steiner tree (?)

- introduce `MorphismInstantiation` which instantiates a
  morphism-template using a type-substitution and a halo type.
- find_morphism_path returns list of `MorphismInstatiation`.

src/dict.rs

@@ -24,6 +24,14 @@ pub trait TypeDict : Send + Sync {
             self.insert(new, tyid);
         }
     }
+
+    fn get_typeid_creat(&mut self, tn: &String) -> TypeID {
+        if let Some(id) = self.get_typeid(tn) {
+            id
+        } else {
+            self.add_typename(tn.clone())
+        }
+    }
 }
 
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\

src/lib.rs

@@ -11,6 +11,8 @@ pub mod lnf;
 pub mod pnf;
 pub mod subtype;
 pub mod unification;
+pub mod morphism;
+pub mod steiner_tree;
 
 #[cfg(test)]
 mod test;
@@ -23,4 +25,5 @@ pub use {
     term::*,
     sugar::*,
     unification::*,
+    morphism::*
 };

src/morphism.rs

@@ -0,0 +1,346 @@
+use {
+    crate::{
+        subtype_unify, sugar::SugaredTypeTerm, unification::UnificationProblem, unparser::*, TypeDict, TypeID, TypeTerm
+    },
+    std::{collections::HashMap, u64}
+};
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone, PartialEq, Eq, Debug)]
+pub struct MorphismType {
+    pub src_type: TypeTerm,
+    pub dst_type: TypeTerm,
+}
+
+impl MorphismType {
+    pub fn normalize(self) -> Self {
+        MorphismType {
+            src_type: self.src_type.normalize().param_normalize(),
+            dst_type: self.dst_type.normalize().param_normalize()
+        }
+    }
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+pub trait Morphism : Sized {
+    fn get_type(&self) -> MorphismType;
+    fn map_morphism(&self, seq_type: TypeTerm) -> Option< Self >;
+
+    fn weight(&self) -> u64 {
+        1
+    }
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone, Debug, PartialEq)]
+pub struct MorphismInstance<M: Morphism + Clone> {
+    pub halo: TypeTerm,
+    pub m: M,
+    pub σ: HashMap<TypeID, TypeTerm>
+}
+
+impl<M: Morphism + Clone> MorphismInstance<M> {
+    pub fn get_type(&self) -> MorphismType {
+        MorphismType {
+            src_type: TypeTerm::Ladder(vec![
+                self.halo.clone(),
+                self.m.get_type().src_type.clone()
+            ]).apply_substitution(&|k| self.σ.get(k).cloned())
+            .clone(),
+
+            dst_type: TypeTerm::Ladder(vec![
+                self.halo.clone(),
+                self.m.get_type().dst_type.clone()
+            ]).apply_substitution(&|k| self.σ.get(k).cloned())
+            .clone()
+        }.normalize()
+    }
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone)]
+pub struct MorphismPath<M: Morphism + Clone> {
+    pub weight: u64,
+    pub cur_type: TypeTerm,
+    pub morphisms: Vec< MorphismInstance<M> >
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone)]
+pub struct MorphismBase<M: Morphism + Clone> {
+    morphisms: Vec< M >,
+    seq_types: Vec< TypeTerm >
+}
+
+impl<M: Morphism + Clone> MorphismBase<M> {
+    pub fn new(seq_types: Vec<TypeTerm>) -> Self {
+        MorphismBase {
+            morphisms: Vec::new(),
+            seq_types
+        }
+    }
+
+    pub fn add_morphism(&mut self, m: M) {
+        self.morphisms.push( m );
+    }
+
+    pub fn enum_direct_morphisms(&self, src_type: &TypeTerm)
+    -> Vec< MorphismInstance<M> >
+    {
+        let mut dst_types = Vec::new();
+        for m in self.morphisms.iter() {
+            if let Ok((halo, σ)) = crate::unification::subtype_unify(
+                &src_type.clone().param_normalize(),
+                &m.get_type().src_type.param_normalize(),
+            ) {
+                let dst_type =
+                    m.get_type().dst_type.clone()
+                    .apply_substitution( &|x| σ.get(x).cloned() )
+                    .clone();
+
+                dst_types.push( (σ, dst_type) );
+            }
+        }
+        dst_types
+    }
+
+    pub fn enum_map_morphisms(&self, src_type: &TypeTerm)
+    -> Vec< MorphismInstance<M> > {
+        let src_type = src_type.clone().param_normalize();
+        let mut dst_types = Vec::new();
+
+        // Check if we have a List type, and if so, see what the Item type is
+        // TODO: function for generating fresh variables
+        let item_variable = TypeID::Var(800);
+
+        for seq_type in self.seq_types.iter() {
+            if let Ok((halo, σ)) = crate::unification::subtype_unify(
+                &src_type,
+                &TypeTerm::App(vec![
+                    seq_type.clone(),
+                    TypeTerm::TypeID(item_variable)
+                ])
+            ) {
+                let src_item_type = σ.get(&item_variable).expect("var not in unificator").clone();
+                for item_morph_inst in self.enum_morphisms( &src_item_type ) {
+
+                    let mut dst_halo_ladder = vec![ halo.clone() ];
+                    if item_morph_inst.halo != TypeTerm::unit() {
+                        dst_halo_ladder.push(
+                            TypeTerm::App(vec![
+                                seq_type.clone().get_lnf_vec().first().unwrap().clone(),
+                                item_morph_inst.halo.clone()
+                            ]));
+                    }
+
+                    if let Some( map_morph ) = item_morph_inst.m.map_morphism( seq_type.clone() ) {
+                        dst_types.push(
+                            MorphismInstance {
+                                halo: TypeTerm::Ladder(dst_halo_ladder).normalize(),
+                                m: map_morph,
+                                σ: item_morph_inst.σ
+                            }
+                        );
+                    } else {
+                        eprintln!("could not get map morphism");
+                    }
+                }
+            }
+        }
+
+        dst_types
+    }
+
+    pub fn enum_morphisms(&self, src_type: &TypeTerm) -> Vec< MorphismInstance<M> > {
+        let mut dst_types = Vec::new();
+        dst_types.append(&mut self.enum_direct_morphisms(src_type));
+        dst_types.append(&mut self.enum_map_morphisms(src_type));
+        dst_types
+    }
+
+    /* try to find shortest morphism-path for a given type
+     */
+    pub fn find_morphism_path(&self, ty: MorphismType
+        /*, type_dict: &mut impl TypeDict*/
+    )
+    -> Option< Vec<MorphismInstance<M>> >
+    {
+        let ty = ty.normalize();
+        let mut queue = vec![
+            MorphismPath { weight: 0, cur_type: ty.src_type.clone(), morphisms: vec![] }
+        ];
+
+        while ! queue.is_empty() {
+            queue.sort_by( |p1,p2| p2.weight.cmp(&p1.weight));
+
+            if let Some(mut cur_path) = queue.pop() {
+                if let Ok((halo, σ)) = crate::unification::subtype_unify( &cur_path.cur_type, &ty.dst_type ) {
+                    // found path
+                    for n in cur_path.morphisms.iter_mut() {
+                        let mut new_σ = HashMap::new();
+                        for (k,v) in σ.iter() {
+                            new_σ.insert(
+                                k.clone(),
+                                v.clone().apply_substitution(&|k| σ.get(k).cloned()).clone()
+                            );
+                        }
+                        for (k,v) in n.σ.iter() {
+                            new_σ.insert(
+                                k.clone(),
+                                v.clone().apply_substitution(&|k| σ.get(k).cloned()).clone()
+                            );
+                        }
+
+                        n.σ = new_σ;
+                    }
+
+                    return Some(cur_path.morphisms);
+                }
+
+                //eprintln!("cur path (w ={}) : @ {}", cur_path.weight, cur_path.cur_type.clone().sugar(type_dict).pretty(type_dict, 0) );
+                for next_morph_inst in self.enum_morphisms(&cur_path.cur_type) {
+                    let dst_type = next_morph_inst.get_type().dst_type;
+
+                    let mut creates_loop = false;
+
+                    let mut new_path = cur_path.clone();
+                    for n in new_path.morphisms.iter_mut() {
+                        let mut new_σ = HashMap::new();
+                        for (k,v) in n.σ.iter() {
+                            new_σ.insert(
+                                k.clone(),
+                                v.clone().apply_substitution(&|k| next_morph_inst.σ.get(k).cloned()).clone()
+                            );
+                        }
+                        for (k,v) in next_morph_inst.σ.iter() {
+                            new_σ.insert(
+                                k.clone(),
+                                v.clone().apply_substitution(&|k| next_morph_inst.σ.get(k).cloned()).clone()
+                            );
+                        }
+                        n.σ = new_σ;
+                    }
+
+                    for m in new_path.morphisms.iter() {
+                        if m.get_type().src_type == dst_type {
+                            creates_loop = true;
+                            //eprintln!("creates loop..");
+                            break;
+                        }
+                    }
+
+                    if ! creates_loop {
+                        /*eprintln!("next morph ? \n    {}\n--> {} ",
+                            next_morph_inst.get_type().src_type.sugar(type_dict).pretty(type_dict, 0),
+                            next_morph_inst.get_type().dst_type.sugar(type_dict).pretty(type_dict, 0)
+                        );
+                        eprintln!("....take!\n  :: halo = {}\n  :: σ = {:?}", next_morph_inst.halo.clone().sugar(type_dict).pretty(type_dict, 0), next_morph_inst.σ);
+                        */
+                        new_path.weight += next_morph_inst.m.weight();
+                        new_path.cur_type = dst_type;
+                        new_path.morphisms.push(next_morph_inst);
+                        queue.push(new_path);
+                    }
+                }
+            }
+        }
+        None
+    }
+
+
+    pub fn find_direct_morphism(&self,
+        ty: &MorphismType,
+        dict: &mut impl TypeDict
+    ) -> Option< MorphismInstance<M> > {
+        eprintln!("find direct morph");
+        for m in self.morphisms.iter() {
+            let ty = ty.clone().normalize();
+            let morph_type = m.get_type().normalize();
+
+            eprintln!("find direct morph:\n   {}  <=   {}",
+                            dict.unparse(&ty.src_type), dict.unparse(&morph_type.src_type),
+                        );
+
+            if let Ok((halo, σ)) = subtype_unify(&ty.src_type, &morph_type.src_type) {
+                eprintln!("halo: {}", dict.unparse(&halo));
+
+                let dst_type = TypeTerm::Ladder(vec![
+                    halo.clone(),
+                    morph_type.dst_type.clone()
+                ]);
+                eprintln!("----------->   {}  <=   {}",
+                    dict.unparse(&dst_type), dict.unparse(&ty.dst_type)
+                );
+
+                /*
+                let unification_problem = UnificationProblem::new(
+                                vec![
+                                    ( ty.src_type, morph_type.src_type ),
+                                    ( morph_type.dst_type, ty.dst_type )
+                                ]
+                            );*/
+
+                if let Ok((halo2, σ2)) = subtype_unify(&dst_type, &ty.dst_type) {
+                    eprintln!("match. halo2 = {}", dict.unparse(&halo2));
+                    return Some(MorphismInstance {
+                        m: m.clone(),
+                        halo: halo2,
+                        σ,
+                    });
+                }
+            }
+        }
+        None
+    }
+
+    pub fn find_map_morphism(&self, ty: &MorphismType, dict: &mut impl TypeDict) -> Option< MorphismInstance<M> > {
+        for seq_type in self.seq_types.iter() {
+            if let Ok((halo, σ)) = UnificationProblem::new(vec![
+                (ty.src_type.clone().param_normalize(),
+                    TypeTerm::App(vec![ seq_type.clone(), TypeTerm::TypeID(TypeID::Var(100)) ])),
+
+                (TypeTerm::App(vec![ seq_type.clone(), TypeTerm::TypeID(TypeID::Var(101)) ]),
+                    ty.dst_type.clone().param_normalize()),
+            ]).solve_subtype() {
+                // TODO: use real fresh variable names
+                let item_morph_type = MorphismType {
+                    src_type: σ.get(&TypeID::Var(100)).unwrap().clone(),
+                    dst_type: σ.get(&TypeID::Var(101)).unwrap().clone(),
+                }.normalize();
+
+                //eprintln!("Map Morph: try to find item-morph with type {:?}", item_morph_type);
+                if let Some(item_morph_inst) = self.find_morphism( &item_morph_type, dict ) {
+                    if let Some( list_morph ) = item_morph_inst.m.map_morphism( seq_type.clone() ) {
+                        return Some( MorphismInstance {
+                            m: list_morph,
+                            σ,
+                            halo
+                        } );
+                    }
+                }
+            }
+        }
+
+        None
+    }
+
+    pub fn find_morphism(&self, ty: &MorphismType,
+        dict: &mut impl TypeDict
+    )
+    -> Option< MorphismInstance<M> > {
+        if let Some(m) = self.find_direct_morphism(ty, dict) {
+            return Some(m);
+        }
+        if let Some(m) = self.find_map_morphism(ty, dict) {
+            return Some(m);
+        }
+        None
+    }
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\

src/steiner_tree.rs

@@ -0,0 +1,255 @@
+use {
+    crate::{
+        morphism::{
+            Morphism, MorphismBase, MorphismType
+        }, MorphismInstance, TypeID, TypeTerm
+    }, std::collections::HashMap
+};
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone)]
+pub struct SteinerTree {
+    weight: u64,
+    goals: Vec< TypeTerm >,
+    pub edges: Vec< MorphismType >,
+}
+
+impl SteinerTree {
+    pub fn into_edges(self) -> Vec< MorphismType > {
+        self.edges
+    }
+
+    fn add_edge(&mut self, ty: MorphismType) {
+        self.weight += 1;
+
+        let ty = ty.normalize();
+
+        // check if by adding this new edge, we reach a goal
+        let mut new_goals = Vec::new();
+        let mut added = false;
+
+        for g in self.goals.clone() {
+            if let Ok(σ) = crate::unify(&ty.dst_type, &g) {
+                if !added {
+                    self.edges.push(ty.clone());
+
+                    // goal reached.
+                    for e in self.edges.iter_mut() {
+                        e.src_type = e.src_type.apply_substitution(&|x| σ.get(x).cloned()).clone();
+                        e.dst_type = e.dst_type.apply_substitution(&|x| σ.get(x).cloned()).clone();
+                    }
+                    added = true;
+                } else {
+                    new_goals.push(g);
+                }
+            } else {
+                new_goals.push(g);
+            }
+        }
+
+        if !added {
+            self.edges.push(ty.clone());
+        }
+
+        self.goals = new_goals;
+    }
+
+    fn is_solved(&self) -> bool {
+        self.goals.len() == 0
+    }
+
+    fn contains(&self, t: &TypeTerm) -> Option< HashMap<TypeID, TypeTerm> > {
+        for e in self.edges.iter() {
+            if let Ok(σ) = crate::unify(&e.dst_type, t) {
+                return Some(σ)
+            }
+        }
+
+        None
+    }
+}
+
+
+pub struct PathApproxSteinerTreeSolver {
+    root: TypeTerm,
+    leaves: Vec< TypeTerm >
+}
+
+impl PathApproxSteinerTreeSolver {
+    pub fn new(
+        root: TypeTerm,
+        leaves: Vec<TypeTerm>
+    ) -> Self {
+        PathApproxSteinerTreeSolver {
+            root, leaves
+        }
+    }
+
+    pub fn solve<M: Morphism + Clone + PartialEq>(self, morphisms: &MorphismBase<M>) -> Option< SteinerTree > {
+        let mut edges = Vec::<MorphismType>::new();
+
+        for goal in self.leaves {
+            eprintln!("solve steiner tree: find path to goal {:?}", goal);
+            // try to find shortest path from root to current leaf
+            if let Some(new_path) = morphisms.find_morphism_path(
+                MorphismType {
+                    src_type: self.root.clone(),
+                    dst_type: goal.clone()
+                }
+            ) {
+                eprintln!("path to {:?} has len {}", goal.clone(), new_path.len());
+                for morph_inst in new_path {
+                    let t = morph_inst.get_type();
+                    if ! edges.contains(&t) {
+                        eprintln!("add edge {:?}", t);
+                        edges.push(t);
+                    }
+                }
+/*
+                // reduce new path so that it does not collide with any existing path
+                let mut src_type = self.root.clone();
+                let mut new_path_iter = new_path.into_iter().peekable();
+
+                // check all existing nodes..
+
+                if new_path_iter.peek().unwrap().get_type().src_type == src_type {
+                    eprintln!("skip initial node..");
+                    new_path_iter.next();
+                }
+
+                for mt in tree.iter() {
+                    //assert!( mt.src_type == &src_type );
+                    if let Some(t) = new_path_iter.peek() {
+                        eprintln!("");
+                        if &mt.dst_type == &t.get_type().src_type {
+                            // eliminate this node from new path
+                            src_type = new_path_iter.next().unwrap().clone();
+                        }
+                    } else {
+                        break;
+                    }
+                }
+
+                for dst_type in new_path_iter {
+                    tree.push(MorphismType {
+                        src_type: src_type.clone(),
+                        dst_type: dst_type.clone()
+                    });
+                    src_type = dst_type;
+                }
+*/
+            } else {
+                eprintln!("could not find path\nfrom {:?}\nto {:?}", &self.root, &goal);
+                return None;
+            }
+        }
+
+        Some(SteinerTree {
+            weight: 0,
+            goals: vec![],
+            edges
+        })
+    }
+}
+
+
+/* given a representation tree with the available
+ * represenatations `src_types`, try to find
+ * a sequence of morphisms that span up all
+ * representations in `dst_types`.
+ */
+pub struct SteinerTreeProblem {
+    src_types: Vec< TypeTerm >,
+    queue: Vec< SteinerTree >
+}
+
+impl SteinerTreeProblem {
+    pub fn new(
+        src_types: Vec< TypeTerm >,
+        dst_types: Vec< TypeTerm >
+    ) -> Self {
+        SteinerTreeProblem {
+            src_types: src_types.into_iter().map(|t| t.normalize()).collect(),
+            queue: vec![
+                SteinerTree {
+                    weight: 0,
+                    goals: dst_types.into_iter().map(|t| t.normalize()).collect(),
+                    edges: Vec::new()
+                }
+            ]
+        }
+    }
+
+    pub fn next(&mut self) -> Option< SteinerTree > {
+        eprintln!("queue size = {}", self.queue.len());
+
+        /* FIXME: by giving the highest priority to
+         * candidate tree with the least remaining goals,
+         * the optimality of the search algorithm
+         * is probably destroyed, but it dramatically helps
+         * to tame the combinatorical explosion in this algorithm.
+         */
+        self.queue.sort_by(|t1, t2|
+            if t1.goals.len() < t2.goals.len() {
+                std::cmp::Ordering::Greater
+            } else if t1.goals.len() == t2.goals.len() {
+                if t1.weight < t2.weight {
+                    std::cmp::Ordering::Greater
+                } else {
+                    std::cmp::Ordering::Less
+                }
+            } else {
+                std::cmp::Ordering::Less
+            }
+        );
+        self.queue.pop()
+    }
+/*
+    pub fn solve_approx_path<M: Morphism + Clone>(&mut self, morphisms: &MorphismBase<M>) -> Option< SteinerTree > {
+        if let Some(master) = self.src_types.first() {
+
+
+        }
+    }
+*/
+    pub fn solve_bfs<M: Morphism + Clone>(&mut self, morphisms: &MorphismBase<M>) -> Option< SteinerTree > {
+
+        // take the currently smallest tree and extend it by one step
+        while let Some( mut current_tree ) = self.next() {
+
+            // check if current tree is a solution
+            if current_tree.goals.len() == 0 {
+                return Some(current_tree);
+            }
+
+            // get all vertices spanned by this tree
+            let mut current_nodes = self.src_types.clone();
+            for e in current_tree.edges.iter() {
+                current_nodes.push( e.dst_type.clone() );
+            }
+
+            // extend the tree by one edge and add it to the queue
+            for src_type in current_nodes {
+                for (dst_halo,dst_ty) in morphisms.enum_morphisms_with_subtypig( &src_type ) {
+                    let dst_type = TypeTerm::Ladder(vec![
+                        dst_halo, dst_ty
+                    ]).normalize();
+
+                    if current_tree.contains( &dst_type ).is_none() {
+                        let mut new_tree = current_tree.clone();
+                        {
+                            let src_type = src_type.clone();
+                            let dst_type = dst_type.clone();
+                            new_tree.add_edge(MorphismType { src_type, dst_type }.normalize());
+                        }
+
+                        self.queue.push( new_tree );
+                    }
+                }
+            }
+        }
+
+        None
+    }
+}

src/sugar.rs

@@ -92,5 +92,23 @@ impl SugaredTypeTerm {
                 ).collect()),
         }
     }
+
+    pub fn is_empty(&self) -> bool {
+        match self {
+            SugaredTypeTerm::TypeID(_) => false,
+            SugaredTypeTerm::Num(_) => false,
+            SugaredTypeTerm::Char(_) => false,
+            SugaredTypeTerm::Univ(t) => t.is_empty(),
+            SugaredTypeTerm::Spec(ts) |
+            SugaredTypeTerm::Ladder(ts) |
+            SugaredTypeTerm::Func(ts) |
+            SugaredTypeTerm::Morph(ts) |
+            SugaredTypeTerm::Struct(ts) |
+            SugaredTypeTerm::Enum(ts) |
+            SugaredTypeTerm::Seq(ts) => {
+                ts.iter().fold(true, |s,t|s&&t.is_empty())
+            }
+        }
+    }
 }
 

src/term.rs

@@ -14,8 +14,6 @@ pub enum TypeTerm {
     Num(i64),
     Char(char),
 
-    
-
     /* Complex Terms */
 
     // Type Parameters
@@ -47,10 +45,9 @@ impl TypeTerm {
                 *self = TypeTerm::App(vec![
                     self.clone(),
                     t.into()
-                ])                
+                ])
             }
         }
-
         self
     }
 
@@ -124,6 +121,64 @@ impl TypeTerm {
 
         self
     }
+
+    /* strip away empty ladders
+     * & unwrap singletons
+     */
+    pub fn strip(self) -> Self {
+        match self {
+            TypeTerm::Ladder(rungs) => {
+                let mut rungs :Vec<_> = rungs.into_iter()
+                    .filter_map(|mut r| {
+                        r = r.strip();
+                        if r != TypeTerm::unit() {
+                            Some(match r {
+                                TypeTerm::Ladder(r) => r,
+                                a => vec![ a ]
+                            })
+                        }
+                        else { None }
+                    })
+                .flatten()
+                .collect();
+
+                if rungs.len() == 1 {
+                    rungs.pop().unwrap()
+                } else {
+                    TypeTerm::Ladder(rungs)
+                }
+            },
+            TypeTerm::App(args) => {
+                let mut args :Vec<_> = args.into_iter().map(|arg| arg.strip()).collect();
+                if args.len() == 0 {
+                    TypeTerm::unit()
+                } else if args.len() == 1 {
+                    args.pop().unwrap()
+                } else {
+                    TypeTerm::App(args)
+                }
+            }
+            atom => atom
+        }
+    }
+
+
+
+    pub fn get_interface_type(&self) -> TypeTerm {
+        match self {
+            TypeTerm::Ladder(rungs) => {
+                if let Some(top) = rungs.first() {
+                    top.get_interface_type()
+                } else {
+                    TypeTerm::unit()
+                }
+            }
+            TypeTerm::App(args) => {
+                TypeTerm::App(args.iter().map(|a| a.get_interface_type()).collect())
+            }
+            atom => atom.clone()
+        }
+    }
 }
 
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\

src/test/mod.rs

@@ -7,4 +7,5 @@ pub mod pnf;
 pub mod subtype;
 pub mod substitution;
 pub mod unification;
+pub mod morphism;
 

src/test/morphism.rs

@@ -0,0 +1,246 @@
+use {
+    crate::{dict::*, morphism::*, parser::*, unparser::*, steiner_tree::*, TypeTerm}
+};
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone, Debug, PartialEq)]
+struct DummyMorphism(MorphismType);
+
+impl Morphism for DummyMorphism {
+    fn get_type(&self) -> MorphismType {
+        self.0.clone().normalize()
+    }
+
+    fn map_morphism(&self, seq_type: TypeTerm) -> Option<DummyMorphism> {
+        Some(DummyMorphism(MorphismType {
+            src_type: TypeTerm::App(vec![
+                seq_type.clone(),
+                self.0.src_type.clone()
+            ]),
+
+            dst_type: TypeTerm::App(vec![
+                seq_type.clone(),
+                self.0.dst_type.clone()
+            ])
+        }))
+    }
+}
+
+fn morphism_test_setup() -> ( BimapTypeDict, MorphismBase<DummyMorphism> ) {
+    let mut dict = BimapTypeDict::new();
+    let mut base = MorphismBase::<DummyMorphism>::new( vec![ dict.parse("Seq").expect("") ] );
+
+    dict.add_varname("Radix".into());
+    dict.add_varname("SrcRadix".into());
+    dict.add_varname("DstRadix".into());
+
+    base.add_morphism(
+        DummyMorphism(MorphismType{
+            src_type: dict.parse("<Digit Radix> ~ Char").unwrap(),
+            dst_type: dict.parse("<Digit Radix> ~ ℤ_2^64 ~ machine.UInt64").unwrap()
+        })
+    );
+    base.add_morphism(
+        DummyMorphism(MorphismType{
+            src_type: dict.parse("<Digit Radix> ~ ℤ_2^64 ~ machine.UInt64").unwrap(),
+            dst_type: dict.parse("<Digit Radix> ~ Char").unwrap()
+        })
+    );
+    base.add_morphism(
+        DummyMorphism(MorphismType{
+            src_type: dict.parse("ℕ ~ <PosInt Radix BigEndian> ~ <Seq <Digit Radix>~ℤ_2^64~machine.UInt64>").unwrap(),
+            dst_type: dict.parse("ℕ ~ <PosInt Radix LittleEndian> ~ <Seq <Digit Radix>~ℤ_2^64~machine.UInt64>").unwrap()
+        })
+    );
+    base.add_morphism(
+        DummyMorphism(MorphismType{
+            src_type: dict.parse("ℕ ~ <PosInt Radix LittleEndian> ~ <Seq <Digit Radix>~ℤ_2^64~machine.UInt64>").unwrap(),
+            dst_type: dict.parse("ℕ ~ <PosInt Radix BigEndian> ~ <Seq <Digit Radix>~ℤ_2^64~machine.UInt64>").unwrap()
+        })
+    );
+    base.add_morphism(
+        DummyMorphism(MorphismType{
+            src_type: dict.parse("ℕ ~ <PosInt SrcRadix LittleEndian> ~ <Seq <Digit SrcRadix>~ℤ_2^64~machine.UInt64>").unwrap(),
+            dst_type: dict.parse("ℕ ~ <PosInt DstRadix LittleEndian> ~ <Seq <Digit DstRadix>~ℤ_2^64~machine.UInt64>").unwrap()
+        })
+    );
+
+    (dict, base)
+}
+
+#[test]
+fn test_morphism_path() {
+    let (mut dict, mut base) = morphism_test_setup();
+
+    let path = base.find_morphism_path(MorphismType {
+        src_type: dict.parse("ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ Char>").unwrap(),
+        dst_type: dict.parse("ℕ ~ <PosInt 16 BigEndian> ~ <Seq <Digit 16> ~ Char>").unwrap(),
+    });
+
+    fn print_subst(m: &std::collections::HashMap<TypeID, TypeTerm>, dict: &mut impl TypeDict) {
+        eprintln!("{{");
+
+        for (k,v) in m.iter() {
+            eprintln!("    {} --> {}",
+                dict.get_typename(k).unwrap(),
+                dict.unparse(v)
+            );
+        }
+
+        eprintln!("}}");
+    }
+
+    if let Some(path) = path.as_ref() {
+        for n in path.iter() {
+            eprintln!("
+ψ = {}
+morph {}
+  --> {}
+with
+            ",
+            n.halo.clone().sugar(&mut dict).pretty(&mut dict, 0),
+            n.m.get_type().src_type.sugar(&mut dict).pretty(&mut dict, 0),
+            n.m.get_type().dst_type.sugar(&mut dict).pretty(&mut dict, 0),
+            );
+            print_subst(&n.σ, &mut dict)
+        }
+    }
+
+    assert_eq!(
+        path,
+        Some(
+            vec![
+                MorphismInstance {
+                    σ: vec![
+                        (dict.get_typeid(&"Radix".into()).unwrap(), TypeTerm::Num(10)),
+                        (dict.get_typeid(&"SrcRadix".into()).unwrap(), TypeTerm::Num(10)),
+                        (dict.get_typeid(&"DstRadix".into()).unwrap(), TypeTerm::Num(16))
+                    ].into_iter().collect(),
+                    halo: dict.parse("ℕ ~ <PosInt 10 BigEndian>").unwrap(),
+                    m: DummyMorphism(MorphismType {
+                        src_type: dict.parse("<Seq <Digit Radix> ~ Char>").unwrap(),
+                        dst_type: dict.parse("<Seq <Digit Radix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap()
+                    }),
+                },
+                MorphismInstance {
+                    σ: vec![
+                        (dict.get_typeid(&"Radix".into()).unwrap(), TypeTerm::Num(10)),
+                        (dict.get_typeid(&"SrcRadix".into()).unwrap(), TypeTerm::Num(10)),
+                        (dict.get_typeid(&"DstRadix".into()).unwrap(), TypeTerm::Num(16))
+                    ].into_iter().collect(),
+                    halo: TypeTerm::unit(),
+                    m: DummyMorphism(MorphismType{
+                        src_type: dict.parse("ℕ ~ <PosInt Radix BigEndian> ~ <Seq <Digit Radix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap(),
+                        dst_type: dict.parse("ℕ ~ <PosInt Radix LittleEndian> ~ <Seq <Digit Radix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap()
+                    }),
+                },
+                MorphismInstance {
+                    σ: vec![
+                        (dict.get_typeid(&"SrcRadix".into()).unwrap(), TypeTerm::Num(10)),
+                        (dict.get_typeid(&"DstRadix".into()).unwrap(), TypeTerm::Num(16)),
+                        (dict.get_typeid(&"Radix".into()).unwrap(), TypeTerm::Num(16)),
+                    ].into_iter().collect(),
+                    halo: TypeTerm::unit(),
+                    m: DummyMorphism(MorphismType{
+                        src_type: dict.parse("ℕ ~ <PosInt SrcRadix LittleEndian> ~ <Seq <Digit SrcRadix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap(),
+                        dst_type: dict.parse("ℕ ~ <PosInt DstRadix LittleEndian> ~ <Seq <Digit DstRadix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap()
+                    }),
+                },
+                MorphismInstance {
+                    σ: vec![
+                        (dict.get_typeid(&"Radix".into()).unwrap(), TypeTerm::Num(16)),
+                        (dict.get_typeid(&"DstRadix".into()).unwrap(), TypeTerm::Num(16))
+                    ].into_iter().collect(),
+                    halo: TypeTerm::unit(),
+                    m: DummyMorphism(MorphismType{
+                        src_type: dict.parse("ℕ ~ <PosInt Radix LittleEndian> ~ <Seq <Digit Radix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap(),
+                        dst_type: dict.parse("ℕ ~ <PosInt Radix BigEndian> ~ <Seq <Digit Radix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap(),
+                    }),
+                },
+                MorphismInstance {
+                    σ: vec![
+                        (dict.get_typeid(&"Radix".into()).unwrap(), TypeTerm::Num(16))
+                    ].into_iter().collect(),
+                    halo: dict.parse("ℕ ~ <PosInt Radix BigEndian>").unwrap(),
+                    m: DummyMorphism(MorphismType{
+                        src_type: dict.parse("<Seq <Digit Radix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap(),
+                        dst_type: dict.parse("<Seq <Digit Radix> ~ Char>").unwrap()
+                    })
+                }
+            ]
+        )
+    );
+/*
+    assert_eq!(
+        base.find_morphism_path(MorphismType {
+            src_type: dict.parse("Symbol ~ ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ Char>").unwrap(),
+            dst_type: dict.parse("Symbol ~ ℕ ~ <PosInt 16 BigEndian> ~ <Seq <Digit 16> ~ Char>").unwrap()
+        }),
+        Some(
+            vec![
+                dict.parse("Symbol ~ ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ Char>").unwrap().normalize(),
+                dict.parse("Symbol ~ ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("Symbol ~ ℕ ~ <PosInt 10 LittleEndian> ~ <Seq <Digit 10> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("Symbol ~ ℕ ~ <PosInt 16 LittleEndian> ~ <Seq <Digit 16> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("Symbol ~ ℕ ~ <PosInt 16 BigEndian> ~ <Seq <Digit 16> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("Symbol ~ ℕ ~ <PosInt 16 BigEndian> ~ <Seq <Digit 16> ~ Char>").unwrap().normalize(),
+            ]
+        )
+    );
+    */
+
+
+/*
+    assert_eq!(
+        base.find_morphism_with_subtyping(
+            &MorphismType {
+                src_type: dict.parse("Symbol ~ ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ Char>").unwrap(),
+                dst_type: dict.parse("Symbol ~ ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ ℤ_2^64 ~ machine.UInt64>").unwrap()
+            }
+        ),
+
+        Some((
+                DummyMorphism(MorphismType{
+                    src_type: dict.parse("<Seq <Digit Radix> ~ Char>").unwrap(),
+                    dst_type: dict.parse("<Seq <Digit Radix> ~ ℤ_2^64 ~ machine.UInt64>").unwrap()
+                }),
+
+                dict.parse("Symbol ~ ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10>>").unwrap(),
+
+                vec![
+                    (dict.get_typeid(&"Radix".into()).unwrap(),
+                    dict.parse("10").unwrap())
+                ].into_iter().collect::<std::collections::HashMap<TypeID, TypeTerm>>()
+        ))
+    );
+    */
+}
+
+#[test]
+fn test_steiner_tree() {
+    let (mut dict, mut base) = morphism_test_setup();
+
+
+    let mut steiner_tree_problem = SteinerTreeProblem::new(
+        // source reprs
+        vec![
+            dict.parse("ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ Char>").unwrap(),
+        ],
+
+        // destination reprs
+        vec![
+            dict.parse("ℕ ~ <PosInt 2 BigEndian> ~ <Seq <Digit 2> ~ Char>").unwrap(),
+            dict.parse("ℕ ~ <PosInt 10 LittleEndian> ~ <Seq <Digit 10> ~ Char>").unwrap(),
+            dict.parse("ℕ ~ <PosInt 16 LittleEndian> ~ <Seq <Digit 16> ~ Char>").unwrap()
+        ]
+    );
+
+    if let Some(solution) = steiner_tree_problem.solve_bfs( &base ) {
+        for e in solution.edges.iter() {
+            eprintln!(" :: {}\n--> {}", dict.unparse(&e.src_type), dict.unparse(&e.dst_type));
+        }
+    } else {
+        eprintln!("no solution");
+    }
+}