diff --git a/executor/src/witgen/jit/cell.rs b/executor/src/witgen/jit/cell.rs
new file mode 100644
index 0000000000..090e277048
--- /dev/null
+++ b/executor/src/witgen/jit/cell.rs
@@ -0,0 +1,59 @@
+use std::{
+    fmt::{self, Display, Formatter},
+    hash::{Hash, Hasher},
+};
+
+use powdr_ast::analyzed::AlgebraicReference;
+
+/// The identifier of a witness cell in the trace table.
+/// The `row_offset` is relative to a certain "zero row" defined
+/// by the component that uses this data structure.
+#[derive(Debug, Clone, Eq)]
+pub struct Cell {
+    /// Name of the column, used only for display purposes.
+    pub column_name: String,
+    pub id: u64,
+    pub row_offset: i32,
+}
+
+impl Hash for Cell {
+    fn hash<H: Hasher>(&self, state: &mut H) {
+        self.id.hash(state);
+        self.row_offset.hash(state);
+    }
+}
+
+impl PartialEq for Cell {
+    fn eq(&self, other: &Self) -> bool {
+        self.id == other.id && self.row_offset == other.row_offset
+    }
+}
+
+impl Ord for Cell {
+    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
+        (self.id, self.row_offset).cmp(&(other.id, other.row_offset))
+    }
+}
+
+impl PartialOrd for Cell {
+    fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
+        Some(self.cmp(other))
+    }
+}
+
+impl Cell {
+    pub fn from_reference(r: &AlgebraicReference, row_offset: i32) -> Self {
+        assert!(r.is_witness());
+        Self {
+            column_name: r.name.clone(),
+            id: r.poly_id.id,
+            row_offset: r.next as i32 + row_offset,
+        }
+    }
+}
+
+impl Display for Cell {
+    fn fmt(&self, f: &mut Formatter) -> fmt::Result {
+        write!(f, "{}[{}]", self.column_name, self.row_offset)
+    }
+}
diff --git a/executor/src/witgen/jit/mod.rs b/executor/src/witgen/jit/mod.rs
index a6d9dabf39..c3151f1638 100644
--- a/executor/src/witgen/jit/mod.rs
+++ b/executor/src/witgen/jit/mod.rs
@@ -1,2 +1,4 @@
-mod affine_symbolic_expression;
+pub(crate) mod affine_symbolic_expression;
+mod cell;
 mod symbolic_expression;
+mod witgen_inference;
diff --git a/executor/src/witgen/jit/witgen_inference.rs b/executor/src/witgen/jit/witgen_inference.rs
new file mode 100644
index 0000000000..c395654dd4
--- /dev/null
+++ b/executor/src/witgen/jit/witgen_inference.rs
@@ -0,0 +1,289 @@
+use std::{
+    collections::{BTreeSet, HashMap, HashSet},
+    iter::once,
+};
+
+use itertools::Itertools;
+use powdr_ast::{
+    analyzed::{
+        AlgebraicBinaryOperation, AlgebraicBinaryOperator, AlgebraicExpression as Expression,
+        AlgebraicReference, AlgebraicUnaryOperation, AlgebraicUnaryOperator, Identity,
+        LookupIdentity, PhantomLookupIdentity, PolyID, PolynomialIdentity, PolynomialType,
+        SelectedExpressions,
+    },
+    indent,
+    parsed::visitor::AllChildren,
+};
+use powdr_number::FieldElement;
+
+use crate::witgen::global_constraints::RangeConstraintSet;
+
+use super::{
+    super::{machines::MachineParts, range_constraints::RangeConstraint, FixedData},
+    affine_symbolic_expression::{AffineSymbolicExpression, Effect},
+    cell::Cell,
+    symbolic_expression::SymbolicExpression,
+};
+
+/// This component can generate code that solves identities.
+/// It needs a driver that tells it which identities to process on which rows.
+pub struct WitgenInference<'a, T: FieldElement> {
+    fixed_data: &'a FixedData<'a, T>,
+    range_constraints: HashMap<Cell, RangeConstraint<T>>,
+    known_cells: HashSet<Cell>,
+    code: Vec<String>, // TODO make this a proper expression
+}
+
+impl<'a, T: FieldElement> WitgenInference<'a, T> {
+    pub fn new(
+        fixed_data: &'a FixedData<'a, T>,
+        known_cells: impl IntoIterator<Item = Cell>,
+    ) -> Self {
+        Self {
+            fixed_data,
+            range_constraints: Default::default(),
+            known_cells: known_cells.into_iter().collect(),
+            code: Default::default(),
+        }
+    }
+
+    fn cell_at_row(&self, id: u64, row_offset: i32) -> Cell {
+        let poly_id = PolyID {
+            id,
+            ptype: PolynomialType::Committed,
+        };
+        Cell {
+            column_name: self.fixed_data.column_name(&poly_id).to_string(),
+            id,
+            row_offset,
+        }
+    }
+
+    fn known_cells(&self) -> &HashSet<Cell> {
+        &self.known_cells
+    }
+
+    pub fn process_identity(&mut self, id: &Identity<T>, row_offset: i32) {
+        let effects = match id {
+            Identity::Polynomial(PolynomialIdentity { expression, .. }) => {
+                self.process_polynomial_identity(expression, row_offset)
+            }
+            Identity::Lookup(LookupIdentity {
+                id,
+                source: _,
+                left,
+                right,
+            })
+            | Identity::PhantomLookup(PhantomLookupIdentity {
+                id,
+                source: _,
+                left,
+                right,
+                multiplicity: _,
+            }) => {
+                // TODO multiplicity?
+                self.process_lookup(*id, left, right, row_offset)
+            }
+            _ => {
+                // TODO
+                vec![]
+            }
+        };
+        self.ingest_effects(effects);
+    }
+
+    fn process_polynomial_identity(
+        &self,
+        expression: &'a Expression<T>,
+        offset: i32,
+    ) -> Vec<Effect<T, Cell>> {
+        if let Some(r) = self.evaluate(expression, offset) {
+            // TODO remove unwrap
+            r.solve(self).unwrap()
+        } else {
+            vec![]
+        }
+    }
+
+    fn process_lookup(
+        &self,
+        lookup_id: u64,
+        left: &SelectedExpressions<T>,
+        right: &SelectedExpressions<T>,
+        offset: i32,
+    ) -> Vec<Effect<T, Cell>> {
+        // TODO: In the future, call the 'mutable state' to check if the
+        // lookup can always be answered.
+
+        // If the RHS is fully fixed columns...
+        if right.expressions.iter().all(|e| match e {
+            Expression::Reference(r) => r.is_fixed(),
+            Expression::Number(_) => true,
+            _ => false,
+        }) {
+            // and the selector is known to be 1 and all except one expression is known on the LHS.
+            if self
+                .evaluate(&left.selector, offset)
+                .map(|s| s.is_known_one())
+                == Some(true)
+            {
+                if let Some(inputs) = left
+                    .expressions
+                    .iter()
+                    .map(|e| self.evaluate(e, offset))
+                    .collect::<Option<Vec<_>>>()
+                {
+                    if inputs.iter().filter(|i| i.is_known()).count() == inputs.len() - 1 {
+                        let mut var_decl = String::new();
+                        let mut output_var = String::new();
+                        let query = inputs
+                            .iter()
+                            .enumerate()
+                            .map(|(i, e)| {
+                                if e.is_known() {
+                                    format!("LookupCell::Input(&({e}))")
+                                } else {
+                                    let var_name = format!("lookup_{lookup_id}_{i}");
+                                    output_var = var_name.clone();
+                                    var_decl.push_str(&format!(
+                                        "let mut {var_name}: FieldElement = Default::default();"
+                                    ));
+                                    format!("LookupCell::Output(&mut {var_name})")
+                                }
+                            })
+                            .format(", ");
+                        let machine_call = format!(
+                            "assert!(process_lookup(mutable_state, {lookup_id}, &mut [{query}]));"
+                        );
+                        // TODO range constraints?
+                        let output_expr = inputs.iter().find(|i| !i.is_known()).unwrap();
+                        return once(Effect::Code(var_decl))
+                            .chain(once(Effect::Code(machine_call)))
+                            .chain(
+                                (output_expr
+                                    - &KnownValue::from_known_local_var(&output_var).into())
+                                    .solve(self),
+                            )
+                            .collect();
+                    }
+                }
+            }
+        }
+        vec![]
+    }
+
+    fn ingest_effects(&mut self, effects: Vec<Effect<T>>) {
+        for e in effects {
+            match e {
+                Effect::Assignment(cell, assignment) => {
+                    // TODO also use raneg constraint?
+                    self.known_cells.insert(cell.clone());
+                    self.code.push(assignment);
+                }
+                Effect::RangeConstraint(cell, rc) => {
+                    self.add_range_constraint(cell, rc);
+                }
+                Effect::Code(code) => {
+                    self.code.push(code);
+                }
+            }
+        }
+    }
+
+    fn add_range_constraint(&mut self, cell: Cell, rc: RangeConstraint<T>) {
+        let rc = self
+            .range_constraint(cell.clone())
+            .map_or(rc.clone(), |existing_rc| existing_rc.conjunction(&rc));
+        // TODO if the conjuntion results in a single value, make the cell known.
+        self.range_constraints.insert(cell, rc);
+    }
+
+    fn evaluate(
+        &self,
+        expr: &Expression<T>,
+        offset: i32,
+    ) -> Option<AffineSymbolicExpression<T, Cell>> {
+        Some(match expr {
+            Expression::Reference(r) => {
+                if r.is_fixed() {
+                    todo!()
+                    // let mut row = self.latch_row as i64 + offset as i64;
+                    // while row < 0 {
+                    //     row += self.block_size as i64;
+                    // }
+                    // // TODO at some point we should check that all of the fixed columns are periodic.
+                    // // TODO We can only do this for block machines.
+                    // // For dynamic machines, fixed columns are "known but symbolic"
+                    // let v = self.fixed_data.fixed_cols[&r.poly_id].values_max_size()[row as usize];
+                    // EvalResult::from_number(v)
+                } else {
+                    let cell = Cell::from_reference(r, offset);
+                    // If a cell is known and has a compile-time constant value,
+                    // that value is stored in the range constraints.
+                    if let Some(v) = self
+                        .range_constraint(cell.clone())
+                        .and_then(|rc| rc.try_to_single_value())
+                    {
+                        AffineSymbolicExpression::from_number(v)
+                    } else if self.known_cells.contains(&cell) {
+                        AffineSymbolicExpression::from_known_variable(cell)
+                    } else {
+                        AffineSymbolicExpression::from_unknown_variable(cell)
+                    }
+                }
+            }
+            Expression::PublicReference(_) => return None, // TODO
+            Expression::Challenge(_) => return None,       // TODO
+            Expression::Number(n) => AffineSymbolicExpression::from_number(*n),
+            Expression::BinaryOperation(op) => self.evaulate_binary_operation(op, offset)?,
+            Expression::UnaryOperation(op) => self.evaluate_unary_operation(op, offset)?,
+        })
+    }
+
+    fn evaulate_binary_operation(
+        &self,
+        op: &AlgebraicBinaryOperation<T>,
+        offset: i32,
+    ) -> Option<AffineSymbolicExpression<T, Cell>> {
+        let left = self.evaluate(&op.left, offset)?;
+        let right = self.evaluate(&op.right, offset)?;
+        match op.op {
+            AlgebraicBinaryOperator::Add => Some(&left + &right),
+            AlgebraicBinaryOperator::Sub => Some(&left - &right),
+            AlgebraicBinaryOperator::Mul => left.try_mul(&right),
+            _ => todo!(),
+        }
+    }
+
+    fn evaluate_unary_operation(
+        &self,
+        op: &AlgebraicUnaryOperation<T>,
+        offset: i32,
+    ) -> Option<AffineSymbolicExpression<T, Cell>> {
+        let expr = self.evaluate(&op.expr, offset)?;
+        match op.op {
+            AlgebraicUnaryOperator::Minus => Some(-&expr),
+        }
+    }
+}
+
+impl<T: FieldElement> RangeConstraintSet<Cell, T> for WitgenInference<'_, T> {
+    // TODO would be nice to use &Cell, but this leads to lifetime trouble
+    // in the solve() function.
+    fn range_constraint(&self, cell: Cell) -> Option<RangeConstraint<T>> {
+        self.fixed_data
+            .global_range_constraints
+            .range_constraint(&AlgebraicReference {
+                name: Default::default(),
+                poly_id: PolyID {
+                    id: cell.id,
+                    ptype: PolynomialType::Committed,
+                },
+                next: false,
+            })
+            .iter()
+            .chain(self.range_constraints.get(&cell))
+            .cloned()
+            .reduce(|gc, rc| gc.conjunction(&rc))
+    }
+}
diff --git a/executor/src/witgen/range_constraints.rs b/executor/src/witgen/range_constraints.rs
index 8e6d9748a8..15b99e9eca 100644
--- a/executor/src/witgen/range_constraints.rs
+++ b/executor/src/witgen/range_constraints.rs
@@ -157,6 +157,14 @@ impl<T: FieldElement> RangeConstraint<T> {
             mask: mask.unwrap_or_else(|| Self::from_range(min, max).mask),
         }
     }
+
+    pub fn try_to_single_value(&self) -> Option<T> {
+        if self.min == self.max {
+            Some(self.min)
+        } else {
+            None
+        }
+    }
 }
 
 /// The number of elements in an (inclusive) min/max range.