf($label)
+        }
+    );
+
+    #[test]
+    fn test_forest_api() -> Result<(), Box<dyn std::error::Error>> {
+        let forest: DefaultForest<usize> = Default::default();
+
+        // empty forest
+
+        assert!(forest.is_empty());
+
+        // leaf forest
+
+        let mut forest = leaf!(0, usize);
+
+        assert_eq!(forest.nodes_len(), 1);
+        assert_eq!(forest.root(), Some(0));
+
+        // add some child
+
+        forest.plant(0, leaf!(1), false)?;
+
+        assert_eq!(forest.nodes_len(), 2);
+        let mut children = forest.children_of(0)?;
+        assert_eq!(children.next(), Some(1));
+        assert_eq!(children.next(), None);
+
+        // add more children
+
+        forest.plant(0, leaf!(2), false)?;
+        forest.plant(0, leaf!(3), false)?;
+        forest.plant(0, leaf!(4), false)?;
+        forest.plant(2, leaf!(5), false)?;
+
+        assert_eq!(forest.nodes_len(), 6);
+        let mut children = forest.children_of(0)?;
+        assert_eq!(children.next(), Some(1));
+        assert_eq!(children.next(), Some(2));
+        assert_eq!(children.next(), Some(3));
+        assert_eq!(children.next(), Some(4));
+        let mut children = forest.children_of(2)?;
+        assert_eq!(children.next(), Some(5));
+        assert_eq!(children.next(), None);
+
+        let mut test_forest = leaf!(0);
+        test_forest.plant(0, leaf!(1), false)?;
+        test_forest.plant(0, leaf!(2), false)?;
+        test_forest.plant(0, leaf!(3), false)?;
+        test_forest.plant(2, leaf!(5), false)?;
+
+        assert!(forest.is_prefix(0, &test_forest)?);
+
+        let mut test_forest = leaf!(0);
+        test_forest.plant(0, leaf!(1), false)?;
+        test_forest.plant(0, leaf!(2), false)?;
+        // this child of the root should have label 3 in order to be a
+        // prefix.
+        test_forest.plant(0, leaf!(4), false)?;
+        test_forest.plant(2, leaf!(5), false)?;
+
+        assert!(!forest.is_prefix(0, &test_forest)?);
+
+        let mut test_forest = leaf!(2);
+        test_forest.plant(0, leaf!(5), false)?;
+
+        assert!(forest.is_prefix(2, &test_forest)?);
+
+        // now test cloning
+
+        // add a duplicate label
+        forest.plant(3, leaf!(5), false)?;
+
+        let _len = forest.nodes_len();
+
+        forest.clone_node(5)?;
+
+        assert_eq!(forest.nodes_len(), 7);
+
+        #[cfg(feature = "test-print-viz")]
+        forest.print_viz("forest.gv")?;
+
+        Ok(())
+    }
+}
diff --git a/chain/src/item/genins.rs b/chain/src/item/genins.rs
new file mode 100644
index 0000000..99d9202
--- /dev/null
+++ b/chain/src/item/genins.rs
@@ -0,0 +1,231 @@
+//! This module implements the generation and insertion of item
+//! derivation forests.
+//!
+//! This is used for the chain-rule machine to conveniently produce
+//! item derivations into a forest.  This forest can serve as a rough
+//! approximation of the parse forests, and can be executed in other
+//! semirings later on.
+
+use super::*;
+use crate::{atom::DefaultAtom, default::Error, item::default::DefaultForest, Edge};
+use grammar::{Error as GrammarError, GrammarLabel, GrammarLabelType, TNT};
+use graph::Graph;
+
+use core::borrow::Borrow;
+use std::collections::HashSet as Set;
+
+/// A helper function to generate a fragment of forest.
+///
+/// It simply constructs a root node and then appends
+/// successive nodes as successive children of the previous
+/// node.  Also the starting positions will all be set to the
+/// same position.
+///
+/// If the input is empty, this returns an empty forest;
+/// otherwise the result is not empty.
+pub fn generate_fragment(
+    labels: impl AsRef<[GrammarLabelType]>,
+    pos: usize,
+) -> Result<DefaultForest<ForestLabel<GrammarLabel>>, crate::default::Error> {
+    let mut labels_iter = labels.as_ref().iter();
+    let mut result: DefaultForest<ForestLabel<GrammarLabel>>;
+
+    if let Some(first_label) = labels_iter.next() {
+        result = DefaultForest::new_leaf(GrammarLabel::new(*first_label, pos));
+
+        let mut index = 0;
+
+        for label in labels_iter {
+            result.plant(
+                index,
+                DefaultForest::new_leaf(GrammarLabel::new(*label, pos)),
+                false,
+            )?;
+
+            // To prevent duplication of labels causing
+            // panics, we query the index of the new node.
+            index = result
+                .query_label(GrammarLabel::new(*label, pos).into())
+                // REVIEW: Perhaps a LabelNoNode error?
+                .ok_or(Error::Invalid)?;
+        }
+    } else {
+        result = Default::default();
+    }
+
+    Ok(result)
+}
+
+impl DefaultForest<ForestLabel<GrammarLabel>> {
+    /// Insert an item derivation forest into a recording forest.
+    ///
+    /// We need the help of other things just for finding the correct
+    /// places to insert these item fragments.
+    pub fn insert_item(
+        &mut self,
+        label: Edge,
+        fragment: impl Borrow<DefaultForest<ForestLabel<GrammarLabel>>>,
+        atom_child_iter: impl Iterator<Item = usize>,
+        atom: &DefaultAtom,
+    ) -> Result<PaSe, Error> {
+        /// Convert an error telling us that an index is out of bounds.
+        ///
+        /// # Panics
+        ///
+        /// The function panics if the error is not of the expected
+        /// kind.
+        fn index_out_of_bounds_conversion(ge: GrammarError) -> Error {
+            match ge {
+                GrammarError::IndexOutOfBounds(index, bound) => {
+                    Error::IndexOutOfBounds(index, bound)
+                }
+                _ => Error::Invalid,
+            }
+        }
+
+        let fragment = fragment.borrow();
+
+        let pase = label.forest_source();
+
+        let parents: Vec<Parent> = {
+            let mut result = Vec::new();
+
+            if let Some(parent) = pase.parent() {
+                result.push(parent);
+            } else {
+                let (root, leaf) = pase.segment().unwrap();
+                let mut seen_nodes = Set::new();
+
+                let mut stack = vec![root];
+
+                while let Some(top) = stack.pop() {
+                    if seen_nodes.contains(&top) {
+                        continue;
+                    }
+
+                    seen_nodes.insert(top);
+
+                    for (index, child) in self.children_of(top)?.enumerate() {
+                        if child == leaf {
+                            result.push(Parent::new(top, index));
+                        } else {
+                            stack.push(child);
+                        }
+                    }
+                }
+            }
+
+            result
+        };
+
+        for atom_child in atom_child_iter {
+            // Find reduction information.
+            let reduction_info = atom
+                .query_reduction(label.label(), atom_child)
+                .map_err(index_out_of_bounds_conversion)?;
+
+            let mut stack = parents.clone();
+            let mut second_stack = Vec::new();
+
+            // locate the nodes
+            for reduction_nt in reduction_info.iter().copied().flatten().rev() {
+                while let Some(node) = stack.pop() {
+                    // REVIEW: A lot of labels appear here.
+                    // Perhaps I need to refactor something?
+                    if matches!(
+                        self
+                            .vertex_label(node.node())?
+                            .ok_or_else(|| Error::NodeNoLabel(node.node()))?
+                            .label()
+                            .label(),
+                        GrammarLabelType::TNT(TNT::Non(nt)) if nt == *reduction_nt
+                    ) {
+                        for parent in self.parents_of(node.node())? {
+                            debug_assert!(matches!(
+                                    self.vertex_label(parent.node()),
+                                    Ok(Some(label)) if
+                                        label
+                                        .label()
+                                        .label()
+                                        .rule()
+                                        .is_some()));
+
+                            second_stack.extend(self.parents_of(parent.node())?.filter(|n| {
+                                matches!(self.vertex_label(n.node()),
+                                                 Ok(Some(label))
+                                                 if matches!(
+                                                     label.label().label().tnt(),
+                                                     Some(TNT::Non(_))))
+                            }));
+                        }
+                    }
+                }
+
+                std::mem::swap(&mut stack, &mut second_stack);
+
+                if stack.is_empty() {
+                    break;
+                }
+            }
+
+            if stack.is_empty() {
+                return Err(Error::CannotPlant);
+            }
+
+            for parent in stack.into_iter() {
+                if parent.edge() + 1 >= self.degree(parent.node())? {
+                    self.plant(parent.node(), fragment, false)?;
+                } else {
+                    let nth_child = self.nth_child(parent.node(), parent.edge() + 1)?;
+
+                    if self.is_prefix(nth_child, fragment)? {
+                        // do nothing
+                        continue;
+                    }
+
+                    let cloned_node = self.clone_node(nth_child)?;
+
+                    self.plant(cloned_node, fragment, false)?;
+                }
+            }
+        }
+
+        let result = if fragment.nodes_len() == 2 {
+            let root_label = fragment.vertex_label(0)?.unwrap();
+            let leaf_label = fragment.vertex_label(1)?.unwrap();
+
+            // it has been planted, so should be safe.
+            let node = self.query_label(root_label).unwrap();
+
+            let edge = {
+                let mut result = None;
+
+                for (index, child) in self.children_of(node)?.enumerate() {
+                    if matches!(self.vertex_label(child)?, Some(child_label) if child_label == leaf_label)
+                    {
+                        result = Some(index);
+                        break;
+                    }
+                }
+
+                if let Some(result) = result {
+                    result
+                } else {
+                    unreachable!("the forest is wrongly planted");
+                }
+            };
+
+            PaSe::Parent(node, edge)
+        } else {
+            let root_label = fragment.vertex_label(0)?.unwrap();
+            let leaf_label = fragment.vertex_label(fragment.nodes_len() - 1)?.unwrap();
+
+            PaSe::Segment(
+                self.query_label(root_label).unwrap(),
+                self.query_label(leaf_label).unwrap(),
+            )
+        };
+
+        Ok(result)
+    }
+}
diff --git a/chain/src/item/mod.rs b/chain/src/item/mod.rs
new file mode 100644
index 0000000..7fafcc8
--- /dev/null
+++ b/chain/src/item/mod.rs
@@ -0,0 +1,252 @@
+#![warn(missing_docs)]
+//! This module implements the type of items for the chain-rule
+//! machine.
+//!
+//! More specifically, it implements the iten derivation forests for
+//! the machine.
+
+// TODO: Implement an enumeration for Parent or Segment, perhaps
+// called PaSe.
+
+// TODO: Move functions for handling forests, currently contained
+// within the method derive to this module, and make them aware of the
+// enumeration PaSe.
+
+// TODO: Implement the Item trait from semirings for the item
+// derivation forest, and then we can easily execute items later on.
+
+use graph::{error::Error as GError, GraphLabel, LabelGraph, Parent, ParentsGraph};
+
+use core::borrow::Borrow;
+
+/// A parent or a segment.
+///
+/// A parent is a node with an edge index, which represents a certain
+/// edge.
+///
+/// A segment represents every edge from the root node to the single
+/// terminating node.
+#[allow(unused)]
+#[derive(Debug, Copy, Clone, Eq, PartialEq, Ord, PartialOrd, Hash)]
+pub enum PaSe {
+    /// An edge from a node, as the n-th child.
+    Parent(usize, usize),
+    /// A segment from a root to the single terminating node.
+    Segment(usize, usize),
+}
+
+impl From<Parent> for PaSe {
+    fn from(value: Parent) -> Self {
+        Self::Parent(value.node(), value.edge())
+    }
+}
+
+impl Default for PaSe {
+    fn default() -> Self {
+        Self::Segment(0, 0)
+    }
+}
+
+impl core::fmt::Display for PaSe {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        match self {
+            Self::Parent(node, edge) => write!(f, "the {edge}-th edge from {node}"),
+            Self::Segment(root, leaf) => write!(f, "a segment from {root} to {leaf}"),
+        }
+    }
+}
+
+impl PaSe {
+    fn parent(self) -> Option<Parent> {
+        if let Self::Parent(node, edge) = self {
+            Some(Parent::new(node, edge))
+        } else {
+            None
+        }
+    }
+
+    fn segment(self) -> Option<(usize, usize)> {
+        if let Self::Segment(root, leaf) = self {
+            Some((root, leaf))
+        } else {
+            None
+        }
+    }
+}
+
+/// An internal type that indicates the status of a node as either a
+/// packed, cloned, or plain node.
+#[derive(Debug, Copy, Clone, Eq, PartialEq, Ord, PartialOrd, Hash, Default)]
+enum ForestLabelType {
+    Packed,
+    #[default]
+    Plain,
+    Cloned(usize),
+}
+
+impl core::fmt::Display for ForestLabelType {
+    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
+        match self {
+            Self::Packed => write!(f, "packed"),
+            Self::Plain => write!(f, "plain"),
+            Self::Cloned(index) => write!(f, "the {index}-th clone"),
+        }
+    }
+}
+
+/// A type that encodes the properties demanded by a forest.
+#[derive(Debug, Copy, Clone, Eq, PartialEq, Ord, PartialOrd, Hash)]
+pub struct ForestLabel<T: GraphLabel> {
+    label: T,
+    status: ForestLabelType,
+}
+
+impl<T: GraphLabel> core::fmt::Display for ForestLabel<T> {
+    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
+        writeln!(f, "label: {}, status: {}", self.label, self.status)
+    }
+}
+
+/// The type of erros for converting forest labels.
+#[non_exhaustive]
+#[derive(Debug, Copy, Clone, Ord, PartialOrd, Eq, PartialEq)]
+pub enum ForestLabelError {
+    /// Try to pack a cloned node.
+    PackClone,
+    /// Try to clone a packed node.
+    ClonePack,
+}
+
+impl core::fmt::Display for ForestLabelError {
+    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
+        match self {
+            Self::PackClone => write!(f, "cannot pack a cloned node"),
+            Self::ClonePack => write!(f, "cannot clone a packed node"),
+        }
+    }
+}
+
+impl std::error::Error for ForestLabelError {}
+
+impl<T: GraphLabel> ForestLabel<T> {
+    /// Retrieve the label.
+    pub fn label(&self) -> T {
+        self.label
+    }
+
+    /// Return true if and only if this node is packed.
+    pub fn is_packed(&self) -> bool {
+        matches!(self.status, ForestLabelType::Packed)
+    }
+
+    /// Retrieve the optional clone index.
+    pub fn clone_index(&self) -> Option<usize> {
+        match self.status {
+            ForestLabelType::Cloned(index) => Some(index),
+            _ => None,
+        }
+    }
+
+    /// Try to clone a node.
+    pub fn clone<F>(self, forest: &F) -> Result<Self, ForestLabelError>
+    where
+        F: Forest<T>,
+    {
+        if self.is_packed() {
+            Err(ForestLabelError::ClonePack)
+        } else {
+            let clone_index = if let Some(old_index) = self.clone_index() {
+                let mut new_index: usize = old_index + 1;
+
+                let mut new_label = Self {
+                    status: ForestLabelType::Cloned(new_index),
+                    ..self
+                };
+
+                while forest.query_label(new_label).is_some() {
+                    new_index += 1;
+                    new_label = Self {
+                        status: ForestLabelType::Cloned(new_index),
+                        ..self
+                    };
+                }
+
+                new_index
+            } else {
+                0
+            };
+
+            Ok(Self {
+                status: ForestLabelType::Cloned(clone_index),
+                ..self
+            })
+        }
+    }
+
+    /// Try to pack a node.
+    pub fn pack(self) -> Result<Self, ForestLabelError> {
+        if self.clone_index().is_some() {
+            Err(ForestLabelError::PackClone)
+        } else {
+            let new_label = Self {
+                status: ForestLabelType::Packed,
+                ..self
+            };
+
+            Ok(new_label)
+        }
+    }
+}
+
+impl<T: GraphLabel> From<T> for ForestLabel<T> {
+    fn from(label: T) -> Self {
+        let status = ForestLabelType::default();
+        Self { label, status }
+    }
+}
+
+/// The expected behaviours of an item derivation forest.
+///
+/// Note that it requires only a subset of the functionalities of
+/// labelled graphs.
+pub trait Forest<T: GraphLabel>: ParentsGraph + LabelGraph<ForestLabel<T>> {
+    /// The type of errors for operations on the forest.
+    type Error: std::error::Error + From<GError> + From<ForestLabelError>;
+
+    /// Return the root of the forest.
+    ///
+    /// A forest without a root is regarded as empty.
+    fn root(&self) -> Option<usize>;
+
+    /// Construct a forest consisting of one leaf node with the given
+    /// label.
+    fn new_leaf(label: T) -> Self;
+
+    /// Detect if the fragment is a prefix of the sub-forest rooted at
+    /// `node_id`.
+    fn is_prefix<F>(&self, node_id: usize, fragment: F) -> Result<bool, Self::Error>
+    where
+        F: Borrow<Self>;
+
+    /// Extend the forest by adjoining another forest at a given node.
+    fn plant<F>(&mut self, node_id: usize, fragment: F, planted: bool) -> Result<(), Self::Error>
+    where
+        F: Borrow<Self>;
+
+    /// Clone a node by making a new node and making all the nodes
+    /// that previously pointed to the old node now point to the new
+    /// node, and the new node points to the old node.  Return the
+    /// index of the new node.  In addition, if the old node had
+    /// already been cloned, just return the index of its
+    /// representative.
+    ///
+    /// The labels of the representing node and of the cloned node
+    /// will be handled automatically.
+    fn clone_node(&mut self, node_id: usize) -> Result<usize, Self::Error>;
+}
+
+pub mod default;
+
+pub mod genins;
+
+pub use genins::generate_fragment;
diff --git a/chain/src/lib.rs b/chain/src/lib.rs
index a3d420b..916678f 100644
--- a/chain/src/lib.rs
+++ b/chain/src/lib.rs
@@ -10,9 +10,16 @@
 
 pub mod atom;
 
-use graph::{error::Error as GError, LabelExtGraph, Parent};
+pub mod item;
 
-use forest::Error as ForestError;
+// TODO: We need a module for items, that serve the role of the
+// forests now.
+
+use graph::{error::Error as GError, LabelExtGraph};
+
+use item::default::Error as ForestError;
+
+use item::PaSe;
 
 /// An edge in the Chain-Rule machine.
 #[derive(Debug, Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash)]
@@ -20,14 +27,14 @@ pub struct Edge {
     /// The position in the atomic languages.
     label: usize,
     /// The source of the associated forest edge.
-    forest_source: Parent,
+    forest_source: PaSe,
     /// Whether or not this edge is "accepting".
     accepting: bool,
 }
 
 impl Edge {
     /// Construct a new edge.
-    pub fn new(label: usize, forest_source: Parent, accepting: bool) -> Self {
+    pub fn new(label: usize, forest_source: PaSe, accepting: bool) -> Self {
         Self {
             label,
             forest_source,
@@ -46,7 +53,7 @@ impl Edge {
     }
 
     /// Return the associated forest edge of the edge.
-    pub fn forest_source(&self) -> Parent {
+    pub fn forest_source(&self) -> PaSe {
         self.forest_source
     }
 }
@@ -56,12 +63,7 @@ impl core::fmt::Display for Edge {
         let label = self.label();
         let forest_source = self.forest_source();
 
-        write!(
-            f,
-            "edge label {label} with forest source {} and edge index {}",
-            forest_source.node(),
-            forest_source.edge()
-        )
+        write!(f, "edge label {label} with item {}", forest_source)
     }
 }
 
@@ -84,7 +86,7 @@ pub enum TwoLayers {
     /// the source of the associated forest edge of the second layer,
     /// and the third is a list of edges, which are the common first
     /// layers.
-    Two(usize, Parent, bool, Vec<(Edge, usize)>),
+    Two(usize, PaSe, bool, Vec<(Edge, usize)>),
 }
 
 /// The type of a collapsed iterator.
diff --git a/chain/src/plan.org b/chain/src/plan.org
index c2a9037..1da33cc 100644
--- a/chain/src/plan.org
+++ b/chain/src/plan.org
@@ -2,7 +2,7 @@
 #+AUTHOR: Durand
 #+DATE: <2022-11-18 Ven 19:57>
 
-* Things to do [7/10]
+* Things to do [6/9]
 
 - [X] Implement builders for graphs
 - [X] Find sets of the first terminals for each non-terminal, in the
@@ -79,15 +79,13 @@
     lack of this step might be the root cause of the failure of the
     previous version of this project.
   + [X] Test atoms
-- [X] Implement forests [4/4]
-  + [X] Design a format of forests.  This should be the most crucial
-    thing to do, in order to have a better understanding of the whole
-    picture.  I need to have a clear understanding of the details of
-    the binarised shared packed parsed forests, that reflects the
-    regular expressions in the grammar equations.
-  + [X] Define a trait with the expected behaviours.
-  + [X] Implement them as parents-knowing graphs.
-  + [X] Test
+- [ ] Implement semiring. [0/5]
+  + [ ] Define the trait.
+  + [ ] Define items and rules for the chain-rule parser, as an
+    item-based description.
+  + [ ] Implement the boolean semiring.
+  + [ ] Implement the natural number semiring.
+  + [ ] Implement the free semiring.
 - [-] Implement languages. [4/6]
   + [X] First define a trait with the expected behaviours.
   + [X] Then implement them as doubly labelled graphs.
@@ -95,19 +93,29 @@
     with a position in the forest.  This perfectly solves the problem
     of those "plugs"!
   + [X] Then implement finding derivatives by use of the chain rule.
-  + [ ] Handle Forests
+  + [ ] Handle Semirings
   + [-] Tests
-- [ ] 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.
 - [X] Miscellaneous [1/1]
   + [X] Implement a function for NFA that checks if a given predicate
     is satisfied for each edge label.
 
+** Forests
+
+This was a failed attempt to handle forests.  I decided to handle
+forests as a special case of semi-rings.  This is not only more
+systematic, but has also the potential of handling more general
+semi-rings later, like the Viterbi semirings, for example.
+
+- [X] Implement forests [4/4]
+  + [X] Design a format of forests.  This should be the most crucial
+    thing to do, in order to have a better understanding of the whole
+    picture.  I need to have a clear understanding of the details of
+    the binarised shared packed parsed forests, that reflects the
+    regular expressions in the grammar equations.
+  + [X] Define a trait with the expected behaviours.
+  + [X] Implement them as parents-knowing graphs.
+  + [X] Test
+
 * Atomic Languages
 
 This describes the behaviours of atomic languages.  The atomic