diff --git a/examples/dag.rs b/examples/dag.rs
index a5d188c..4ef9222 100644
--- a/examples/dag.rs
+++ b/examples/dag.rs
@@ -1,11 +1,12 @@
 use std::cell::RefCell;
 use std::rc::Rc;
 
-use circuit_cas::circuit::dag::{Circuit, CircuitExt};
+use circuit_cas::circuit::traits::Circuit;
+use circuit_cas::circuit::dag::{ProbCircuit, CircuitExt};
 use circuit_cas::var;
 
 fn main() {
-    let circuit = Rc::new(RefCell::new(Circuit::new()));
+    let mut circuit = ProbCircuit::new();
 
     // Build (x + y) * (x + z)
     let x = circuit.leaf(var!("x"));
diff --git a/src/circuit/dag.rs b/src/circuit/dag.rs
index dc2d4cf..ac5c335 100644
--- a/src/circuit/dag.rs
+++ b/src/circuit/dag.rs
@@ -1,57 +1,27 @@
 use slotmap::{SlotMap, new_key_type};
-use std::cell::RefCell;
 use std::collections::HashMap;
-use std::ops::{Add, Mul};
-use std::rc::Rc;
 
-use crate::poly::var::Var;
+use super::traits::{Circuit,Node};
 
 new_key_type! { pub struct NodeId; }
 
-#[derive(Clone, Debug, PartialEq, Eq, Hash)]
-pub enum Node<V: Var> {
-    Leaf(V),
-    Scale(NodeId, i32),
-    Sum(NodeId, NodeId),
-    Prod(NodeId, NodeId),
-}
-
-impl<V: Var> Node<V> {
-    pub fn children(&self) -> impl Iterator<Item = NodeId> {
-        match self {
-            Node::Leaf(_) => [None, None],
-            Node::Scale(n, _) => [Some(*n), None],
-            Node::Sum(l, r) | Node::Prod(l, r) => [Some(*l), Some(*r)],
-        }
-        .into_iter()
-        .flatten()
-    }
-}
-
 #[derive(Clone, Debug)]
-pub struct Circuit<V: Var> {
-    nodes: SlotMap<NodeId, Node<V>>,
-    intern: HashMap<Node<V>, NodeId>,
+pub struct Dag<N: Node> {
+    nodes: SlotMap<NodeId, N>,
+    intern: HashMap<N, NodeId>,
 }
 
-impl<V: Var> Default for Circuit<V> {
+impl<N: Node> Default for Dag<N> {
     fn default() -> Self {
-        Circuit {
+        Dag {
             nodes: Default::default(),
             intern: Default::default(),
         }
     }
 }
 
-impl<V: Var> Circuit<V> {
-    pub fn new() -> Self {
-        Circuit {
-            nodes: SlotMap::with_key(),
-            intern: HashMap::new(),
-        }
-    }
-
-    pub fn node(&mut self, n: Node<V>) -> NodeId {
+impl<N:Node> Dag<N>{
+    fn node(&mut self, n: N) -> NodeId {
         if let Some(&id) = self.intern.get(&n) {
             return id;
         }
@@ -60,90 +30,44 @@ impl<V: Var> Circuit<V> {
         id
     }
 
-    pub fn get(&self, id: NodeId) -> Option<&Node<V>> {
+    fn get(&self, id: NodeId) -> Option<&N> {
         self.nodes.get(id)
     }
 
-    pub fn add(&mut self, left: NodeId, right: NodeId) -> NodeId {
-        let (l, r) = if left <= right {
-            (left, right)
-        } else {
-            (right, left)
-        };
-        self.node(Node::Sum(l, r))
-    }
-
-    pub fn mul(&mut self, left: NodeId, right: NodeId) -> NodeId {
-        let (l, r) = if left <= right {
-            (left, right)
-        } else {
-            (right, left)
-        };
-        self.node(Node::Prod(l, r))
-    }
-
-    pub fn len(&self) -> usize {
+    fn len(&self) -> usize {
         self.nodes.len()
     }
 
-    pub fn children(&self, id: NodeId) -> impl Iterator<Item = NodeId> + '_ {
+    fn children(&self, id: NodeId) -> impl Iterator<Item = NodeId> + '_ {
         self.nodes.get(id).into_iter().flat_map(Node::children)
     }
 
-    pub fn remove(&mut self, id: NodeId) {
+    fn remove(&mut self, id: NodeId) {
         if let Some(node) = self.nodes.remove(id) {
             self.intern.remove(&node);
         }
     }
 }
 
-pub struct CircuitNode<V: Var> {
-    pub id: NodeId,
-    circuit: Rc<RefCell<Circuit<V>>>,
-}
-pub trait CircuitExt<V: Var> {
-    fn leaf(&self, v: V) -> CircuitNode<V>;
-    fn get_node(&self, id: NodeId) -> Option<CircuitNode<V>>;
+pub struct RefNode<T:DerefMut<Target=Dag>>{
+    pub id:NodeId,
+    circuit: Rc<RefCell<T>>
 }
 
-impl<V: Var> CircuitExt<V> for Rc<RefCell<Circuit<V>>> {
-    fn leaf(&self, v: V) -> CircuitNode<V> {
-        let id = self.borrow_mut().node(Node::Leaf(v));
-        CircuitNode {
+impl<C:Circuit> RefNode<C>{
+    fn leaf(&mut self, variable: C::Var)->Self{
+        let mut c = self.circuit.borrow_mut();
+        let id = c.leaf(variable);
+        RefNode {
             id,
-            circuit: self.clone(),
+            circuit: self.circuit.clone(),
         }
     }
-
-    fn get_node(&self, id: NodeId) -> Option<CircuitNode<V>> {
-        self.borrow().get(id)?;
-        Some(CircuitNode {
+    fn get_node(&self, id:NodeId)->Option<Self>{
+        self.circuit.borrow().get(id)?;
+        Some(RefNode {
             id,
-            circuit: self.clone(),
+            circuit: self.circuit.clone(),
         })
     }
 }
-
-impl<V: Var> Add for CircuitNode<V> {
-    type Output = Self;
-
-    fn add(self, rhs: Self) -> Self {
-        let id = self.circuit.borrow_mut().add(self.id, rhs.id);
-        CircuitNode {
-            id,
-            circuit: self.circuit,
-        }
-    }
-}
-
-impl<V: Var> Mul for CircuitNode<V> {
-    type Output = Self;
-
-    fn mul(self, rhs: Self) -> Self {
-        let id = self.circuit.borrow_mut().mul(self.id, rhs.id);
-        CircuitNode {
-            id,
-            circuit: self.circuit,
-        }
-    }
-}
diff --git a/src/circuit/mod.rs b/src/circuit/mod.rs
index e0e3241..efe1130 100644
--- a/src/circuit/mod.rs
+++ b/src/circuit/mod.rs
@@ -1,4 +1,6 @@
+pub mod traits;
 pub mod dag;
+pub mod probabilistic;
 pub mod quotient;
 
 #[cfg(test)]
diff --git a/src/circuit/probabilistic.rs b/src/circuit/probabilistic.rs
new file mode 100644
index 0000000..b0e7652
--- /dev/null
+++ b/src/circuit/probabilistic.rs
@@ -0,0 +1,79 @@
+use std::ops::{Deref, DerefMut};
+use std::rc::Rc;
+
+use crate::poly::var::Var;
+use super::dag::{Dag,NodeId};
+use super::traits::{Circuit,Node,RefNode};
+
+
+#[derive(Clone, Debug, PartialEq, Eq, Hash)]
+pub enum PNode<V: Var> {
+    Leaf(V),
+    Sum(NodeId, NodeId),
+    Prod(NodeId, NodeId),
+}
+
+
+impl<V:Var> Node for PNode<V>{
+    fn children(&self) -> impl Iterator<Item = NodeId> {
+        match self {
+            Self::Leaf(_) => [None, None],
+            Self::Sum(l, r) | Self::Prod(l, r) => [Some(*l), Some(*r)],
+        }
+        .into_iter()
+        .flatten()
+    }
+}
+
+#[derive(Clone, Debug, Default)]
+pub struct ProbCircuit<V: Var> {
+    dag: Dag<PNode<V>>,
+}
+
+impl<V:Var> Deref for ProbCircuit<V>{
+    type Target = Dag<PNode<V>>;
+    fn deref(&self)->&Self::Target{
+        &self.dag
+    }
+}
+
+impl<V:Var> DerefMut for ProbCircuit<V>{
+    fn deref_mut(&mut self)->&mut Self::Target{
+        &mut self.dag
+    }
+}
+
+
+impl<V: Var> Rc<ProbCircuit<V>> {
+    pub fn var<T:Into<V>>(variable:T)->RefNode<Self>{
+        todo!()
+    }
+}
+
+impl<V:Var> Circuit for ProbCircuit<V>{
+    type Node=PNode<V>;
+    type Var=V;
+
+    fn leaf(&mut self, variable:V)->NodeId{
+        self.node(Self::Node::Leaf(variable))
+    }
+
+    fn add(&mut self, left: NodeId, right: NodeId) -> NodeId {
+        let (l, r) = if left <= right {
+            (left, right)
+        } else {
+            (right, left)
+        };
+        self.node(Self::Node::Sum(l, r))
+    }
+
+    fn mul(&mut self, left: NodeId, right: NodeId) -> NodeId {
+        let (l, r) = if left <= right {
+            (left, right)
+        } else {
+            (right, left)
+        };
+        self.node(Self::Node::Prod(l, r))
+    }
+}
+
diff --git a/src/circuit/quotient.rs b/src/circuit/quotient.rs
index f5c86c4..dcd2b56 100644
--- a/src/circuit/quotient.rs
+++ b/src/circuit/quotient.rs
@@ -1,52 +1,94 @@
-use super::dag::{Circuit, Node, NodeId};
-use crate::poly::{
-    flat::Poly,
-    ideal::{Generators, GroebnerBasis, Ideal},
-    var::Var,
-};
+use std::ops::{Deref, DerefMut};
 
-use std::fmt::{self, Display};
+use crate::poly::var::Var;
+use crate::poly::flat::Poly;
+use crate::poly::ideal::{Generators, GroebnerBasis, Ideal};
+use super::dag::{Dag,NodeId};
+use super::traits::{Circuit,Node};
+
+
+#[derive(Clone, Debug, PartialEq, Eq, Hash)]
+pub enum QNode<V: Var> {
+    Leaf(V),
+    Sum(NodeId, NodeId),
+    Prod(NodeId, NodeId),
+    DivStep(NodeId, NodeId)
+}
+
+
+impl<V:Var> Node for QNode<V>{
+    fn children(&self) -> impl Iterator<Item = NodeId> {
+        match self {
+            Self::Leaf(_) => [None, None],
+            Self::Sum(l, r) | Self::Prod(l, r) | Self::DivStep(l,r) => [Some(*l), Some(*r)],
+        }
+        .into_iter()
+        .flatten()
+    }
+}
 
 #[derive(Clone, Debug)]
-pub struct Quotient<V: Var> {
+pub struct QuotientCircuit<V: Var> {
     basis: Ideal<V, GroebnerBasis>,
-    circuit: Circuit<V>,
+    dag: Dag<QNode<V>>,
 }
 
-impl<V: Var> From<Ideal<V, GroebnerBasis>> for Quotient<V> {
+impl<V: Var> From<Ideal<V, GroebnerBasis>> for QuotientCircuit<V> {
     fn from(basis: Ideal<V, GroebnerBasis>) -> Self {
-        Quotient {
+        Self {
             basis,
-            circuit: Default::default(),
+            dag: Default::default(),
         }
     }
 }
 
-impl<V: Var> FromIterator<Poly<V>> for Quotient<V> {
+impl<V: Var> FromIterator<Poly<V>> for QuotientCircuit<V> {
     fn from_iter<T: IntoIterator<Item = Poly<V>>>(iter: T) -> Self {
         let ideal: Ideal<V, Generators> = iter.into_iter().collect();
-        Quotient {
+        Self {
             basis: ideal.groebner_basis(),
-            circuit: Default::default(),
+            dag: Default::default(),
         }
     }
 }
 
-impl<V: Var> Display for Quotient<V> {
-    fn fmt(&self, fmt: &mut fmt::Formatter<'_>) -> Result<(), fmt::Error> {
-        write!(fmt, "C/{}", self.basis)
+impl<V:Var> Deref for QuotientCircuit<V>{
+    type Target = Dag<QNode<V>>;
+    fn deref(&self)->&Self::Target{
+        &self.dag
     }
 }
 
-impl<V: Var> Quotient<V> {
-    pub fn node(&mut self, n: Node<V>) -> NodeId {
-        self.circuit.node(n)
-    }
-
-    pub fn add(&mut self, left: NodeId, right: NodeId) -> NodeId {
-        self.circuit.add(left, right)
-    }
-    pub fn mul(&mut self, left: NodeId, right: NodeId) -> NodeId {
-        self.circuit.mul(left, right)
+impl<V:Var> DerefMut for QuotientCircuit<V>{
+    fn deref_mut(&mut self)->&mut Self::Target{
+        &mut self.dag
     }
 }
+
+impl<V:Var> Circuit for QuotientCircuit<V>{
+    type Node=QNode<V>;
+    type Var=V;
+
+    fn leaf(&mut self, variable:V)->NodeId{
+        self.node(Self::Node::Leaf(variable))
+    }
+
+    fn add(&mut self, left: NodeId, right: NodeId) -> NodeId {
+        let (l, r) = if left <= right {
+            (left, right)
+        } else {
+            (right, left)
+        };
+        self.node(Self::Node::Sum(l, r))
+    }
+
+    fn mul(&mut self, left: NodeId, right: NodeId) -> NodeId {
+        let (l, r) = if left <= right {
+            (left, right)
+        } else {
+            (right, left)
+        };
+        self.node(Self::Node::Prod(l, r))
+    }
+}
+
diff --git a/src/circuit/tests.rs b/src/circuit/tests.rs
index 201dfa7..158614d 100644
--- a/src/circuit/tests.rs
+++ b/src/circuit/tests.rs
@@ -1,12 +1,13 @@
 use std::cell::RefCell;
 use std::rc::Rc;
 
-use super::dag::{Circuit, CircuitExt};
+use super::dag::{ProbCircuit, CircuitExt};
+use super::traits::Circuit;
 use crate::poly::var::StaticVar;
 
 #[test]
 fn test_deduplication() {
-    let circuit = Rc::new(RefCell::new(Circuit::new()));
+    let mut circuit = ProbCircuit::new();
 
     // Same leaf constructed twice returns the same NodeId
     let x1 = circuit.leaf(StaticVar::from("x"));
@@ -48,5 +49,5 @@ fn test_deduplication() {
     assert_eq!(xyz1.id, xyz2.id);
     let _sum = xyz1 + xyz2;
     // x, y, z, x+y(==y+x), (x+y)*z, (x+y)+z, y+z, x+(y+z), (x+y)*z+(x+y)*z, sq
-    assert_eq!(circuit.borrow().len(), 10);
+    assert_eq!(circuit.len(), 10);
 }
diff --git a/src/circuit/traits.rs b/src/circuit/traits.rs
new file mode 100644
index 0000000..75b7bf0
--- /dev/null
+++ b/src/circuit/traits.rs
@@ -0,0 +1,51 @@
+use std::hash::Hash;
+use std::ops::{Add,Mul};
+use std::{rc::Rc,cell::RefCell};
+
+use super::dag::NodeId;
+
+pub trait Node:Clone+PartialEq+Eq+Hash{
+    fn children(&self) -> impl Iterator<Item = NodeId>;
+}
+
+pub trait Circuit:Clone{
+    type Var;
+    type Node;
+    
+    fn node(&mut self, n:Self::Node)->NodeId;
+    fn remove(&mut self, id:NodeId);
+    fn get(&self, id:NodeId)->Option<&Self::Node>;
+
+    fn leaf(&mut self, var: Self::Var)->NodeId;
+    fn add(&mut self, left: NodeId, right: NodeId)->NodeId;
+    fn mul(&mut self, left: NodeId, right: NodeId)->NodeId;
+    
+    fn len(&self)->usize;
+    fn children(&self, id:NodeId)->impl Iterator<Item=NodeId>+'_;
+}
+
+
+impl<C:Circuit> Add for RefNode<C> {
+    type Output = Self;
+
+    fn add(self, rhs: Self) -> Self {
+        let id = self.circuit.borrow_mut().add(self.id, rhs.id);
+        RefNode {
+            id,
+            circuit: self.circuit,
+        }
+    }
+}
+
+impl<C:Circuit> Mul for RefNode<C> {
+    type Output = Self;
+
+    fn mul(self, rhs: Self) -> Self {
+        let id = self.circuit.borrow_mut().mul(self.id, rhs.id);
+        RefNode {
+            id,
+            circuit: self.circuit,
+        }
+    }
+}
+