diff --git a/executor/src/witgen/generator.rs b/executor/src/witgen/generator.rs
index 8a48941f8..ad29ac0ce 100644
--- a/executor/src/witgen/generator.rs
+++ b/executor/src/witgen/generator.rs
@@ -333,7 +333,7 @@ where
     fn process_polynomial_identity(&self, identity: &Expression) -> EvalResult {
         // If there is no "next" reference in the expression,
         // we just evaluate it directly on the "next" row.
-        let row = if contains_next_witness_ref(identity, self.fixed_data) {
+        let row = if contains_next_witness_ref(identity) {
             // TODO this is the only situation where we use "current"
             // TODO this is the only that actually uses a window.
             EvaluationRow::Current
diff --git a/executor/src/witgen/machines/fixed_lookup_machine.rs b/executor/src/witgen/machines/fixed_lookup_machine.rs
index 254a184e7..374326ab2 100644
--- a/executor/src/witgen/machines/fixed_lookup_machine.rs
+++ b/executor/src/witgen/machines/fixed_lookup_machine.rs
@@ -178,10 +178,7 @@ impl FixedLookup {
         // This is a matching machine if it is a plookup and the RHS is fully constant.
         if kind != IdentityKind::Plookup
             || right.selector.is_some()
-            || right
-                .expressions
-                .iter()
-                .any(|e| contains_witness_ref(e, fixed_data))
+            || right.expressions.iter().any(contains_witness_ref)
         {
             return None;
         }
diff --git a/executor/src/witgen/util.rs b/executor/src/witgen/util.rs
index 6654590c2..635eec622 100644
--- a/executor/src/witgen/util.rs
+++ b/executor/src/witgen/util.rs
@@ -1,7 +1,5 @@
 use pil_analyzer::{util::expr_any, Expression, PolynomialReference};
 
-use super::FixedData;
-
 pub trait WitnessColumnNamer {
     fn name(&self, i: usize) -> String;
 }
@@ -16,21 +14,17 @@ pub fn contains_next_ref(expr: &Expression) -> bool {
 }
 
 /// @returns true if the expression contains a reference to a next value of a witness column.
-pub fn contains_next_witness_ref(expr: &Expression, fixed_data: &FixedData) -> bool {
+pub fn contains_next_witness_ref(expr: &Expression) -> bool {
     expr_any(expr, |e| match e {
-        Expression::PolynomialReference(poly) => {
-            poly.next && fixed_data.witness_ids.contains_key(poly.name.as_str())
-        }
+        Expression::PolynomialReference(poly) => poly.next && poly.is_witness(),
         _ => false,
     })
 }
 
 /// @returns true if the expression contains a reference to a witness column.
-pub fn contains_witness_ref(expr: &Expression, fixed_data: &FixedData) -> bool {
+pub fn contains_witness_ref(expr: &Expression) -> bool {
     expr_any(expr, |e| match e {
-        Expression::PolynomialReference(poly) => {
-            fixed_data.witness_ids.contains_key(poly.name.as_str())
-        }
+        Expression::PolynomialReference(poly) => poly.is_witness(),
         _ => false,
     })
 }
diff --git a/pil_analyzer/src/lib.rs b/pil_analyzer/src/lib.rs
index a75921c23..42cc47696 100644
--- a/pil_analyzer/src/lib.rs
+++ b/pil_analyzer/src/lib.rs
@@ -197,6 +197,21 @@ pub struct PolynomialReference {
     pub next: bool,
 }
 
+impl PolynomialReference {
+    #[inline]
+    pub fn poly_id(&self) -> u64 {
+        self.poly_id.unwrap().0
+    }
+    #[inline]
+    pub fn is_witness(&self) -> bool {
+        self.poly_id.unwrap().1 == PolynomialType::Committed
+    }
+    #[inline]
+    pub fn is_fixed(&self) -> bool {
+        self.poly_id.unwrap().1 == PolynomialType::Constant
+    }
+}
+
 impl PartialOrd for PolynomialReference {
     fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
         // TODO for efficiency reasons, we should avoid the unwrap check here somehow.