diff --git a/mlir/include/mlir/Analysis/DataFlow/ConstantPropagationAnalysis.h b/mlir/include/mlir/Analysis/DataFlow/ConstantPropagationAnalysis.h
--- a/mlir/include/mlir/Analysis/DataFlow/ConstantPropagationAnalysis.h
+++ b/mlir/include/mlir/Analysis/DataFlow/ConstantPropagationAnalysis.h
@@ -28,15 +28,24 @@
 /// This lattice value represents a known constant value of a lattice.
 class ConstantValue {
 public:
+  /// Construct a constant value as uninitialized.
+  explicit ConstantValue() = default;
+
   /// Construct a constant value with a known constant.
-  ConstantValue(Attribute knownValue = {}, Dialect *dialect = nullptr)
-      : constant(knownValue), dialect(dialect) {}
+  explicit ConstantValue(Attribute constant, Dialect *dialect)
+      : constant(constant), dialect(dialect) {}
 
   /// Get the constant value. Returns null if no value was determined.
-  Attribute getConstantValue() const { return constant; }
+  Attribute getConstantValue() const {
+    assert(!isUninitialized());
+    return *constant;
+  }
 
   /// Get the dialect instance that can be used to materialize the constant.
-  Dialect *getConstantDialect() const { return dialect; }
+  Dialect *getConstantDialect() const {
+    assert(!isUninitialized());
+    return dialect;
+  }
 
   /// Compare the constant values.
   bool operator==(const ConstantValue &rhs) const {
@@ -46,20 +55,35 @@
   /// Print the constant value.
   void print(raw_ostream &os) const;
 
+  /// The state where the constant value is uninitialized. This happens when the
+  /// state hasn't been set during the analysis.
+  static ConstantValue getUninitialized() { return ConstantValue{}; }
+
+  /// Whether the state is uninitialized.
+  bool isUninitialized() const { return !constant.has_value(); }
+
   /// The state where the constant value is unknown.
-  static ConstantValue getUnknownConstant() { return {}; }
+  static ConstantValue getUnknownConstant() {
+    return ConstantValue{/*constant=*/nullptr, /*dialect=*/nullptr};
+  }
 
   /// The union with another constant value is null if they are different, and
   /// the same if they are the same.
   static ConstantValue join(const ConstantValue &lhs,
                             const ConstantValue &rhs) {
-    return lhs == rhs ? lhs : ConstantValue();
+    if (lhs.isUninitialized())
+      return rhs;
+    if (rhs.isUninitialized())
+      return lhs;
+    if (lhs == rhs)
+      return lhs;
+    return getUnknownConstant();
   }
 
 private:
   /// The constant value.
-  Attribute constant;
-  /// An dialect instance that can be used to materialize the constant.
+  Optional<Attribute> constant;
+  /// A dialect instance that can be used to materialize the constant.
   Dialect *dialect;
 };
 
diff --git a/mlir/include/mlir/Analysis/DataFlow/DeadCodeAnalysis.h b/mlir/include/mlir/Analysis/DataFlow/DeadCodeAnalysis.h
--- a/mlir/include/mlir/Analysis/DataFlow/DeadCodeAnalysis.h
+++ b/mlir/include/mlir/Analysis/DataFlow/DeadCodeAnalysis.h
@@ -38,9 +38,6 @@
 public:
   using AnalysisState::AnalysisState;
 
-  /// The state is initialized by default.
-  bool isUninitialized() const override { return false; }
-
   /// Set the state of the program point to live.
   ChangeResult setToLive();
 
@@ -95,9 +92,6 @@
 public:
   using AnalysisState::AnalysisState;
 
-  /// The state is initialized by default.
-  bool isUninitialized() const override { return false; }
-
   /// Print the known predecessors.
   void print(raw_ostream &os) const override;
 
diff --git a/mlir/include/mlir/Analysis/DataFlow/IntegerRangeAnalysis.h b/mlir/include/mlir/Analysis/DataFlow/IntegerRangeAnalysis.h
--- a/mlir/include/mlir/Analysis/DataFlow/IntegerRangeAnalysis.h
+++ b/mlir/include/mlir/Analysis/DataFlow/IntegerRangeAnalysis.h
@@ -30,10 +30,18 @@
   static IntegerValueRange getMaxRange(Value value);
 
   /// Create an integer value range lattice value.
-  IntegerValueRange(ConstantIntRanges value) : value(std::move(value)) {}
+  IntegerValueRange(Optional<ConstantIntRanges> value = None)
+      : value(std::move(value)) {}
+
+  /// Whether the range is uninitialized. This happens when the state hasn't
+  /// been set during the analysis.
+  bool isUninitialized() const { return !value.has_value(); }
 
   /// Get the known integer value range.
-  const ConstantIntRanges &getValue() const { return value; }
+  const ConstantIntRanges &getValue() const {
+    assert(!isUninitialized());
+    return *value;
+  }
 
   /// Compare two ranges.
   bool operator==(const IntegerValueRange &rhs) const {
@@ -43,7 +51,11 @@
   /// Take the union of two ranges.
   static IntegerValueRange join(const IntegerValueRange &lhs,
                                 const IntegerValueRange &rhs) {
-    return lhs.value.rangeUnion(rhs.value);
+    if (lhs.isUninitialized())
+      return rhs;
+    if (rhs.isUninitialized())
+      return lhs;
+    return IntegerValueRange{lhs.getValue().rangeUnion(rhs.getValue())};
   }
 
   /// Print the integer value range.
@@ -51,7 +63,7 @@
 
 private:
   /// The known integer value range.
-  ConstantIntRanges value;
+  Optional<ConstantIntRanges> value;
 };
 
 /// This lattice element represents the integer value range of an SSA value.
diff --git a/mlir/include/mlir/Analysis/DataFlow/SparseAnalysis.h b/mlir/include/mlir/Analysis/DataFlow/SparseAnalysis.h
--- a/mlir/include/mlir/Analysis/DataFlow/SparseAnalysis.h
+++ b/mlir/include/mlir/Analysis/DataFlow/SparseAnalysis.h
@@ -81,27 +81,17 @@
 
   /// Return the value held by this lattice. This requires that the value is
   /// initialized.
-  ValueT &getValue() {
-    assert(!isUninitialized() && "expected known lattice element");
-    return *value;
-  }
+  ValueT &getValue() { return value; }
   const ValueT &getValue() const {
     return const_cast<Lattice<ValueT> *>(this)->getValue();
   }
 
-  /// Returns true if the value of this lattice hasn't yet been initialized.
-  bool isUninitialized() const override { return !value.has_value(); }
-
   /// Join the information contained in the 'rhs' lattice into this
   /// lattice. Returns if the state of the current lattice changed.
   ChangeResult join(const AbstractSparseLattice &rhs) override {
     const Lattice<ValueT> &rhsLattice =
         static_cast<const Lattice<ValueT> &>(rhs);
 
-    // If rhs is uninitialized, there is nothing to do.
-    if (rhsLattice.isUninitialized())
-      return ChangeResult::NoChange;
-
     // Join the rhs value into this lattice.
     return join(rhsLattice.getValue());
   }
@@ -109,15 +99,9 @@
   /// Join the information contained in the 'rhs' value into this
   /// lattice. Returns if the state of the current lattice changed.
   ChangeResult join(const ValueT &rhs) {
-    // If the current lattice is uninitialized, copy the rhs value.
-    if (isUninitialized()) {
-      value = rhs;
-      return ChangeResult::Change;
-    }
-
     // Otherwise, join rhs with the current optimistic value.
-    ValueT newValue = ValueT::join(*value, rhs);
-    assert(ValueT::join(newValue, *value) == newValue &&
+    ValueT newValue = ValueT::join(value, rhs);
+    assert(ValueT::join(newValue, value) == newValue &&
            "expected `join` to be monotonic");
     assert(ValueT::join(newValue, rhs) == newValue &&
            "expected `join` to be monotonic");
@@ -131,17 +115,11 @@
   }
 
   /// Print the lattice element.
-  void print(raw_ostream &os) const override {
-    if (value)
-      value->print(os);
-    else
-      os << "<NULL>";
-  }
+  void print(raw_ostream &os) const override { value.print(os); }
 
 private:
-  /// The currently computed value that is optimistically assumed to be true,
-  /// or None if the lattice element is uninitialized.
-  Optional<ValueT> value;
+  /// The currently computed value that is optimistically assumed to be true.
+  ValueT value;
 };
 
 //===----------------------------------------------------------------------===//
diff --git a/mlir/include/mlir/Analysis/DataFlowFramework.h b/mlir/include/mlir/Analysis/DataFlowFramework.h
--- a/mlir/include/mlir/Analysis/DataFlowFramework.h
+++ b/mlir/include/mlir/Analysis/DataFlowFramework.h
@@ -291,9 +291,6 @@
   /// Returns the program point this static is located at.
   ProgramPoint getPoint() const { return point; }
 
-  /// Returns true if the analysis state is uninitialized.
-  virtual bool isUninitialized() const = 0;
-
   /// Print the contents of the analysis state.
   virtual void print(raw_ostream &os) const = 0;
 
diff --git a/mlir/lib/Analysis/DataFlow/ConstantPropagationAnalysis.cpp b/mlir/lib/Analysis/DataFlow/ConstantPropagationAnalysis.cpp
--- a/mlir/lib/Analysis/DataFlow/ConstantPropagationAnalysis.cpp
+++ b/mlir/lib/Analysis/DataFlow/ConstantPropagationAnalysis.cpp
@@ -20,9 +20,15 @@
 //===----------------------------------------------------------------------===//
 
 void ConstantValue::print(raw_ostream &os) const {
-  if (constant)
-    return constant.print(os);
-  os << "<NO VALUE>";
+  if (isUninitialized()) {
+    os << "<UNINITIALIZED>";
+    return;
+  }
+  if (getConstantValue() == nullptr) {
+    os << "<UNKNOWN>";
+    return;
+  }
+  return getConstantValue().print(os);
 }
 
 //===----------------------------------------------------------------------===//
@@ -45,8 +51,11 @@
 
   SmallVector<Attribute, 8> constantOperands;
   constantOperands.reserve(op->getNumOperands());
-  for (auto *operandLattice : operands)
+  for (auto *operandLattice : operands) {
+    if (operandLattice->getValue().isUninitialized())
+      return;
     constantOperands.push_back(operandLattice->getValue().getConstantValue());
+  }
 
   // Save the original operands and attributes just in case the operation
   // folds in-place. The constant passed in may not correspond to the real
diff --git a/mlir/lib/Analysis/DataFlow/DeadCodeAnalysis.cpp b/mlir/lib/Analysis/DataFlow/DeadCodeAnalysis.cpp
--- a/mlir/lib/Analysis/DataFlow/DeadCodeAnalysis.cpp
+++ b/mlir/lib/Analysis/DataFlow/DeadCodeAnalysis.cpp
@@ -318,7 +318,7 @@
   for (Value operand : op->getOperands()) {
     const Lattice<ConstantValue> *cv = getLattice(operand);
     // If any of the operands' values are uninitialized, bail out.
-    if (cv->isUninitialized())
+    if (cv->getValue().isUninitialized())
       return {};
     operands.push_back(cv->getValue().getConstantValue());
   }
diff --git a/mlir/lib/Analysis/DataFlow/DenseAnalysis.cpp b/mlir/lib/Analysis/DataFlow/DenseAnalysis.cpp
--- a/mlir/lib/Analysis/DataFlow/DenseAnalysis.cpp
+++ b/mlir/lib/Analysis/DataFlow/DenseAnalysis.cpp
@@ -74,9 +74,6 @@
     before = getLatticeFor(op, prev);
   else
     before = getLatticeFor(op, op->getBlock());
-  // If the incoming lattice is uninitialized, bail out.
-  if (before->isUninitialized())
-    return;
 
   // Invoke the operation transfer function.
   visitOperationImpl(op, *before, after);
diff --git a/mlir/lib/Analysis/DataFlow/IntegerRangeAnalysis.cpp b/mlir/lib/Analysis/DataFlow/IntegerRangeAnalysis.cpp
--- a/mlir/lib/Analysis/DataFlow/IntegerRangeAnalysis.cpp
+++ b/mlir/lib/Analysis/DataFlow/IntegerRangeAnalysis.cpp
@@ -29,7 +29,7 @@
   APInt umax = APInt::getMaxValue(width);
   APInt smin = width != 0 ? APInt::getSignedMinValue(width) : umin;
   APInt smax = width != 0 ? APInt::getSignedMaxValue(width) : umax;
-  return {{umin, umax, smin, smax}};
+  return IntegerValueRange{ConstantIntRanges{umin, umax, smin, smax}};
 }
 
 void IntegerValueRangeLattice::onUpdate(DataFlowSolver *solver) const {
@@ -57,6 +57,13 @@
 void IntegerRangeAnalysis::visitOperation(
     Operation *op, ArrayRef<const IntegerValueRangeLattice *> operands,
     ArrayRef<IntegerValueRangeLattice *> results) {
+  // If the lattice on any operand is unitialized, bail out.
+  if (llvm::any_of(operands, [](const IntegerValueRangeLattice *lattice) {
+        return lattice->getValue().isUninitialized();
+      })) {
+    return;
+  }
+
   // Ignore non-integer outputs - return early if the op has no scalar
   // integer results
   bool hasIntegerResult = false;
@@ -91,11 +98,9 @@
 
     LLVM_DEBUG(llvm::dbgs() << "Inferred range " << attrs << "\n");
     IntegerValueRangeLattice *lattice = results[result.getResultNumber()];
-    Optional<IntegerValueRange> oldRange;
-    if (!lattice->isUninitialized())
-      oldRange = lattice->getValue();
+    IntegerValueRange oldRange = lattice->getValue();
 
-    ChangeResult changed = lattice->join(attrs);
+    ChangeResult changed = lattice->join(IntegerValueRange{attrs});
 
     // Catch loop results with loop variant bounds and conservatively make
     // them [-inf, inf] so we don't circle around infinitely often (because
@@ -104,8 +109,8 @@
     bool isYieldedResult = llvm::any_of(v.getUsers(), [](Operation *op) {
       return op->hasTrait<OpTrait::IsTerminator>();
     });
-    if (isYieldedResult && oldRange.has_value() &&
-        !(lattice->getValue() == *oldRange)) {
+    if (isYieldedResult && !oldRange.isUninitialized() &&
+        !(lattice->getValue() == oldRange)) {
       LLVM_DEBUG(llvm::dbgs() << "Loop variant loop result detected\n");
       changed |= lattice->join(IntegerValueRange::getMaxRange(v));
     }
@@ -134,11 +139,9 @@
 
       LLVM_DEBUG(llvm::dbgs() << "Inferred range " << attrs << "\n");
       IntegerValueRangeLattice *lattice = argLattices[arg.getArgNumber()];
-      Optional<IntegerValueRange> oldRange;
-      if (!lattice->isUninitialized())
-        oldRange = lattice->getValue();
+      IntegerValueRange oldRange = lattice->getValue();
 
-      ChangeResult changed = lattice->join(attrs);
+      ChangeResult changed = lattice->join(IntegerValueRange{attrs});
 
       // Catch loop results with loop variant bounds and conservatively make
       // them [-inf, inf] so we don't circle around infinitely often (because
@@ -147,7 +150,8 @@
       bool isYieldedValue = llvm::any_of(v.getUsers(), [](Operation *op) {
         return op->hasTrait<OpTrait::IsTerminator>();
       });
-      if (isYieldedValue && oldRange && !(lattice->getValue() == *oldRange)) {
+      if (isYieldedValue && !oldRange.isUninitialized() &&
+          !(lattice->getValue() == oldRange)) {
         LLVM_DEBUG(llvm::dbgs() << "Loop variant loop result detected\n");
         changed |= lattice->join(IntegerValueRange::getMaxRange(v));
       }
@@ -212,7 +216,7 @@
 
     IntegerValueRangeLattice *ivEntry = getLatticeElement(*iv);
     auto ivRange = ConstantIntRanges::fromSigned(min, max);
-    propagateIfChanged(ivEntry, ivEntry->join(ivRange));
+    propagateIfChanged(ivEntry, ivEntry->join(IntegerValueRange{ivRange}));
     return;
   }
 
diff --git a/mlir/lib/Analysis/DataFlow/SparseAnalysis.cpp b/mlir/lib/Analysis/DataFlow/SparseAnalysis.cpp
--- a/mlir/lib/Analysis/DataFlow/SparseAnalysis.cpp
+++ b/mlir/lib/Analysis/DataFlow/SparseAnalysis.cpp
@@ -117,9 +117,6 @@
   for (Value operand : op->getOperands()) {
     AbstractSparseLattice *operandLattice = getLatticeElement(operand);
     operandLattice->useDefSubscribe(this);
-    // If any of the operand states are not initialized, bail out.
-    if (operandLattice->isUninitialized())
-      return;
     operandLattices.push_back(operandLattice);
   }
 
diff --git a/mlir/lib/Transforms/SCCP.cpp b/mlir/lib/Transforms/SCCP.cpp
--- a/mlir/lib/Transforms/SCCP.cpp
+++ b/mlir/lib/Transforms/SCCP.cpp
@@ -43,7 +43,7 @@
                                          OpBuilder &builder,
                                          OperationFolder &folder, Value value) {
   auto *lattice = solver.lookupState<Lattice<ConstantValue>>(value);
-  if (!lattice || lattice->isUninitialized())
+  if (!lattice || lattice->getValue().isUninitialized())
     return failure();
   const ConstantValue &latticeValue = lattice->getValue();
   if (!latticeValue.getConstantValue())
diff --git a/mlir/test/lib/Analysis/DataFlow/TestDenseDataFlowAnalysis.cpp b/mlir/test/lib/Analysis/DataFlow/TestDenseDataFlowAnalysis.cpp
--- a/mlir/test/lib/Analysis/DataFlow/TestDenseDataFlowAnalysis.cpp
+++ b/mlir/test/lib/Analysis/DataFlow/TestDenseDataFlowAnalysis.cpp
@@ -21,16 +21,29 @@
 class UnderlyingValue {
 public:
   /// Create an underlying value state with a known underlying value.
-  UnderlyingValue(Value underlyingValue) : underlyingValue(underlyingValue) {}
+  explicit UnderlyingValue(Optional<Value> underlyingValue = None)
+      : underlyingValue(underlyingValue) {}
+
+  /// Whether the state is uninitialized.
+  bool isUninitialized() const { return !underlyingValue.has_value(); }
 
   /// Returns the underlying value.
-  Value getUnderlyingValue() const { return underlyingValue; }
+  Value getUnderlyingValue() const {
+    assert(!isUninitialized());
+    return *underlyingValue;
+  }
 
   /// Join two underlying values. If there are conflicting underlying values,
   /// go to the pessimistic value.
   static UnderlyingValue join(const UnderlyingValue &lhs,
                               const UnderlyingValue &rhs) {
-    return lhs.underlyingValue == rhs.underlyingValue ? lhs : Value();
+    if (lhs.isUninitialized())
+      return rhs;
+    if (rhs.isUninitialized())
+      return lhs;
+    return lhs.underlyingValue == rhs.underlyingValue
+               ? lhs
+               : UnderlyingValue(Value{});
   }
 
   /// Compare underlying values.
@@ -41,7 +54,7 @@
   void print(raw_ostream &os) const { os << underlyingValue; }
 
 private:
-  Value underlyingValue;
+  Optional<Value> underlyingValue;
 };
 
 /// This lattice represents, for a given memory resource, the potential last
@@ -52,9 +65,6 @@
 
   using AbstractDenseLattice::AbstractDenseLattice;
 
-  /// The lattice is always initialized.
-  bool isUninitialized() const override { return false; }
-
   /// Clear all modifications.
   ChangeResult reset() {
     if (lastMods.empty())
@@ -169,7 +179,7 @@
   const UnderlyingValueLattice *underlying;
   do {
     underlying = getUnderlyingValueFn(value);
-    if (!underlying || underlying->isUninitialized())
+    if (!underlying || underlying->getValue().isUninitialized())
       return {};
     Value underlyingValue = underlying->getValue().getUnderlyingValue();
     if (underlyingValue == value)
diff --git a/mlir/test/lib/Analysis/TestDataFlowFramework.cpp b/mlir/test/lib/Analysis/TestDataFlowFramework.cpp
--- a/mlir/test/lib/Analysis/TestDataFlowFramework.cpp
+++ b/mlir/test/lib/Analysis/TestDataFlowFramework.cpp
@@ -21,7 +21,7 @@
   using AnalysisState::AnalysisState;
 
   /// Returns true if the state is uninitialized.
-  bool isUninitialized() const override { return !state; }
+  bool isUninitialized() const { return !state; }
 
   /// Print the integer value or "none" if uninitialized.
   void print(raw_ostream &os) const override {
diff --git a/mlir/test/lib/Transforms/TestIntRangeInference.cpp b/mlir/test/lib/Transforms/TestIntRangeInference.cpp
--- a/mlir/test/lib/Transforms/TestIntRangeInference.cpp
+++ b/mlir/test/lib/Transforms/TestIntRangeInference.cpp
@@ -25,7 +25,7 @@
                                          OperationFolder &folder, Value value) {
   auto *maybeInferredRange =
       solver.lookupState<IntegerValueRangeLattice>(value);
-  if (!maybeInferredRange || maybeInferredRange->isUninitialized())
+  if (!maybeInferredRange || maybeInferredRange->getValue().isUninitialized())
     return failure();
   const ConstantIntRanges &inferredRange =
       maybeInferredRange->getValue().getValue();