diff --git a/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.h b/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.h
--- a/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.h
+++ b/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.h
@@ -25,6 +25,81 @@
 class AnalysisState;
 class BufferizableOpInterface;
 
+/// Specifies a fine-grain relationship between buffers to enable more analysis.
+enum class BufferRelation {
+  None,
+  // TODO: ResultContainsOperand,
+  // TODO: OperandContainsResult,
+  Equivalent
+};
+
+/// Specifies the type of a property.
+enum class Certainty { MayBe, MustBe };
+
+/// An aliasing OpOperand.
+struct AliasingOpOperand {
+  AliasingOpOperand(OpOperand *opOperand, BufferRelation relation)
+      : opOperand(opOperand), relation(relation) {}
+
+  OpOperand *opOperand;
+  BufferRelation relation;
+};
+
+/// An aliasing OpResult.
+struct AliasingOpResult {
+  AliasingOpResult(OpResult opResult, BufferRelation relation)
+      : opResult(opResult), relation(relation) {}
+
+  OpResult opResult;
+  BufferRelation relation;
+};
+
+template <typename T> class AliasList {
+public:
+  /// Create an empty MayBe list of aliases.
+  AliasList<T>() : certainty(Certainty::MayBe) {}
+
+  /// Create an empty list of aliases with the specified certainty.
+  explicit AliasList<T>(Certainty certainty) : certainty(certainty) {}
+
+  /// Create a list of aliases with the specified certainty.
+  AliasList<T>(Certainty certainty, std::initializer_list<T> elems)
+      : certainty(certainty) {
+    for (T alias : elems)
+      addAlias(alias);
+  }
+
+  /// Create a list of aliases with the specified certainty.
+  AliasList<T>(Certainty certainty, SmallVector<T> &&aliases)
+      : aliases(std::move(aliases)), certainty(certainty) {}
+
+  ArrayRef<T> getAliases() const { return aliases; }
+
+  Certainty getCertainty() const { return certainty; }
+
+  size_t getNumAliases() const { return aliases.size(); }
+
+  void addAlias(T alias) { aliases.push_back(alias); }
+
+  auto begin() const { return aliases.begin(); }
+  auto end() const { return aliases.end(); }
+
+private:
+  /// The list of aliases.
+  SmallVector<T> aliases;
+
+  /// Specifies the type of the property.
+  /// * MayBe: Either one of the entries may turn out to alias at runtime. The
+  ///   runtime alias is guaranteed to be in the list.
+  /// * MustBe: Each entry is guaranteed to alias at runtime.
+  const Certainty certainty;
+};
+
+/// A list of aliasing OpOperands.
+using AliasingOpOperandList = AliasList<AliasingOpOperand>;
+/// A list of aliasing OpResults.
+using AliasingOpResultList = AliasList<AliasingOpResult>;
+
 class OpFilter {
 public:
   /// An op filter entry. Filters can be used to specify which ops should be
@@ -300,14 +375,6 @@
   SmallVector<AnalysisStateInitFn> stateInitializers;
 };
 
-/// Specify fine-grain relationship between buffers to enable more analysis.
-enum class BufferRelation {
-  None,
-  // TODO: ResultContainsOperand,
-  // TODO: OperandContainsResult,
-  Equivalent
-};
-
 /// Return `true` if the given value is a BlockArgument of a func::FuncOp.
 bool isFunctionArgument(Value value);
 
@@ -315,15 +382,15 @@
 /// tensor values.
 class AnalysisState {
 public:
-  /// Determine which OpOperand* will alias with `opResult` if the op is
+  /// Determine which OpOperand* will alias with `result` if the op is
   /// bufferized in place. Return all tensor OpOperand* if the op is not
   /// bufferizable.
-  SmallVector<OpOperand *> getAliasingOpOperand(OpResult opResult) const;
+  AliasingOpOperandList getAliasingOpOperands(OpResult result) const;
 
   /// Determine which OpResult will alias with `opOperand` if the op is
   /// bufferized in place. Return all tensor OpResults if the op is not
   /// bufferizable.
-  SmallVector<OpResult> getAliasingOpResult(OpOperand &opOperand) const;
+  AliasingOpResultList getAliasingOpResults(OpOperand &opOperand) const;
 
   /// Return true if `opOperand` bufferizes to a memory read. Return `true` if
   /// the op is not bufferizable.
@@ -539,6 +606,12 @@
                                      const BufferizationOptions &options);
 
 namespace detail {
+/// This is the default implementation of
+/// BufferizableOpInterface::getAliasingOpOperands. Should not be called from
+/// other places.
+AliasingOpOperandList defaultGetAliasingOpOperands(OpResult opResult,
+                                                   const AnalysisState &state);
+
 /// This is the default implementation of
 /// BufferizableOpInterface::getBufferType. Should not be called from other
 /// places.
@@ -552,13 +625,13 @@
 bool defaultIsRepetitiveRegion(BufferizableOpInterface bufferizableOp,
                                unsigned index);
 
-/// This is the default implementation of getAliasingOpOperand in case the
-/// defining op does not implement the BufferizableOpInterface.
-SmallVector<OpOperand *> unknownGetAliasingOpOperand(OpResult opResult);
+/// This is the default implementation of getAliasingOpOperands in case the
+/// owner op does not implement the BufferizableOpInterface.
+AliasingOpOperandList unknownGetAliasingOpOperands(OpResult opResult);
 
-/// This is the default implementation of getAliasingOpResult in case the
+/// This is the default implementation of getAliasingOpResults in case the
 /// defining op does not implement the BufferizableOpInterface.
-SmallVector<OpResult> unknownGetAliasingOpResult(OpOperand &opOperand);
+AliasingOpResultList unknownGetAliasingOpResults(OpOperand &opOperand);
 } // namespace detail
 
 } // namespace bufferization
diff --git a/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.td b/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.td
--- a/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.td
+++ b/mlir/include/mlir/Dialect/Bufferization/IR/BufferizableOpInterface.td
@@ -110,15 +110,15 @@
           /*defaultImplementation=*/[{
             auto bufferizableOp =
                 cast<BufferizableOpInterface>($_op.getOperation());
-            SmallVector<OpOperand*> opOperands =
-              bufferizableOp.getAliasingOpOperand(opResult, state);
-            if (opOperands.empty())
+            AliasingOpOperandList aliases =
+              bufferizableOp.getAliasingOpOperands(opResult, state);
+            if (aliases.getNumAliases() == 0)
               return true;
             return llvm::any_of(
-                opOperands,
-                [&](OpOperand *operand) {
-                  return bufferizableOp.bufferizesToMemoryWrite(*operand,
-                                                                state);
+                aliases,
+                [&](AliasingOpOperand alias) {
+                  return bufferizableOp.bufferizesToMemoryWrite(
+                      *alias.opOperand, state);
                 });
           }]
       >,
@@ -140,22 +140,71 @@
       >,
       InterfaceMethod<
         /*desc=*/[{
-          Return the OpResult that aliases with a given OpOperand when
+          Return the OpResults that may alias with a given OpOperand when
           bufferized in-place. This method will never be called on OpOperands
           that do not have a tensor type.
 
-          Note: This method can return multiple OpResults, indicating that a
-          given OpOperand may at runtime alias with any (or multiple) of the
-          returned OpResults.
+          This method can return multiple OpResults, indicating that a given
+          OpOperand may at runtime alias with any (or multiple) of the returned
+          OpResults.
+
+          This property is specified with a degree of certainty:
+
+          * Certainty::MayBe: The returned set of OpResults is an exhaustive
+            set of possible aliases. At runtime, buffer(opOperand) may alias
+            with any (or none) of the buffers of the returned OpResults. False
+            positives are allowed. On the contrary, omitting potential aliases
+            is incorrect.
+          * Certainty::MustBe: The buffers of returned set of OpResults are
+            guaranteed to alias with buffer(opOperand). False positives are not
+            allowed. Omitting potential aliases is also incorrect.
+
+          Certainty::MustBe is a stronger property than Certainty::MayBe and can
+          enable more efficient bufferization. When possible, aliasing should be
+          specified as a MustBe relationship.
+
+          For each alias, a BufferRelation can be specified:
+
+          * BufferRelation::Equivalent: Both aliases are the exact same buffer.
+            I.e., same size, no offset, same strides.
+          * BufferRelation::None: There is no further information apart from the
+            fact that both buffers alias.
+
+          One possible (conservative) implementation of this interface method
+          that is always safe is to return all tensor OpResults with
+          BufferRelation::None and Certainty::Maybe.
+
+          Examples:
+
+          ```
+          // aliasingOpResults(%t) = MustBe {Equivalent %r}
+          %r = tensor.insert_slice %f into %t : tensor<10xf32>
+
+          // aliasingOpResults(%t) = MustBe {None %r}
+          %r = tensor.extract_slice %t[0]][5][1]
+              : tensor<10xf32> to tensor<5xf32>
+
+          // aliasingOpResults(%t1) = MayBe {Equivalent %r}
+          // aliasingOpResults(%t2) = MayBe {Equivalent %r}
+          %r = arith.select %c, %t1, %t2 : tensor<10xf32>
+
+          // A hypothetical op that bufferizes to roll a dice and based on the
+          // result ot either return buffer(%t) or a newly allocated copy
+          // thereof.
+          // aliasingOpResults(%t1) = MayBe {Equivalent %r}
+          %r = "dummy.alias_or_copy(%t) : (tensor<10xf32>) -> (tensor<10xf32>)"
+          ```
         }],
-        /*retType=*/"::mlir::SmallVector<::mlir::OpResult>",
-        /*methodName=*/"getAliasingOpResult",
+        /*retType=*/"::mlir::bufferization::AliasingOpResultList",
+        /*methodName=*/"getAliasingOpResults",
         /*args=*/(ins "::mlir::OpOperand &":$opOperand,
                       "const ::mlir::bufferization::AnalysisState &":$state),
         /*methodBody=*/"",
         /*defaultImplementation=*/[{
           // Does not have to be implemented for ops without tensor OpOperands.
-          llvm_unreachable("getAliasingOpResult not implemented");
+          assert(opOperand.get().getType().isa<TensorType>() &&
+                 "expected OpOperand with tensor type");
+          llvm_unreachable("getAliasingOpResults not implemented");
         }]
       >,
       InterfaceMethod<
@@ -164,57 +213,37 @@
           bufferized in-place. This method will never be called on OpResults
           that do not have a tensor type.
 
-          By default, this method is the inverse of `getAliasingOpResult`. Ops
+          By default, this method is the inverse of `getAliasingOpResults`. Ops
           with a region that yield values may want to override this method to
           return the OpOperands that are yielded by the terminator.
 
-          Note: This method can return multiple OpOperands, indicating that the
-          given OpResult may at runtime alias with any (or multiple) of the
-          returned OpOperands. This can be useful for branches and for ops such
-          as `arith.select`.
+          Examples:
+
+          ```
+          // aliasingOpOperands(%r) = MustBe {Equivalent %t}
+          %r = tensor.insert_slice %f into %t : tensor<10xf32>
+
+          // aliasingOpOperands(%r) = MustBe {None %t}
+          %r = tensor.extract_slice %t[0]][5][1]
+              : tensor<10xf32> to tensor<5xf32>
+
+          // aliasingOpOperands(%r) = MayBe {Equivalent %t1, Equivalent %t2}
+          %r = arith.select %c, %t1, %t2 : tensor<10xf32>
+
+          // aliasingOpOperands(%r) = MayBe {}
+          %r = tensor.empty() : tensor<10xf32>
+          ```
         }],
-        /*retType=*/"::mlir::SmallVector<::mlir::OpOperand *>",
-        /*methodName=*/"getAliasingOpOperand",
+        /*retType=*/"::mlir::bufferization::AliasingOpOperandList",
+        /*methodName=*/"getAliasingOpOperands",
         /*args=*/(ins "::mlir::OpResult":$opResult,
                       "const ::mlir::bufferization::AnalysisState &":$state),
         /*methodBody=*/"",
         /*defaultImplementation=*/[{
           assert(opResult.getType().isa<TensorType>() &&
                  "expected OpResult with tensor type");
-          SmallVector<OpOperand *> result;
-          auto bufferizableOp =
-              cast<BufferizableOpInterface>($_op.getOperation());
-          for (OpOperand &opOperand : $_op.getOperation()->getOpOperands()) {
-            if (!opOperand.get().getType().isa<TensorType>())
-              continue;
-            SmallVector<OpResult> aliasingOpResults =
-                bufferizableOp.getAliasingOpResult(opOperand, state);
-            if (llvm::is_contained(aliasingOpResults, opResult))
-              result.push_back(&opOperand);
-          }
-          return result;
-        }]
-      >,
-      InterfaceMethod<
-        /*desc=*/[{
-          Return the buffer relation between the given OpResult and its aliasing
-          OpOperands when bufferized in-place. Most OpOperands have an
-          "equivalence" relation. This method will never be called on OpResults
-          that do not have a tensor type. It will also never be called on
-          OpResults that do not have at least one aliasing OpOperand.
-
-          TODO: Support other relations such as "OpOperand is included in
-          OpResult".
-        }],
-        /*retType=*/"::mlir::bufferization::BufferRelation",
-        /*methodName=*/"bufferRelation",
-        /*args=*/(ins "::mlir::OpResult":$opResult,
-                      "const ::mlir::bufferization::AnalysisState &":$state),
-        /*methodBody=*/"",
-        /*defaultImplementation=*/[{
-          // Does not have to be implemented for ops without tensor OpResults
-          // that have an aliasing OpOperand.
-          llvm_unreachable("bufferRelation not implemented");
+          return ::mlir::bufferization::detail::defaultGetAliasingOpOperands(
+              opResult, state);
         }]
       >,
       InterfaceMethod<
@@ -253,11 +282,15 @@
           conversion.
 
           The implementation of this method must be consistent with the
-          remaining methods, in particular `getAliasingOpOperand`. I.e., a
+          remaining methods, in particular `getAliasingOpOperands`. I.e., a
           tensor result `r` may only be replaced with:
-          a) A buffer that aliases one of buffers in getAliasingOpOperand(r).
+
+          a) A buffer that aliases one of buffers in getAliasingOpOperands(r).
           b) Or: A newly allocated buffer.
 
+          If `getAliasingOpOperands` is specified as Certainty::MustBe, the
+          second option is not available.
+
           This method will never be called on ops that do not have at least one
           tensor operand/result.
 
@@ -408,7 +441,8 @@
           cast<BufferizableOpInterface>(getOperation());
       return !bufferizableOp.bufferizesToMemoryRead(opOperand, state)
           && !bufferizableOp.bufferizesToMemoryWrite(opOperand, state)
-          && !bufferizableOp.getAliasingOpResult(opOperand, state).empty();
+          && bufferizableOp.getAliasingOpResults(opOperand, state)
+              .getNumAliases() != 0;
     }
   }];
 }
diff --git a/mlir/include/mlir/Dialect/Bufferization/IR/BufferizationOps.td b/mlir/include/mlir/Dialect/Bufferization/IR/BufferizationOps.td
--- a/mlir/include/mlir/Dialect/Bufferization/IR/BufferizationOps.td
+++ b/mlir/include/mlir/Dialect/Bufferization/IR/BufferizationOps.td
@@ -99,7 +99,7 @@
     bool bufferizesToMemoryWrite(OpOperand &opOperand,
                                  const AnalysisState &state);
 
-    SmallVector<OpResult> getAliasingOpResult(
+    AliasingOpResultList getAliasingOpResults(
         OpOperand &opOperand, const AnalysisState &state);
 
     FailureOr<BaseMemRefType> getBufferType(
@@ -247,7 +247,7 @@
       return false;
     }
 
-    SmallVector<OpResult> getAliasingOpResult(
+    AliasingOpResultList getAliasingOpResults(
         OpOperand &opOperand, const AnalysisState &state) const {
       return {};
     }
@@ -408,7 +408,7 @@
       return true;
     }
 
-    SmallVector<OpResult> getAliasingOpResult(
+    AliasingOpResultList getAliasingOpResults(
         OpOperand &opOperand, const AnalysisState &state) const {
       return {};
     }
diff --git a/mlir/include/mlir/Dialect/Bufferization/IR/DstBufferizableOpInterfaceImpl.h b/mlir/include/mlir/Dialect/Bufferization/IR/DstBufferizableOpInterfaceImpl.h
--- a/mlir/include/mlir/Dialect/Bufferization/IR/DstBufferizableOpInterfaceImpl.h
+++ b/mlir/include/mlir/Dialect/Bufferization/IR/DstBufferizableOpInterfaceImpl.h
@@ -36,21 +36,16 @@
     return dstOp.isDpsInit(&opOperand);
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     // Output operands alias with their respective tied OpResults.
     auto dstOp = cast<DestinationStyleOpInterface>(op);
     if (dstOp.isDpsInit(&opOperand))
-      return {dstOp.getTiedOpResult(&opOperand)};
+      return {
+          Certainty::MustBe,
+          {{dstOp.getTiedOpResult(&opOperand), BufferRelation::Equivalent}}};
     return {};
   }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    assert(isa<DestinationStyleOpInterface>(op) &&
-           "expected that op implements DestinationStyleOpInterface");
-    return BufferRelation::Equivalent;
-  }
 };
 
 } // namespace bufferization
diff --git a/mlir/lib/Dialect/Arith/Transforms/BufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/Arith/Transforms/BufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/Arith/Transforms/BufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/Arith/Transforms/BufferizableOpInterfaceImpl.cpp
@@ -74,14 +74,10 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    return {op->getResult(0)};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
+    return {Certainty::MustBe,
+            {{op->getResult(0), BufferRelation::Equivalent}}};
   }
 
   LogicalResult bufferize(Operation *op, RewriterBase &rewriter,
@@ -126,16 +122,10 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    return {op->getOpResult(0) /*result*/};
-  }
-
-  SmallVector<OpOperand *>
-  getAliasingOpOperand(Operation *op, OpResult opResult,
-                       const AnalysisState &state) const {
-    return {&op->getOpOperand(1) /*true_value*/,
-            &op->getOpOperand(2) /*false_value*/};
+    return {Certainty::MayBe,
+            {{op->getOpResult(0) /*result*/, BufferRelation::Equivalent}}};
   }
 
   LogicalResult bufferize(Operation *op, RewriterBase &rewriter,
@@ -199,11 +189,6 @@
                               memrefType.getElementType()),
         memrefType.getMemorySpace());
   }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::None;
-  }
 };
 
 } // namespace
diff --git a/mlir/lib/Dialect/Bufferization/IR/BufferizableOpInterface.cpp b/mlir/lib/Dialect/Bufferization/IR/BufferizableOpInterface.cpp
--- a/mlir/lib/Dialect/Bufferization/IR/BufferizableOpInterface.cpp
+++ b/mlir/lib/Dialect/Bufferization/IR/BufferizableOpInterface.cpp
@@ -177,28 +177,30 @@
     if (operandType.isa<UnrankedTensorType>())
       return op->emitError("copies of unranked tensors are not supported");
 
-    SmallVector<OpResult> aliasingOpResults =
-        state.getAliasingOpResult(opOperand);
+    AliasingOpResultList aliasingOpResults =
+        state.getAliasingOpResults(opOperand);
     // Is the result yielded from a block? Or are deallocations turned off
     // entirely? In either case, mark the allocation as "escaping", so that it
     // will not be deallocated.
     bool escape = !state.getOptions().createDeallocs ||
-                  llvm::any_of(aliasingOpResults, [&](Value v) {
-                    return state.isTensorYielded(v);
+                  llvm::any_of(aliasingOpResults, [&](AliasingOpResult a) {
+                    return state.isTensorYielded(a.opResult);
                   });
 
-    if (aliasingOpResults.size() == 1 &&
+    if (aliasingOpResults.getNumAliases() == 1 &&
         !state.bufferizesToMemoryWrite(opOperand) &&
-        state.getAliasingOpOperand(aliasingOpResults.front()).size() == 1) {
+        state.getAliasingOpOperands(aliasingOpResults.getAliases()[0].opResult)
+                .getNumAliases() == 1) {
       // The op itself does not write but may create exactly one alias. Instead
       // of copying the OpOperand, copy the OpResult. The OpResult can sometimes
       // be smaller than the OpOperand (e.g., in the case of an extract_slice,
       // where the result is usually a smaller part of the source).
-      outOfPlaceOpResults.push_back(aliasingOpResults.front());
+      OpResult opResult = aliasingOpResults.getAliases()[0].opResult;
+      outOfPlaceOpResults.push_back(opResult);
       if (!state.canOmitTensorCopy(opOperand))
-        copiedOpResults.insert(aliasingOpResults.front());
+        copiedOpResults.insert(opResult);
       if (escape)
-        escapingOpResultCopies.insert(aliasingOpResults.front());
+        escapingOpResultCopies.insert(opResult);
     } else {
       // In all other cases, make a copy of the OpOperand.
       outOfPlaceOpOperands.push_back(&opOperand);
@@ -352,26 +354,26 @@
 /// Determine which OpOperand* will alias with `opResult` if the op is
 /// bufferized in place. Return all tensor OpOperand* if the op is not
 /// bufferizable.
-SmallVector<OpOperand *>
-AnalysisState::getAliasingOpOperand(OpResult opResult) const {
+AliasingOpOperandList
+AnalysisState::getAliasingOpOperands(OpResult opResult) const {
   if (Operation *op = opResult.getDefiningOp())
     if (auto bufferizableOp = getOptions().dynCastBufferizableOp(op))
-      return bufferizableOp.getAliasingOpOperand(opResult, *this);
+      return bufferizableOp.getAliasingOpOperands(opResult, *this);
 
   // The op is not bufferizable.
-  return detail::unknownGetAliasingOpOperand(opResult);
+  return detail::unknownGetAliasingOpOperands(opResult);
 }
 
 /// Determine which OpResult will alias with `opOperand` if the op is bufferized
 /// in place. Return all tensor OpResults if the op is not bufferizable.
-SmallVector<OpResult>
-AnalysisState::getAliasingOpResult(OpOperand &opOperand) const {
+AliasingOpResultList
+AnalysisState::getAliasingOpResults(OpOperand &opOperand) const {
   if (auto bufferizableOp =
           getOptions().dynCastBufferizableOp(opOperand.getOwner()))
-    return bufferizableOp.getAliasingOpResult(opOperand, *this);
+    return bufferizableOp.getAliasingOpResults(opOperand, *this);
 
   // The op is not bufferizable.
-  return detail::unknownGetAliasingOpResult(opOperand);
+  return detail::unknownGetAliasingOpResults(opOperand);
 }
 
 /// Return true if `opOperand` bufferizes to a memory read. Return `true` if the
@@ -423,8 +425,8 @@
     OpOperand *uMaybeReading = workingSet.pop_back_val();
     // Skip over all ops that neither read nor write (but create an alias).
     if (bufferizesToAliasOnly(*uMaybeReading))
-      for (OpResult opResult : getAliasingOpResult(*uMaybeReading))
-        for (OpOperand &use : opResult.getUses())
+      for (AliasingOpResult alias : getAliasingOpResults(*uMaybeReading))
+        for (OpOperand &use : alias.opResult.getUses())
           workingSet.push_back(&use);
     if (bufferizesToMemoryRead(*uMaybeReading))
       return true;
@@ -453,23 +455,25 @@
     OpResult opResult = value.cast<OpResult>();
     BufferizableOpInterface bufferizableOp =
         options.dynCastBufferizableOp(opResult.getDefiningOp());
-    SmallVector<OpOperand *> opOperands = getAliasingOpOperand(opResult);
+    AliasingOpOperandList aliases = getAliasingOpOperands(opResult);
 
     // Stop iterating in either one of these cases:
     // * The current op is not bufferizable or excluded in the filter.
     // * There are no OpOperands to follow.
-    // * There is an OpOperand, but it is not an equivalent tensor (only if
-    //   `followEquivalentOnly` is set).
-    if (!bufferizableOp || opOperands.empty() ||
-        (followEquivalentOnly &&
-         bufferizableOp.bufferRelation(opResult, *this) !=
-             BufferRelation::Equivalent)) {
+    if (!bufferizableOp || aliases.getNumAliases() == 0) {
       result.insert(value);
       continue;
     }
 
-    for (OpOperand *o : opOperands)
-      workingSet.insert(o->get());
+    for (AliasingOpOperand a : aliases) {
+      if (followEquivalentOnly && a.relation != BufferRelation::Equivalent) {
+        // Stop iterating if `followEquivalentOnly` is set but the alias is not
+        // equivalent.
+        result.insert(value);
+      } else {
+        workingSet.insert(a.opOperand->get());
+      }
+    }
   }
 
   return result;
@@ -510,10 +514,10 @@
     return true;
 
   // Do not copy if the tensor is never read.
-  SmallVector<OpResult> aliasingOpResults = getAliasingOpResult(opOperand);
+  AliasingOpResultList aliases = getAliasingOpResults(opOperand);
   if (!bufferizesToMemoryRead(opOperand) &&
-      llvm::none_of(aliasingOpResults,
-                    [&](OpResult opResult) { return isValueRead(opResult); }))
+      llvm::none_of(
+          aliases, [&](AliasingOpResult a) { return isValueRead(a.opResult); }))
     return true;
 
   // Default: Cannot omit the copy.
@@ -581,8 +585,8 @@
     // Note: In the absence of detailed analysis information (e.g., there may be
     // no function call analysis information), this `getAliasingOpResult` is
     // conservative and may report additional OpResults as potentially aliasing.
-    for (OpResult opResult : getAliasingOpResult(*operand))
-      for (OpOperand &use : opResult.getUses())
+    for (AliasingOpResult alias : getAliasingOpResults(*operand))
+      for (OpOperand &use : alias.opResult.getUses())
         worklist.push_back(&use);
   }
 
@@ -806,6 +810,31 @@
 // Default implementations of interface methods
 //===----------------------------------------------------------------------===//
 
+// Compute the AliasingOpOperandList for a given OpResult based on
+// getAliasingOpResults.
+AliasingOpOperandList bufferization::detail::defaultGetAliasingOpOperands(
+    OpResult opResult, const AnalysisState &state) {
+  Operation *op = opResult.getDefiningOp();
+  SmallVector<AliasingOpOperand> result;
+  std::optional<Certainty> certainty = {};
+  for (OpOperand &opOperand : op->getOpOperands()) {
+    if (!opOperand.get().getType().isa<TensorType>())
+      continue;
+    AliasingOpResultList aliasingOpResults =
+        state.getAliasingOpResults(opOperand);
+    for (const auto &it : aliasingOpResults) {
+      if (it.opResult == opResult) {
+        result.emplace_back(&opOperand, it.relation);
+        if (!certainty.has_value())
+          certainty = aliasingOpResults.getCertainty();
+      }
+      break;
+    }
+  }
+  return AliasingOpOperandList(
+      certainty.has_value() ? *certainty : Certainty::MayBe, std::move(result));
+}
+
 FailureOr<BaseMemRefType> bufferization::detail::defaultGetBufferType(
     Value value, const BufferizationOptions &options,
     const DenseMap<Value, BaseMemRefType> &fixedTypes) {
@@ -818,15 +847,13 @@
   // Value is an OpResult.
   Operation *op = getOwnerOfValue(value);
   auto opResult = value.cast<OpResult>();
-  auto bufferizableOp = cast<BufferizableOpInterface>(op);
   AnalysisState state(options);
-  auto aliasingOperands = bufferizableOp.getAliasingOpOperand(opResult, state);
-  if (!aliasingOperands.empty() &&
-      bufferizableOp.bufferRelation(opResult, state) ==
-          BufferRelation::Equivalent) {
+  AliasingOpOperandList aliases = state.getAliasingOpOperands(opResult);
+  if (aliases.getNumAliases() > 0 &&
+      aliases.getAliases()[0].relation == BufferRelation::Equivalent) {
     // If the OpResult has an equivalent OpOperand, both OpResult and
     // OpOperand bufferize to the exact same buffer type.
-    Value equivalentOperand = aliasingOperands.front()->get();
+    Value equivalentOperand = aliases.getAliases().front().opOperand->get();
     return getBufferType(equivalentOperand, options, fixedTypes);
   }
 
@@ -849,22 +876,22 @@
   return regionInterface.isRepetitiveRegion(index);
 }
 
-SmallVector<OpOperand *>
-bufferization::detail::unknownGetAliasingOpOperand(OpResult opResult) {
+AliasingOpOperandList
+bufferization::detail::unknownGetAliasingOpOperands(OpResult opResult) {
   // Conservatively assume that everything is aliasing.
-  SmallVector<OpOperand *> r;
+  AliasingOpOperandList r(Certainty::MayBe);
   for (OpOperand &operand : opResult.getDefiningOp()->getOpOperands())
     if (operand.get().getType().isa<TensorType>())
-      r.push_back(&operand);
+      r.addAlias({&operand, BufferRelation::None});
   return r;
 }
 
-SmallVector<OpResult>
-bufferization::detail::unknownGetAliasingOpResult(OpOperand &opOperand) {
+AliasingOpResultList
+bufferization::detail::unknownGetAliasingOpResults(OpOperand &opOperand) {
   // Conservatively assume that everything is aliasing.
-  SmallVector<OpResult> r;
+  AliasingOpResultList r(Certainty::MayBe);
   for (OpResult result : opOperand.getOwner()->getOpResults())
     if (result.getType().isa<TensorType>())
-      r.push_back(result);
+      r.addAlias({result, BufferRelation::None});
   return r;
 }
diff --git a/mlir/lib/Dialect/Bufferization/IR/BufferizationOps.cpp b/mlir/lib/Dialect/Bufferization/IR/BufferizationOps.cpp
--- a/mlir/lib/Dialect/Bufferization/IR/BufferizationOps.cpp
+++ b/mlir/lib/Dialect/Bufferization/IR/BufferizationOps.cpp
@@ -226,9 +226,9 @@
   return false;
 }
 
-SmallVector<OpResult>
-AllocTensorOp::getAliasingOpResult(OpOperand &opOperand,
-                                   const AnalysisState &state) {
+AliasingOpResultList
+AllocTensorOp::getAliasingOpResults(OpOperand &opOperand,
+                                    const AnalysisState &state) {
   // This is a new allocation. It does not alias with any other buffer.
   return {};
 }
diff --git a/mlir/lib/Dialect/Bufferization/Transforms/FuncBufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/Bufferization/Transforms/FuncBufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/Bufferization/Transforms/FuncBufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/Bufferization/Transforms/FuncBufferizableOpInterfaceImpl.cpp
@@ -172,52 +172,37 @@
         opOperand.getOperandNumber());
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     func::CallOp callOp = cast<func::CallOp>(op);
     FuncOp funcOp = getCalledFunction(callOp);
     assert(funcOp && "expected CallOp to a FuncOp");
     if (getFuncOpAnalysisState(state, funcOp) != FuncOpAnalysisState::Analyzed)
       // FuncOp not analyzed yet. Any OpResult may be aliasing.
-      return detail::unknownGetAliasingOpResult(opOperand);
+      return detail::unknownGetAliasingOpResults(opOperand);
 
     // Get aliasing results from state.
     const FuncAnalysisState &funcState = getFuncAnalysisState(state);
     auto aliasingReturnVals =
         funcState.aliasingReturnVals.lookup(funcOp).lookup(
             opOperand.getOperandNumber());
-    SmallVector<OpResult> result;
-    for (int64_t resultIdx : aliasingReturnVals)
-      result.push_back(callOp->getOpResult(resultIdx));
-    return result;
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    func::CallOp callOp = cast<func::CallOp>(op);
-    FuncOp funcOp = getCalledFunction(callOp);
-    assert(funcOp && "expected CallOp to a FuncOp");
-    if (getFuncOpAnalysisState(state, funcOp) !=
-        FuncOpAnalysisState::Analyzed) {
-      // Function not analyzed yet. The conservative answer is "None".
-      return BufferRelation::None;
-    }
 
-    const FuncAnalysisState &funcState = getFuncAnalysisState(state);
-    std::optional<int64_t> maybeEquiv =
-        getEquivalentFuncArgIdx(funcOp, funcState, opResult.getResultNumber());
-    if (maybeEquiv) {
-#ifndef NDEBUG
-      SmallVector<OpOperand *> aliasingOpOperands =
-          getAliasingOpOperand(op, opResult, state);
-      assert(aliasingOpOperands.size() == 1 &&
-             "expected exactly 1 aliasing OpOperand");
-      assert(aliasingOpOperands.front()->getOperandNumber() == *maybeEquiv &&
+    // Check if the aliasing OpResult is equivalent to the OpOperand.
+    std::optional<int64_t> equivalent = {};
+    if (aliasingReturnVals.size() == 1) {
+      equivalent = getEquivalentFuncArgIdx(funcOp, funcState,
+                                           aliasingReturnVals.front());
+      assert((!equivalent.has_value() ||
+              *equivalent == opOperand.getOperandNumber()) &&
              "inconsistent analysis state");
-#endif
-      return BufferRelation::Equivalent;
     }
-    return BufferRelation::None;
+    AliasingOpResultList result(equivalent.has_value() ? Certainty::MustBe
+                                                       : Certainty::MayBe);
+    for (int64_t resultIdx : aliasingReturnVals)
+      result.addAlias({callOp->getOpResult(resultIdx),
+                       equivalent.has_value() ? BufferRelation::Equivalent
+                                              : BufferRelation::None});
+    return result;
   }
 
   /// All function arguments are writable. It is the responsibility of the
@@ -329,7 +314,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
diff --git a/mlir/lib/Dialect/Bufferization/Transforms/OneShotAnalysis.cpp b/mlir/lib/Dialect/Bufferization/Transforms/OneShotAnalysis.cpp
--- a/mlir/lib/Dialect/Bufferization/Transforms/OneShotAnalysis.cpp
+++ b/mlir/lib/Dialect/Bufferization/Transforms/OneShotAnalysis.cpp
@@ -147,8 +147,8 @@
   if (inplaceBufferized.contains(&operand))
     return;
   markInPlace(operand);
-  for (OpResult result : state.getAliasingOpResult(operand))
-    aliasInfo.unionSets(result, operand.get());
+  for (AliasingOpResult alias : state.getAliasingOpResults(operand))
+    aliasInfo.unionSets(alias.opResult, operand.get());
   ++statNumTensorInPlace;
 }
 
@@ -661,9 +661,10 @@
 
         // No conflict if the conflicting write and the last write are the same
         // use.
-        SmallVector<OpResult> aliasingOpResult =
-            state.getAliasingOpResult(*uConflictingWrite);
-        if (aliasingOpResult.size() == 1 && aliasingOpResult[0] == lastWrite) {
+        AliasingOpResultList aliases =
+            state.getAliasingOpResults(*uConflictingWrite);
+        if (aliases.getNumAliases() == 1 &&
+            aliases.getAliases()[0].opResult == lastWrite) {
           LLVM_DEBUG(llvm::dbgs()
                      << "    no conflict: last write and write are same\n");
           continue;
@@ -722,9 +723,10 @@
       // there would then be no flow of data from the extract_slice operand to
       // its result's uses.)
       if (!state.bufferizesToMemoryWrite(use)) {
-        SmallVector<OpResult> opResults = state.getAliasingOpResult(use);
-        if (llvm::any_of(opResults,
-                         [&](OpResult r) { return state.isValueRead(r); }))
+        AliasingOpResultList aliases = state.getAliasingOpResults(use);
+        if (llvm::any_of(aliases, [&](AliasingOpResult a) {
+              return state.isValueRead(a.opResult);
+            }))
           res.insert(&use);
       }
     }
@@ -768,9 +770,9 @@
   DenseSet<OpOperand *> usesRead, usesWrite;
   getAliasingReads(usesRead, operand.get(), aliasInfo, state);
   getAliasingInplaceWrites(usesWrite, operand.get(), aliasInfo, state);
-  for (OpResult result : state.getAliasingOpResult(operand)) {
-    getAliasingReads(usesRead, result, aliasInfo, state);
-    getAliasingInplaceWrites(usesWrite, result, aliasInfo, state);
+  for (AliasingOpResult alias : state.getAliasingOpResults(operand)) {
+    getAliasingReads(usesRead, alias.opResult, aliasInfo, state);
+    getAliasingInplaceWrites(usesWrite, alias.opResult, aliasInfo, state);
   }
   if (!checkConsistencyOnly && state.bufferizesToMemoryWrite(operand))
     usesWrite.insert(&operand);
@@ -821,11 +823,10 @@
       continue;
 
     // Follow reverse SSA use-def chain.
-    SmallVector<OpOperand *> aliasingOpOperands =
-        state.getAliasingOpOperand(opResult);
-    for (OpOperand *opOperand : aliasingOpOperands)
-      if (aliasInfo.isInPlace(*opOperand) || currentOpOperand == opOperand)
-        worklist.push_back(opOperand->get());
+    for (AliasingOpOperand alias : state.getAliasingOpOperands(opResult))
+      if (aliasInfo.isInPlace(*alias.opOperand) ||
+          currentOpOperand == alias.opOperand)
+        worklist.push_back(alias.opOperand->get());
   }
   return false;
 }
@@ -838,8 +839,8 @@
   // Collect writes of all aliases of OpOperand and OpResult.
   DenseSet<OpOperand *> usesWrite;
   getAliasingInplaceWrites(usesWrite, operand.get(), aliasInfo, state);
-  for (OpResult result : state.getAliasingOpResult(operand)) {
-    getAliasingInplaceWrites(usesWrite, result, aliasInfo, state);
+  for (AliasingOpResult alias : state.getAliasingOpResults(operand)) {
+    getAliasingInplaceWrites(usesWrite, alias.opResult, aliasInfo, state);
   }
   if (!checkConsistencyOnly && state.bufferizesToMemoryWrite(operand))
     usesWrite.insert(&operand);
@@ -974,16 +975,25 @@
 static void equivalenceAnalysis(SmallVector<Operation *> &ops,
                                 BufferizationAliasInfo &aliasInfo,
                                 AnalysisState &state) {
-  for (Operation *op : ops)
-    if (auto bufferizableOp = state.getOptions().dynCastBufferizableOp(op))
-      for (OpResult opResult : op->getOpResults())
-        if (opResult.getType().isa<TensorType>())
-          for (OpOperand *opOperand :
-               bufferizableOp.getAliasingOpOperand(opResult, state))
-            if (state.isInPlace(*opOperand))
-              if (bufferizableOp.bufferRelation(opResult, state) ==
-                  BufferRelation::Equivalent)
-                aliasInfo.unionEquivalenceClasses(opResult, opOperand->get());
+  for (Operation *op : ops) {
+    for (OpOperand &opOperand : op->getOpOperands()) {
+      if (opOperand.get().getType().isa<TensorType>()) {
+        if (!state.isInPlace(opOperand))
+          // Out-of-place OpOperands bufferize to new allocations and do not
+          // union equivalence sets.
+          continue;
+        AliasingOpResultList aliases = state.getAliasingOpResults(opOperand);
+        if (aliases.getCertainty() == Certainty::MayBe)
+          // If it is not certain that the alias will materialize at runtime,
+          // we have to be conservative.
+          continue;
+        for (AliasingOpResult alias : aliases) {
+          if (alias.relation == BufferRelation::Equivalent)
+            aliasInfo.unionEquivalenceClasses(alias.opResult, opOperand.get());
+        }
+      }
+    }
+  }
 }
 
 /// Analyze equivalence of tied OpResult/OpOperand pairs of all ops contained
@@ -995,7 +1005,7 @@
   SmallVector<Operation *> ops;
   op->walk<WalkOrder::PostOrder>([&](Operation *op) {
     // No tensors => no buffers.
-    if (none_of(op->getResultTypes(), isaTensor))
+    if (none_of(op->getOperandTypes(), isaTensor))
       return;
     ops.push_back(op);
   });
diff --git a/mlir/lib/Dialect/Linalg/Transforms/BufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/Linalg/Transforms/BufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/Linalg/Transforms/BufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/Linalg/Transforms/BufferizableOpInterfaceImpl.cpp
@@ -102,8 +102,8 @@
   bool bufferizesToMemoryWrite(Operation *op, OpOperand &opOperand,
                                const AnalysisState &state) const {
     // Operand is written to if it has an aliasing OpResult.
-    auto bufferizableOp = cast<BufferizableOpInterface>(op);
-    return !bufferizableOp.getAliasingOpResult(opOperand, state).empty();
+    auto dpsOp = cast<DestinationStyleOpInterface>(op);
+    return dpsOp.isDpsInit(&opOperand);
   }
 
   LogicalResult bufferize(Operation *op, RewriterBase &rewriter,
diff --git a/mlir/lib/Dialect/SCF/Transforms/BufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/SCF/Transforms/BufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/SCF/Transforms/BufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/SCF/Transforms/BufferizableOpInterfaceImpl.cpp
@@ -57,7 +57,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -104,9 +104,9 @@
 struct ExecuteRegionOpInterface
     : public BufferizableOpInterface::ExternalModel<ExecuteRegionOpInterface,
                                                     scf::ExecuteRegionOp> {
-  SmallVector<OpOperand *>
-  getAliasingOpOperand(Operation *op, OpResult opResult,
-                       const AnalysisState &state) const {
+  AliasingOpOperandList
+  getAliasingOpOperands(Operation *op, OpResult opResult,
+                        const AnalysisState &state) const {
     // ExecuteRegionOps do not have tensor OpOperands. The yielded value can be
     // any SSA value that is in scope. To allow for use-def chain traversal
     // through ExecuteRegionOps in the analysis, the corresponding yield value
@@ -120,7 +120,8 @@
     auto yieldOp = dyn_cast<scf::YieldOp>(
         executeRegionOp.getRegion().front().getTerminator());
     assert(yieldOp && "expected scf.yield terminator in scf.execute_region");
-    return {&yieldOp->getOpOperand(resultNum)};
+    return {Certainty::MustBe,
+            {{&yieldOp->getOpOperand(resultNum), BufferRelation::Equivalent}}};
   }
 
   // TODO: For better bufferization results, this could return `true` only if
@@ -166,19 +167,14 @@
 
     return success();
   }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
-  }
 };
 
 /// Bufferization of scf.if. Replace with a new scf.if that yields memrefs.
 struct IfOpInterface
     : public BufferizableOpInterface::ExternalModel<IfOpInterface, scf::IfOp> {
-  SmallVector<OpOperand *>
-  getAliasingOpOperand(Operation *op, OpResult opResult,
-                       const AnalysisState &state) const {
+  AliasingOpOperandList
+  getAliasingOpOperands(Operation *op, OpResult opResult,
+                        const AnalysisState &state) const {
     // IfOps do not have tensor OpOperands. The yielded value can be any SSA
     // value that is in scope. To allow for use-def chain traversal through
     // IfOps in the analysis, both corresponding yield values from the then/else
@@ -186,8 +182,13 @@
     auto ifOp = cast<scf::IfOp>(op);
     size_t resultNum = std::distance(op->getOpResults().begin(),
                                      llvm::find(op->getOpResults(), opResult));
-    return {&ifOp.thenYield()->getOpOperand(resultNum),
-            &ifOp.elseYield()->getOpOperand(resultNum)};
+    OpOperand *thenOperand = &ifOp.thenYield()->getOpOperand(resultNum);
+    OpOperand *elseOperand = &ifOp.elseYield()->getOpOperand(resultNum);
+    bool equivalentYields = state.areEquivalentBufferizedValues(
+        thenOperand->get(), elseOperand->get());
+    return {equivalentYields ? Certainty::MustBe : Certainty::MayBe,
+            {{thenOperand, BufferRelation::Equivalent},
+             {elseOperand, BufferRelation::Equivalent}}};
   }
 
   // TODO: For better bufferization results, this could return `true` only if
@@ -301,19 +302,6 @@
     return getMemRefTypeWithFullyDynamicLayout(
         opResult.getType().cast<TensorType>(), thenBufferType.getMemorySpace());
   }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    // IfOp results are equivalent to their corresponding yield values if both
-    // yield values are equivalent to each other.
-    auto bufferizableOp = cast<BufferizableOpInterface>(op);
-    SmallVector<OpOperand *> yieldValues =
-        bufferizableOp.getAliasingOpOperand(opResult, state);
-    assert(yieldValues.size() == 2 && "expected 2 yield values");
-    bool equivalentYields = state.areEquivalentBufferizedValues(
-        yieldValues[0]->get(), yieldValues[1]->get());
-    return equivalentYields ? BufferRelation::Equivalent : BufferRelation::None;
-  }
 };
 
 /// Helper function for loop bufferization. Return the indices of all values
@@ -485,10 +473,14 @@
     return true;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     auto forOp = cast<scf::ForOp>(op);
-    return {forOp.getResultForOpOperand(opOperand)};
+    OpResult opResult = forOp.getResultForOpOperand(opOperand);
+    BufferRelation relation = bufferRelation(op, opResult, state);
+    return {relation == BufferRelation::Equivalent ? Certainty::MustBe
+                                                   : Certainty::MayBe,
+            {{opResult, relation}}};
   }
 
   BufferRelation bufferRelation(Operation *op, OpResult opResult,
@@ -698,7 +690,7 @@
     return true;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     auto whileOp = cast<scf::WhileOp>(op);
     unsigned int idx = opOperand.getOperandNumber();
@@ -710,7 +702,11 @@
       return {};
 
     // The only aliasing OpResult may be the one at the same index.
-    return {whileOp->getResult(idx)};
+    OpResult opResult = whileOp->getResult(idx);
+    BufferRelation relation = bufferRelation(op, opResult, state);
+    return {relation == BufferRelation::Equivalent ? Certainty::MustBe
+                                                   : Certainty::MayBe,
+            {{opResult, relation}}};
   }
 
   BufferRelation bufferRelation(Operation *op, OpResult opResult,
@@ -993,12 +989,23 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    if (isa<scf::IfOp>(op->getParentOp()))
-      return {op->getParentOp()->getResult(opOperand.getOperandNumber())};
+    if (auto ifOp = dyn_cast<scf::IfOp>(op->getParentOp())) {
+      Region &otherCase =
+          ifOp->getRegion((op->getParentRegion()->getRegionNumber() + 1) % 2);
+      auto otherYieldOp = cast<scf::YieldOp>(otherCase.back().back());
+      bool equivalentYields = state.areEquivalentBufferizedValues(
+          opOperand.get(),
+          otherYieldOp->getOperand(opOperand.getOperandNumber()));
+      return {equivalentYields ? Certainty::MustBe : Certainty::MayBe,
+              {{op->getParentOp()->getResult(opOperand.getOperandNumber()),
+                BufferRelation::Equivalent}}};
+    }
     if (isa<scf::ExecuteRegionOp>(op->getParentOp()))
-      return {op->getParentOp()->getResult(opOperand.getOperandNumber())};
+      return {Certainty::MustBe,
+              {{op->getParentOp()->getResult(opOperand.getOperandNumber()),
+                BufferRelation::Equivalent}}};
     return {};
   }
 
@@ -1092,15 +1099,12 @@
     return true;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     auto foreachThreadOp = cast<ForeachThreadOp>(op);
-    return {foreachThreadOp.getTiedOpResult(&opOperand)};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
+    return {Certainty::MustBe,
+            {{foreachThreadOp.getTiedOpResult(&opOperand),
+              BufferRelation::Equivalent}}};
   }
 
   bool isWritable(Operation *op, Value value,
diff --git a/mlir/lib/Dialect/Shape/Transforms/BufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/Shape/Transforms/BufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/Shape/Transforms/BufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/Shape/Transforms/BufferizableOpInterfaceImpl.cpp
@@ -27,9 +27,9 @@
 struct AssumingOpInterface
     : public BufferizableOpInterface::ExternalModel<AssumingOpInterface,
                                                     shape::AssumingOp> {
-  SmallVector<OpOperand *>
-  getAliasingOpOperand(Operation *op, OpResult opResult,
-                       const AnalysisState &state) const {
+  AliasingOpOperandList
+  getAliasingOpOperands(Operation *op, OpResult opResult,
+                        const AnalysisState &state) const {
     // AssumingOps do not have tensor OpOperands. The yielded value can be any
     // SSA value that is in scope. To allow for use-def chain traversal through
     // AssumingOps in the analysis, the corresponding yield value is considered
@@ -43,7 +43,8 @@
     auto yieldOp = dyn_cast<shape::AssumingYieldOp>(
         assumingOp.getDoRegion().front().getTerminator());
     assert(yieldOp && "expected shape.assuming_yield terminator");
-    return {&yieldOp->getOpOperand(resultNum)};
+    return {Certainty::MustBe,
+            {{&yieldOp->getOpOperand(resultNum), BufferRelation::Equivalent}}};
   }
 
   // TODO: For better bufferization results, this could return `true` only if
@@ -89,11 +90,6 @@
 
     return success();
   }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
-  }
 };
 
 /// Bufferization of shape.assuming_yield. Bufferized as part of their enclosing
@@ -111,11 +107,13 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     assert(isa<shape::AssumingOp>(op->getParentOp()) &&
            "expected that parent is an AssumingOp");
-    return {op->getParentOp()->getResult(opOperand.getOperandNumber())};
+    OpResult opResult =
+        op->getParentOp()->getResult(opOperand.getOperandNumber());
+    return {Certainty::MustBe, {{opResult, BufferRelation::Equivalent}}};
   }
 
   bool mustBufferizeInPlace(Operation *op, OpOperand &opOperand,
diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/BufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/SparseTensor/Transforms/BufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/SparseTensor/Transforms/BufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/BufferizableOpInterfaceImpl.cpp
@@ -43,7 +43,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -73,7 +73,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -97,14 +97,10 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    return {op->getOpResult(0)};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
+    return {Certainty::MustBe,
+            {{op->getOpResult(0), BufferRelation::Equivalent}}};
   }
 };
 
@@ -136,18 +132,12 @@
     return true;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     // InsertOp returns an alias of its operand.
     assert(op->getNumResults() == 1);
-    return op->getResults();
-  }
-
-  BufferRelation bufferRelation(Operation *oo, OpResult opResult,
-                                const AnalysisState &state) const {
-    // InsertOp returns the same object (realloc should not invalidate
-    // aliases).
-    return BufferRelation::Equivalent;
+    return {Certainty::MustBe,
+            {{op->getOpResult(0), BufferRelation::Equivalent}}};
   }
 };
 
@@ -164,7 +154,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -185,7 +175,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -206,7 +196,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -227,7 +217,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
diff --git a/mlir/lib/Dialect/Tensor/Transforms/BufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/Tensor/Transforms/BufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/Tensor/Transforms/BufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/Tensor/Transforms/BufferizableOpInterfaceImpl.cpp
@@ -41,14 +41,10 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    return {op->getResult(0)};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
+    return {Certainty::MustBe,
+            {{op->getResult(0), BufferRelation::Equivalent}}};
   }
 
   LogicalResult bufferize(Operation *op, RewriterBase &rewriter,
@@ -98,16 +94,12 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    if (&opOperand == &op->getOpOperand(0) /*src*/)
-      return {op->getOpResult(0)};
-    return {};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
+    // CollapseShapeOp may allocate at runtime.
+    // TODO: Predict future buffer type and compute more accurate Certainty.
+    return {Certainty::MayBe,
+            {{op->getOpResult(0), BufferRelation::Equivalent}}};
   }
 
   FailureOr<BaseMemRefType>
@@ -214,7 +206,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -263,16 +255,10 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    if (&opOperand == &op->getOpOperand(0) /*src*/)
-      return {op->getOpResult(0)};
-    return {};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
+    return {Certainty::MustBe,
+            {{op->getOpResult(0), BufferRelation::Equivalent}}};
   }
 
   FailureOr<BaseMemRefType>
@@ -324,16 +310,9 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    if (&opOperand == &op->getOpOperand(0) /*source*/)
-      return {op->getOpResult(0)};
-    return {};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::None;
+    return {Certainty::MustBe, {{op->getOpResult(0), BufferRelation::None}}};
   }
 
   LogicalResult bufferize(Operation *op, RewriterBase &rewriter,
@@ -397,7 +376,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -837,7 +816,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -935,7 +914,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -968,14 +947,10 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-    return {op->getOpResult(0)};
-  }
-
-  BufferRelation bufferRelation(Operation *op, OpResult opResult,
-                                const AnalysisState &state) const {
-    return BufferRelation::Equivalent;
+    return {Certainty::MustBe,
+            {{op->getOpResult(0), BufferRelation::Equivalent}}};
   }
 
   LogicalResult bufferize(Operation *op, RewriterBase &rewriter,
@@ -1000,9 +975,9 @@
 struct ParallelInsertSliceOpInterface
     : public BufferizableOpInterface::ExternalModel<
           ParallelInsertSliceOpInterface, ParallelInsertSliceOp> {
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
-      return {};
+    return {};
   }
 
   bool bufferizesToMemoryRead(Operation *op, OpOperand &opOperand,
diff --git a/mlir/lib/Dialect/Vector/Transforms/BufferizableOpInterfaceImpl.cpp b/mlir/lib/Dialect/Vector/Transforms/BufferizableOpInterfaceImpl.cpp
--- a/mlir/lib/Dialect/Vector/Transforms/BufferizableOpInterfaceImpl.cpp
+++ b/mlir/lib/Dialect/Vector/Transforms/BufferizableOpInterfaceImpl.cpp
@@ -41,7 +41,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }
@@ -110,7 +110,7 @@
     return false;
   }
 
-  SmallVector<OpResult> getAliasingOpResult(Operation *op, OpOperand &opOperand,
+  AliasingOpResultList getAliasingOpResults(Operation *op, OpOperand &opOperand,
                                             const AnalysisState &state) const {
     return {};
   }