add Send+Sync trait bound to TypeDict

* also apply substitution from src-type match
* get this substitution as result from `enum_morphisms_with_subtyping`

Cargo.toml

@@ -4,3 +4,8 @@ edition = "2018"
 name = "laddertypes"
 version = "0.1.0"
 
+[dependencies]
+tiny-ansi = { version = "0.1.0", optional = true }
+
+[features]
+pretty = ["dep:tiny-ansi"]

src/bimap.rs

@@ -2,6 +2,7 @@ use std::{collections::HashMap, hash::Hash};
 
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
 
+#[derive(Debug)]
 pub struct Bimap<V: Eq + Hash, Λ: Eq + Hash> {
     pub mλ: HashMap<V, Λ>,
     pub my: HashMap<Λ, V>,

src/dict.rs

@@ -2,15 +2,34 @@ use crate::bimap::Bimap;
 
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
 
-#[derive(Eq, PartialEq, Hash, Clone, Debug)]
+#[derive(Eq, PartialEq, Hash, Clone, Copy, Debug)]
 pub enum TypeID {
     Fun(u64),
     Var(u64)
 }
 
+pub trait TypeDict : Send + Sync {
+    fn insert(&mut self, name: String, id: TypeID);
+    fn add_varname(&mut self, vn: String) -> TypeID;
+    fn add_typename(&mut self, tn: String) -> TypeID;
+    fn get_typeid(&self, tn: &String) -> Option<TypeID>;
+    fn get_typename(&self, tid: &TypeID) -> Option<String>;
+
+    fn get_varname(&self, var_id: u64) -> Option<String> {
+        self.get_typename(&TypeID::Var(var_id))
+    }
+
+    fn add_synonym(&mut self, new: String, old: String) {
+        if let Some(tyid) = self.get_typeid(&old) {
+            self.insert(new, tyid);
+        }
+    }
+}
+
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
 
-pub struct TypeDict {
+#[derive(Debug)]
+pub struct BimapTypeDict {
     typenames: Bimap<String, TypeID>,
     type_lit_counter: u64,
     type_var_counter: u64,
@@ -18,42 +37,66 @@ pub struct TypeDict {
 
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
 
-impl TypeDict {
+impl BimapTypeDict {
     pub fn new() -> Self {
-        TypeDict {
+        BimapTypeDict {
             typenames: Bimap::new(),
             type_lit_counter: 0,
             type_var_counter: 0,
         }
     }
+}
 
-    pub fn add_varname(&mut self, tn: String) -> TypeID {
+impl TypeDict for BimapTypeDict {
+    fn insert(&mut self, name: String, id: TypeID) {
+        self.typenames.insert(name, id);
+    }
+
+    fn add_varname(&mut self, tn: String) -> TypeID {
         let tyid = TypeID::Var(self.type_var_counter);
         self.type_var_counter += 1;
-        self.typenames.insert(tn, tyid.clone());
+        self.insert(tn, tyid.clone());
         tyid
     }
 
-    pub fn add_typename(&mut self, tn: String) -> TypeID {
+    fn add_typename(&mut self, tn: String) -> TypeID {
         let tyid = TypeID::Fun(self.type_lit_counter);
         self.type_lit_counter += 1;
-        self.typenames.insert(tn, tyid.clone());
+        self.insert(tn, tyid.clone());
         tyid
     }
 
-    pub fn add_synonym(&mut self, new: String, old: String) {
-        if let Some(tyid) = self.get_typeid(&old) {
-            self.typenames.insert(new, tyid);
-        }
-    }
-
-    pub fn get_typename(&self, tid: &TypeID) -> Option<String> {
+    fn get_typename(&self, tid: &TypeID) -> Option<String> {
         self.typenames.my.get(tid).cloned()
     }
 
-    pub fn get_typeid(&self, tn: &String) -> Option<TypeID> {
+    fn get_typeid(&self, tn: &String) -> Option<TypeID> {
         self.typenames.mλ.get(tn).cloned()
     }
 }
 
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>
+
+use std::sync::Arc;
+use std::ops::{Deref, DerefMut};
+use std::sync::RwLock;
+
+impl<T: TypeDict> TypeDict for Arc<RwLock<T>> {
+    fn insert(&mut self, name: String, id: TypeID) {
+        self.write().unwrap().insert(name, id);
+    }
+    fn add_varname(&mut self, vn: String) -> TypeID {
+        self.write().unwrap().add_varname(vn)
+    }
+    fn add_typename(&mut self, tn: String) -> TypeID {
+        self.write().unwrap().add_typename(tn)
+    }
+    fn get_typename(&self, tid: &TypeID)-> Option<String> {
+        self.read().unwrap().get_typename(tid)
+    }
+    fn get_typeid(&self, tn: &String) -> Option<TypeID> {
+        self.read().unwrap().get_typeid(tn)
+    }
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>

src/lib.rs

@@ -5,18 +5,26 @@ pub mod term;
 pub mod lexer;
 pub mod parser;
 pub mod unparser;
+pub mod sugar;
 pub mod curry;
 pub mod lnf;
 pub mod pnf;
 pub mod subtype;
 pub mod unification;
+pub mod morphism;
+pub mod steiner_tree;
 
 #[cfg(test)]
 mod test;
 
+#[cfg(feature = "pretty")]
+mod pretty;
+
 pub use {
     dict::*,
     term::*,
-    unification::*
+    sugar::*,
+    unification::*,
+    morphism::*
 };
 

src/morphism.rs

@@ -0,0 +1,258 @@
+use {
+    crate::{
+        TypeTerm, TypeID,
+        unification::UnificationProblem,
+    },
+    std::collections::HashMap
+};
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone, PartialEq, Eq, Debug)]
+pub struct MorphismType {
+    pub src_type: TypeTerm,
+    pub dst_type: TypeTerm,
+}
+
+pub trait Morphism : Sized {
+    fn get_type(&self) -> MorphismType;
+    fn list_map_morphism(&self, list_typeid: TypeID) -> Option< Self >;
+
+    fn weight(&self) -> u64 {
+        1
+    }
+}
+
+#[derive(Clone)]
+pub struct MorphismBase<M: Morphism + Clone> {
+    morphisms: Vec< M >,
+    list_typeid: TypeID
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+impl MorphismType {
+    pub fn normalize(self) -> Self {
+        MorphismType {
+            src_type: self.src_type.normalize(),
+            dst_type: self.dst_type.normalize()
+        }
+    }
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+impl<M: Morphism + Clone> MorphismBase<M> {
+    pub fn new(list_typeid: TypeID) -> Self {
+        MorphismBase {
+            morphisms: Vec::new(),
+            list_typeid
+        }
+    }
+
+    pub fn add_morphism(&mut self, m: M) {
+        self.morphisms.push( m );
+    }
+
+    pub fn enum_morphisms(&self, src_type: &TypeTerm)
+    -> Vec< (HashMap<TypeID, TypeTerm>, TypeTerm) >
+    {
+        let mut dst_types = Vec::new();
+
+        // first enumerate all "direct" morphisms,
+        for m in self.morphisms.iter() {
+            if let Ok(σ) = crate::unification::unify(
+                &m.get_type().src_type,
+                &src_type.clone().normalize()
+            ) {
+                let dst_type = 
+                    m.get_type().dst_type.clone()
+                    .apply_substitution( &|x| σ.get(x).cloned() )
+                    .clone();
+
+                dst_types.push( (σ, dst_type) );
+            }
+        }
+
+        // ..then all "list-map" morphisms.
+        // 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(100);
+
+        if let Ok(σ) = crate::unification::unify(
+            &TypeTerm::App(vec![
+                TypeTerm::TypeID(self.list_typeid),
+                TypeTerm::TypeID(item_variable)
+            ]),
+            &src_type.clone().param_normalize(),
+        ) {
+            let src_item_type = σ.get(&item_variable).unwrap().clone();
+
+            for (γ, dst_item_type) in self.enum_morphisms( &src_item_type ) {
+                let dst_type =
+                        TypeTerm::App(vec![
+                            TypeTerm::TypeID(self.list_typeid),
+                            dst_item_type.clone()
+                                .apply_substitution(
+                                    &|x| γ.get(x).cloned()
+                                ).clone()
+                        ]).normalize();
+
+                dst_types.push( (γ.clone(), dst_type) );
+            }
+        }
+
+        dst_types
+    }
+
+    pub fn enum_morphisms_with_subtyping(
+        &self,
+        src_type: &TypeTerm,
+    ) -> Vec<(TypeTerm, TypeTerm, HashMap<TypeID, TypeTerm>)> {
+        let mut src_lnf = src_type.clone().get_lnf_vec();
+        let mut halo_lnf = vec![];
+        let mut dst_types = Vec::new();
+
+        while src_lnf.len() > 0 {
+            let src_type = TypeTerm::Ladder(src_lnf.clone());
+            let halo_type = TypeTerm::Ladder(halo_lnf.clone());
+
+            for (σ, t) in self.enum_morphisms(&src_type) {
+                dst_types.push((
+                    halo_type
+                        .clone()
+                        .apply_substitution(&|x| σ.get(x).cloned())
+                        .clone(),
+                    t.clone().apply_substitution(&|x| σ.get(x).cloned()).clone(),
+                    σ,
+                ));
+            }
+
+            // continue with next supertype
+            halo_lnf.push(src_lnf.remove(0));
+        }
+
+        dst_types
+    }
+
+    /* try to find shortest morphism-path for a given type
+     */
+    pub fn find_morphism_path(&self, ty: MorphismType)
+    -> Option< Vec<TypeTerm> >
+    {
+        let ty = ty.normalize();
+
+        let mut queue = vec![
+            (0, vec![ ty.src_type.clone().normalize() ])
+        ];
+
+        while ! queue.is_empty() {
+            queue.sort_by( |&(w1,_),&(w2,_)| w2.cmp(&w1));
+
+            if let Some((current_weight, current_path)) = queue.pop() {
+                let current_type = current_path.last().unwrap();
+                for (h, t, σp) in self.enum_morphisms_with_subtyping(&current_type) {
+                    let tt = TypeTerm::Ladder(vec![h, t]).normalize();
+
+                    if !current_path.contains(&tt) {
+                        let unification_result = crate::unification::unify(&tt, &ty.dst_type);
+                        let morphism_weight = 1;
+                        /*self.find_morphism( &tt ).unwrap().0.get_weight()*/
+
+                        let new_weight = current_weight + morphism_weight;
+                        let mut new_path = current_path.clone();
+
+                        new_path.push(tt);
+
+                        for n in new_path.iter_mut() {
+                            n.apply_substitution(&|x| σp.get(x).cloned());
+                        }
+
+                        if let Ok(σ) = unification_result {
+                            for n in new_path.iter_mut() {
+                                n.apply_substitution(&|x| σ.get(x).cloned());
+                            }
+                            return Some(new_path);
+                        } else {
+                            queue.push((new_weight, new_path));
+                        }
+                    }
+                }
+            }
+        }
+
+        None
+    }
+
+    pub fn find_morphism(&self, ty: &MorphismType)
+    -> Option< ( M, HashMap<TypeID, TypeTerm> ) > {
+
+        // try list-map morphism
+        if let Ok(σ) = UnificationProblem::new(vec![
+            (ty.src_type.clone().param_normalize(), TypeTerm::App(vec![ TypeTerm::TypeID(self.list_typeid), TypeTerm::TypeID(TypeID::Var(100)) ])),
+            (ty.dst_type.clone().param_normalize(), TypeTerm::App(vec![ TypeTerm::TypeID(self.list_typeid), TypeTerm::TypeID(TypeID::Var(101)) ])),
+        ]).solve() {
+
+            // 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();
+
+            if let Some((m, σ)) = self.find_morphism( &item_morph_type ) {
+                if let Some(list_morph) = m.list_map_morphism( self.list_typeid ) {
+                    return Some( (list_morph, σ) );
+                }
+            }
+        }
+
+        // otherwise
+        for m in self.morphisms.iter() {
+            let unification_problem = UnificationProblem::new(
+                vec![
+                    ( ty.src_type.clone().normalize(), m.get_type().src_type.clone() ),
+                    ( ty.dst_type.clone().normalize(), m.get_type().dst_type.clone() )
+                ]
+            );
+
+            let unification_result = unification_problem.solve();
+            if let Ok(σ) = unification_result {
+                return Some((m.clone(), σ));
+            }
+        }
+
+        None
+    }
+
+    pub fn find_morphism_with_subtyping(&self, ty: &MorphismType)
+    -> Option<( M, TypeTerm, HashMap<TypeID, TypeTerm> )> {
+        let mut src_lnf = ty.src_type.clone().get_lnf_vec();
+        let mut dst_lnf = ty.dst_type.clone().get_lnf_vec();
+        let mut halo = vec![];
+
+        while src_lnf.len() > 0 && dst_lnf.len() > 0 {
+            if let Some((m, σ)) = self.find_morphism(&MorphismType{
+                src_type: TypeTerm::Ladder(src_lnf.clone()),
+                dst_type: TypeTerm::Ladder(dst_lnf.clone())
+            }) {
+                halo.push(src_lnf.get(0).unwrap().clone());
+                return Some((m,
+                    TypeTerm::Ladder(halo).apply_substitution(&|x| σ.get(x).cloned()).clone(),
+                    σ));
+            } else {
+                if src_lnf[0] == dst_lnf[0] {
+                    src_lnf.remove(0);
+                    halo.push(dst_lnf.remove(0));
+                } else {
+                    return None;
+                }
+            }
+        }
+
+        None
+    }
+}
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+

src/parser.rs

@@ -18,10 +18,23 @@ pub enum ParseError {
     UnexpectedToken
 }
 
+pub trait ParseLadderType {
+    fn parse(&mut self, s: &str) -> Result<TypeTerm, ParseError>;
+    
+    fn parse_app<It>(&mut self, tokens: &mut Peekable<LadderTypeLexer<It>>) -> Result<TypeTerm, ParseError>
+    where It: Iterator<Item = char>;
+
+    fn parse_rung<It>(&mut self, tokens: &mut Peekable<LadderTypeLexer<It>>) -> Result<TypeTerm, ParseError>
+    where It: Iterator<Item = char>;
+
+    fn parse_ladder<It>(&mut self, tokens: &mut Peekable<LadderTypeLexer<It>>) -> Result<TypeTerm, ParseError>
+    where It: Iterator<Item = char>;
+}
+
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
 
-impl TypeDict {
-    pub fn parse(&mut self, s: &str) -> Result<TypeTerm, ParseError> {
+impl<T: TypeDict> ParseLadderType for T {
+    fn parse(&mut self, s: &str) -> Result<TypeTerm, ParseError> {
         let mut tokens = LadderTypeLexer::from(s.chars()).peekable();
 
         match self.parse_ladder(&mut tokens) {

src/pretty.rs

@@ -0,0 +1,148 @@
+use {
+    crate::TypeDict,
+    crate::sugar::SugaredTypeTerm,
+    tiny_ansi::TinyAnsi
+};
+
+impl SugaredTypeTerm {
+    pub fn pretty(&self, dict: &TypeDict, indent: u64) -> String {
+        let indent_width = 4;
+        match self {
+            SugaredTypeTerm::TypeID(id) => {
+                format!("{}", dict.get_typename(id).unwrap_or("??".bright_red())).bright_blue()
+            },
+
+            SugaredTypeTerm::Num(n) => {
+                format!("{}", n).bright_cyan()
+            }
+
+            SugaredTypeTerm::Char(c) => {
+                format!("'{}'", c)
+            }
+
+            SugaredTypeTerm::Univ(t) => {
+                format!("{} {} . {}",
+                    "∀".yellow().bold(),
+                    dict.get_varname(0).unwrap_or("??".into()).bright_blue(),
+                    t.pretty(dict,indent)
+                )
+            }
+
+            SugaredTypeTerm::Spec(args) => {
+                let mut s = String::new();
+                s.push_str(&"<".yellow().bold());
+                for i in 0..args.len() {
+                    let arg = &args[i];
+                    if i > 0 {
+                        s.push(' ');
+                    }
+                    s.push_str( &arg.pretty(dict,indent+1) );
+                }
+                s.push_str(&">".yellow().bold());
+                s
+            }
+
+            SugaredTypeTerm::Struct(args) => {
+                let mut s = String::new();
+                s.push_str(&"{".yellow().bold());
+                for arg in args {
+                    s.push('\n');
+                    for x in 0..(indent+1)*indent_width {
+                        s.push(' ');
+                    }
+                    s.push_str(&arg.pretty(dict, indent + 1));
+                    s.push_str(&";\n".bright_yellow());
+                }
+
+                s.push('\n');
+                for x in 0..indent*indent_width {
+                    s.push(' ');
+                }
+                s.push_str(&"}".yellow().bold());
+                s
+            }
+
+            SugaredTypeTerm::Enum(args) => {
+                let mut s = String::new();
+                s.push_str(&"(".yellow().bold());
+                for i in 0..args.len() {
+                    let arg = &args[i];
+                    s.push('\n');
+                    for x in 0..(indent+1)*indent_width {
+                        s.push(' ');
+                    }
+                    if i > 0 {
+                        s.push_str(&"| ".yellow().bold());
+                    }
+                    s.push_str(&arg.pretty(dict, indent + 1));
+                }
+
+                s.push('\n');
+                for x in 0..indent*indent_width {
+                    s.push(' ');
+                }
+                s.push_str(&")".yellow().bold());
+                s
+            }
+
+            SugaredTypeTerm::Seq(args) => {
+                let mut s = String::new();
+                s.push_str(&"[ ".yellow().bold());
+                for i in 0..args.len() {
+                    let arg = &args[i];
+                    if i > 0 {
+                        s.push(' ');
+                    }
+                    s.push_str(&arg.pretty(dict, indent+1));
+                }
+                s.push_str(&" ]".yellow().bold());
+                s
+            }
+
+            SugaredTypeTerm::Morph(args) => {
+                let mut s = String::new();
+                for arg in args {
+                    s.push_str(&"  ~~morph~~>  ".bright_yellow());
+                    s.push_str(&arg.pretty(dict, indent));
+                }
+                s
+            }
+
+            SugaredTypeTerm::Func(args) => {
+                let mut s = String::new();
+                for i in 0..args.len() {
+                    let arg = &args[i];
+                    if i > 0{
+                        s.push('\n');
+                        for x in 0..(indent*indent_width) {
+                            s.push(' ');
+                        }                
+                        s.push_str(&"-->  ".bright_yellow());
+                    } else {
+//                        s.push_str("   ");
+                    }
+                    s.push_str(&arg.pretty(dict, indent));
+                }
+                s
+            }
+
+            SugaredTypeTerm::Ladder(rungs) => {
+                let mut s = String::new();
+                for i in 0..rungs.len() {
+                    let rung = &rungs[i];
+                    if i > 0{
+                        s.push('\n');
+                        for x in 0..(indent*indent_width) {
+                            s.push(' ');
+                        }
+                        s.push_str(&"~ ".yellow());
+                    }
+                    s.push_str(&rung.pretty(dict, indent));
+                }
+                s
+            }
+        }
+    }
+}
+
+

src/steiner_tree.rs

@@ -0,0 +1,241 @@
+use {
+    std::collections::HashMap,
+    crate::{
+        TypeID,
+        TypeTerm,
+        morphism::{
+            MorphismType,
+            Morphism,
+            MorphismBase
+        }
+    }
+};
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone)]
+pub struct SteinerTree {
+    weight: u64,
+    goals: Vec< TypeTerm >,
+    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>(self, morphisms: &MorphismBase<M>) -> Option< SteinerTree > {
+        let mut tree = Vec::<MorphismType>::new();
+
+        for goal in self.leaves {
+            // 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()
+                }
+            ) {
+                // 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..
+                for mt in tree.iter() {
+//                    assert!( mt.src_type == &src_type );
+                    if let Some(t) = new_path_iter.peek() {
+                        if &mt.dst_type == t {
+                            // 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: tree
+        })
+    }
+}
+
+
+/* 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, σ) in morphisms.enum_morphisms_with_subtyping(&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

@@ -0,0 +1,114 @@
+use {
+    crate::{TypeTerm, TypeID, parser::ParseLadderType}
+};
+
+#[derive(Clone, PartialEq)]
+pub enum SugaredTypeTerm {
+    TypeID(TypeID),
+    Num(i64),
+    Char(char),
+    Univ(Box< SugaredTypeTerm >),
+    Spec(Vec< SugaredTypeTerm >),
+    Func(Vec< SugaredTypeTerm >),
+    Morph(Vec< SugaredTypeTerm >),
+    Ladder(Vec< SugaredTypeTerm >),
+    Struct(Vec< SugaredTypeTerm >),
+    Enum(Vec< SugaredTypeTerm >),
+    Seq(Vec< SugaredTypeTerm >)
+}
+
+impl TypeTerm {
+    pub fn sugar(self: TypeTerm, dict: &mut impl crate::TypeDict) -> SugaredTypeTerm {
+        match self {
+            TypeTerm::TypeID(id) => SugaredTypeTerm::TypeID(id),
+            TypeTerm::Num(n) => SugaredTypeTerm::Num(n),
+            TypeTerm::Char(c) => SugaredTypeTerm::Char(c),
+            TypeTerm::App(args) => if let Some(first) = args.first() {
+                if first == &dict.parse("Func").unwrap() {
+                    SugaredTypeTerm::Func( args[1..].into_iter().map(|t| t.clone().sugar(dict)).collect() )
+                }
+                else if first == &dict.parse("Morph").unwrap() {
+                    SugaredTypeTerm::Morph( args[1..].into_iter().map(|t| t.clone().sugar(dict)).collect() )
+                }
+                else if first == &dict.parse("Struct").unwrap() {
+                    SugaredTypeTerm::Struct( args[1..].into_iter().map(|t| t.clone().sugar(dict)).collect() )
+                }
+                else if first == &dict.parse("Enum").unwrap() {
+                    SugaredTypeTerm::Enum( args[1..].into_iter().map(|t| t.clone().sugar(dict)).collect() )
+                }
+                else if first == &dict.parse("Seq").unwrap() {
+                    SugaredTypeTerm::Seq( args[1..].into_iter().map(|t| t.clone().sugar(dict)).collect() )
+                }
+                else if first == &dict.parse("Spec").unwrap() {
+                    SugaredTypeTerm::Spec( args[1..].into_iter().map(|t| t.clone().sugar(dict)).collect() )
+                }
+                else if first == &dict.parse("Univ").unwrap() {
+                    SugaredTypeTerm::Univ(Box::new(
+                        SugaredTypeTerm::Spec(
+                            args[1..].into_iter().map(|t| t.clone().sugar(dict)).collect()
+                        )
+                    ))
+                }
+                else {
+                    SugaredTypeTerm::Spec(args.into_iter().map(|t| t.sugar(dict)).collect())
+                }
+            } else {
+                SugaredTypeTerm::Spec(args.into_iter().map(|t| t.sugar(dict)).collect())
+            },
+            TypeTerm::Ladder(rungs) =>            
+               SugaredTypeTerm::Ladder(rungs.into_iter().map(|t| t.sugar(dict)).collect())
+        }
+    }
+}
+
+impl SugaredTypeTerm {
+    pub fn desugar(self, dict: &mut impl crate::TypeDict) -> TypeTerm {
+        match self {
+            SugaredTypeTerm::TypeID(id) => TypeTerm::TypeID(id),
+            SugaredTypeTerm::Num(n) => TypeTerm::Num(n),
+            SugaredTypeTerm::Char(c) => TypeTerm::Char(c),
+            SugaredTypeTerm::Univ(t) => t.desugar(dict),
+            SugaredTypeTerm::Spec(ts) => TypeTerm::App(ts.into_iter().map(|t| t.desugar(dict)).collect()),
+            SugaredTypeTerm::Ladder(ts) => TypeTerm::Ladder(ts.into_iter().map(|t|t.desugar(dict)).collect()),
+            SugaredTypeTerm::Func(ts) => TypeTerm::App(
+                std::iter::once( dict.parse("Func").unwrap() ).chain(
+                    ts.into_iter().map(|t| t.desugar(dict))
+                ).collect()),
+            SugaredTypeTerm::Morph(ts) => TypeTerm::App(
+                std::iter::once( dict.parse("Morph").unwrap() ).chain(
+                    ts.into_iter().map(|t| t.desugar(dict))
+                ).collect()),
+            SugaredTypeTerm::Struct(ts) => TypeTerm::App(
+                std::iter::once( dict.parse("Struct").unwrap() ).chain(
+                    ts.into_iter().map(|t| t.desugar(dict))
+                ).collect()),
+            SugaredTypeTerm::Enum(ts) => TypeTerm::App(
+                std::iter::once( dict.parse("Enum").unwrap() ).chain(
+                    ts.into_iter().map(|t| t.desugar(dict))
+                ).collect()),
+            SugaredTypeTerm::Seq(ts) => TypeTerm::App(
+                std::iter::once( dict.parse("Seq").unwrap() ).chain(
+                    ts.into_iter().map(|t| t.desugar(dict))
+                ).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

@@ -57,7 +57,7 @@ impl TypeTerm {
     pub fn repr_as(&mut self, t: impl Into<TypeTerm>) -> &mut Self {
         match self {
             TypeTerm::Ladder(rungs) => {
-                rungs.push(t.into());                
+                rungs.push(t.into());
             }
 
             _ => {

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 +1,161 @@
+use {
+    crate::{dict::*, morphism::*, steiner_tree::*, TypeTerm}
+};
+
+//<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
+
+#[derive(Clone, Debug, PartialEq)]
+struct DummyMorphism(MorphismType);
+
+impl Morphism for DummyMorphism {
+    fn get_type(&self) -> MorphismType {
+        self.0.clone().normalize()
+    }
+
+    fn list_map_morphism(&self, list_typeid: TypeID) -> Option<DummyMorphism> {
+        Some(DummyMorphism(MorphismType {
+            src_type: TypeTerm::App(vec![
+                TypeTerm::TypeID( list_typeid ),
+                self.0.src_type.clone()
+            ]),
+
+            dst_type: TypeTerm::App(vec![
+                TypeTerm::TypeID( list_typeid ),
+                self.0.dst_type.clone()
+            ])
+        }))
+    }
+}
+
+fn morphism_test_setup() -> ( TypeDict, MorphismBase<DummyMorphism> ) {
+    let mut dict = TypeDict::new();
+    let mut base = MorphismBase::<DummyMorphism>::new( dict.add_typename("Seq".into()) );
+
+    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();
+
+    assert_eq!(
+        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()
+        }),
+        Some(
+            vec![
+                dict.parse("ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ Char>").unwrap().normalize(),
+                dict.parse("ℕ ~ <PosInt 10 BigEndian> ~ <Seq <Digit 10> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("ℕ ~ <PosInt 10 LittleEndian> ~ <Seq <Digit 10> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("ℕ ~ <PosInt 16 LittleEndian> ~ <Seq <Digit 16> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("ℕ ~ <PosInt 16 BigEndian> ~ <Seq <Digit 16> ~ ℤ_2^64 ~ machine.UInt64>").unwrap().normalize(),
+                dict.parse("ℕ ~ <PosInt 16 BigEndian> ~ <Seq <Digit 16> ~ Char>").unwrap().normalize(),
+            ]
+        )
+    );
+
+    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( &dict, &base ) {
+        for e in solution.edges.iter() {
+            eprintln!(" :: {}\n--> {}", dict.unparse(&e.src_type), dict.unparse(&e.dst_type));
+        }
+    } else {
+        eprintln!("no solution");
+    }
+}
+

src/unparser.rs

@@ -2,8 +2,12 @@ use crate::{dict::*, term::*};
 
 //<<<<>>>><<>><><<>><<<*>>><<>><><<>><<<<>>>>\\
 
-impl TypeDict {
-    pub fn unparse(&self, t: &TypeTerm) -> String {
+pub trait UnparseLadderType {
+    fn unparse(&self, t: &TypeTerm) -> String;
+}
+
+impl<T: TypeDict> UnparseLadderType for T {
+    fn unparse(&self, t: &TypeTerm) -> String {
         match t {
             TypeTerm::TypeID(id) => self.get_typename(id).unwrap(),
             TypeTerm::Num(n) => format!("{}", n),