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+# Issue #396: Duplicate Python Values - Analysis and Solutions
+
+## Problem Summary
+
+When Python objects are stored in egglog using `PyObject`, cloudpickle is used to serialize them to bytes. These bytes are then used for equality comparison and hashing in the Rust layer (`PyPickledValue`).
+
+**The Issue**: Cloudpickle does not always produce identical byte sequences for the same object when pickled by-value. This particularly affects:
+- Classes defined in `__main__` module
+- Closures and nested functions
+- Objects that cannot be imported normally
+
+When cloudpickle produces different bytes for the same logical object, the e-graph treats them as distinct nodes, creating duplicates.
+
+## Architecture Analysis
+
+### Current Flow:
+1. Python object → `cloudpickle.dumps()` → bytes
+2. Bytes stored in `PyObjectDecl.pickled` (Python) and `PyPickledValue` (Rust)
+3. Rust uses byte-level equality: `impl Eq for PyPickledValue { ... }`
+4. If bytes differ → different e-graph nodes
+
+### Root Cause:
+The `PyPickledValue` struct in `src/py_object_sort.rs` implements equality based on raw bytes:
+```rust
+#[derive(PartialEq, Eq, Hash, Clone)]
+pub struct PyPickledValue(pub Vec<u8>);
+```
+
+## Proposed Solutions
+
+### Solution 1: Object Identity Cache ⭐ (RECOMMENDED)
+**Approach**: Maintain a Python-side cache that maps object identity to pickled bytes.
+
+**Implementation**:
+- Add a cache in the EGraph: `Dict[int, bytes]` mapping `id(obj)` to pickled bytes
+- When creating a PyObject:
+  1. Check if `id(obj)` exists in cache
+  2. If yes, reuse those bytes
+  3. If no, pickle and cache
+- Use weak references to avoid keeping objects alive unnecessarily
+
+**Pros**:
+- ✅ Fast - no repeated pickling
+- ✅ Solves the common case where same object is added multiple times
+- ✅ Minimal changes to existing architecture
+- ✅ Preserves object identity semantics
+
+**Cons**:
+- ❌ Only works while objects are alive
+- ❌ Doesn't help with value equality (two equal but distinct objects)
+- ❌ Requires Python-side state management
+
+**Example Code Location**: `python/egglog/egraph_state.py` - add cache to EGraphState
+
+---
+
+### Solution 2: Hash-Based Deduplication
+**Approach**: Use Python's `hash()` for hashable objects, fall back to bytes for others.
+
+**Implementation**:
+- Modify `dump()` to compute and store both:
+  - Pickled bytes (for reconstruction)
+  - Python hash (for equality)
+- In Rust, compare using stored hash for hashable objects
+- Keep objects alive in a registry
+
+**Pros**:
+- ✅ Respects Python's hash semantics
+- ✅ Works for hashable objects
+
+**Cons**:
+- ❌ Doesn't work for unhashable objects (dicts, lists, etc.)
+- ❌ Complex lifecycle management
+- ❌ Hash collisions possible
+
+---
+
+### Solution 3: Unpickled Equality Comparison
+**Approach**: When comparing PyPickledValue for equality, unpickle and use Python's `==`.
+
+**Implementation**:
+- Override `PartialEq` in Rust to unpickle both objects
+- Call back to Python to perform equality check
+- Cache result to avoid repeated unpickling
+
+**Pros**:
+- ✅ Respects Python value equality semantics
+- ✅ Conceptually simple
+
+**Cons**:
+- ❌ **Very expensive** - unpickle on every equality check
+- ❌ May not work for objects without `__eq__`
+- ❌ Could be 100-1000x slower
+- ❌ Not suitable for use in hash tables
+
+---
+
+### Solution 4: Canonical Pickle Format
+**Approach**: Normalize pickled bytes to ensure determinism.
+
+**Implementation**:
+- Post-process pickled bytes to remove non-deterministic elements
+- Sort dictionaries, normalize timestamps, etc.
+
+**Pros**:
+- ✅ Would work with existing architecture
+
+**Cons**:
+- ❌ **Very difficult** to implement correctly
+- ❌ May break cloudpickle's reconstruction
+- ❌ Pickle format is complex and opaque
+- ❌ May not be possible for all objects
+
+---
+
+### Solution 5: Hybrid Identity + Value Cache
+**Approach**: Combine object identity cache with value-based deduplication.
+
+**Implementation**:
+- First check identity cache (fast path)
+- For new objects, compute a content-based fingerprint
+- Cache both identity and fingerprint mappings
+
+**Pros**:
+- ✅ Best of both worlds
+- ✅ Handles both identity and value equality
+
+**Cons**:
+- ❌ Complex implementation
+- ❌ Hard to define "content fingerprint" reliably
+- ❌ Still has performance overhead
+
+---
+
+### Solution 6: User-Provided Keys
+**Approach**: Allow users to optionally provide a custom key for deduplication.
+
+**Implementation**:
+```python
+PyObject(obj, key="my_unique_key")
+```
+
+**Pros**:
+- ✅ Simple to implement
+- ✅ User has full control
+- ✅ Works for any object
+
+**Cons**:
+- ❌ Puts burden on user
+- ❌ Not automatic
+- ❌ Easy to misuse
+
+---
+
+## Recommendation
+
+**Implement Solution 1 (Object Identity Cache)** as it:
+1. Solves the most common case (same object added multiple times)
+2. Requires minimal changes
+3. Has good performance characteristics
+4. Can be implemented entirely in Python layer
+
+### Implementation Plan:
+
+1. **Add cache to EGraph** (`python/egglog/egraph_state.py`):
+   ```python
+   class EGraphState:
+       def __init__(self):
+           self._py_object_cache: Dict[int, bytes] = {}
+   ```
+
+2. **Modify PyObject creation** to check cache first:
+   ```python
+   def create_py_object(obj):
+       obj_id = id(obj)
+       if obj_id in cache:
+           return cached_bytes
+       pickled = cloudpickle.dumps(obj)
+       cache[obj_id] = pickled
+       return pickled
+   ```
+
+3. **Use weak references** to avoid keeping objects alive:
+   ```python
+   import weakref
+   self._py_object_refs = weakref.WeakValueDictionary()
+   ```
+
+4. **Add tests** to verify deduplication works
+
+### Future Enhancements:
+- Add optional value-based deduplication for specific types
+- Provide user override for custom equality
+- Add metrics/warnings when duplicates are detected
+
+## Test Results
+
+Created test file: `python/tests/test_issue_396_duplicate_pyobjects.py`
+- Tests demonstrate expected behavior
+- In simple cases, cloudpickle IS deterministic
+- Real-world cases with classes in `__main__` may still have non-determinism
+- Test provides framework to catch regressions
+
+## References
+
+- Issue: https://github.com/egraphs-good/egglog-python/issues/396
+- PR Discussion: https://github.com/egraphs-good/egglog-python/pull/393
+- Cloudpickle source: https://github.com/cloudpipe/cloudpickle/blob/f5199fe2bc102a5ee070c743336699fc885ca966/cloudpickle/cloudpickle.py#L291-L303
diff --git a/python/tests/test_issue_396_duplicate_pyobjects.py b/python/tests/test_issue_396_duplicate_pyobjects.py
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+"""
+Test for issue #396: Duplicate Python Values
+
+This test demonstrates that cloudpickle can produce different byte sequences
+for the same object when pickled by-value, particularly for classes defined
+in __main__, which results in duplicate egglog PyObject nodes.
+"""
+
+from __future__ import annotations
+
+import cloudpickle
+import pytest
+
+from egglog import *
+
+
+def test_duplicate_pyobjects_same_class_instance():
+    """
+    Test that demonstrates the duplicate PyObject issue.
+
+    When a class is defined in __main__ (or in this test module), cloudpickle
+    serializes it by value rather than by reference. According to issue #396,
+    this can produce different byte sequences for the same object across
+    different pickle calls, leading to duplicate nodes in the e-graph.
+
+    EXPECTED BEHAVIOR: The same Python object should create equal PyObject nodes
+    ACTUAL BEHAVIOR: May create duplicate nodes if cloudpickle produces different bytes
+
+    Note: This test may pass in simple cases where cloudpickle is deterministic,
+    but the issue can still occur in real-world scenarios where:
+    - Classes are defined in __main__
+    - Objects are pickled at different times/contexts
+    - There are closures or complex nested structures
+    """
+
+    # Define a simple class in the test module (acts like __main__)
+    class MyClass:
+        def __init__(self, value) -> None:
+            self.value = value
+
+        def __eq__(self, other):
+            return isinstance(other, MyClass) and self.value == other.value
+
+    egraph = EGraph()
+
+    # Create the SAME Python object
+    obj = MyClass(42)
+
+    # Pickle the same object multiple times to check for determinism
+    pickled1 = cloudpickle.dumps(obj)
+    pickled2 = cloudpickle.dumps(obj)
+
+    # Report on pickle determinism
+    print("\nCloudpickle determinism check:")
+    print(f"  First pickle length: {len(pickled1)}")
+    print(f"  Second pickle length: {len(pickled2)}")
+    print(f"  Bytes are identical: {pickled1 == pickled2}")
+
+    if pickled1 != pickled2:
+        print("  ⚠ Cloudpickle produced DIFFERENT bytes for the same object!")
+    else:
+        print("  ✓ Cloudpickle is deterministic in this case")
+
+    # Create PyObject expressions from the same object instance multiple times
+    # In an ideal world, these should be recognized as equal
+    py_obj1 = egraph.let("obj1", PyObject(obj))
+    py_obj2 = egraph.let("obj2", PyObject(obj))
+
+    egraph.run(10)
+
+    # Check if the e-graph recognizes them as equal
+    # This should pass, but might fail if cloudpickle produced different bytes
+    try:
+        egraph.check(eq(py_obj1).to(py_obj2))
+        print("  ✓ E-graph correctly identifies objects as equal")
+    except Exception as e:
+        print("  ✗ E-graph treats objects as DIFFERENT - duplicates detected!")
+        print(f"  Error: {e}")
+        pytest.fail(
+            f"Duplicate PyObject nodes detected for the same Python object.\n"
+            f"This means cloudpickle produced different bytes for identical objects.\n"
+            f"Error: {e}"
+        )
+
+
+def test_duplicate_pyobjects_nested_class():
+    """
+    Test with nested class definitions which are even more likely
+    to be serialized by-value and produce different bytes.
+    """
+
+    class OuterClass:
+        class InnerClass:
+            def __init__(self, x) -> None:
+                self.x = x
+
+    egraph = EGraph()
+
+    obj = OuterClass.InnerClass(100)
+
+    # Create the same object in the egraph multiple times
+    py_obj1 = egraph.let("nested1", PyObject(obj))
+    py_obj2 = egraph.let("nested2", PyObject(obj))
+
+    # Check if they're equal
+    egraph.run(10)
+
+    try:
+        egraph.check(eq(py_obj1).to(py_obj2))
+        print("✓ Nested objects are equal - no duplicates")
+    except Exception as e:
+        print("✗ Nested objects are NOT equal - duplicates detected!")
+        pytest.fail(f"Duplicate PyObject nodes for nested class: {e}")
+
+
+def test_pyobject_with_closure():
+    """
+    Test that functions with closures (which must be serialized by-value)
+    can create duplicate PyObject nodes.
+    """
+
+    def make_adder(x):
+        def adder(y):
+            return x + y
+
+        return adder
+
+    egraph = EGraph()
+
+    add_five = make_adder(5)
+
+    # Create the same closure in the egraph multiple times
+    py_fn1 = egraph.let("fn1", PyObject(add_five))
+    py_fn2 = egraph.let("fn2", PyObject(add_five))
+
+    egraph.run(10)
+
+    try:
+        egraph.check(eq(py_fn1).to(py_fn2))
+        print("✓ Closures are equal - no duplicates")
+    except Exception as e:
+        print("✗ Closures are NOT equal - duplicates detected!")
+        pytest.fail(f"Duplicate PyObject nodes for closure: {e}")
+
+
+if __name__ == "__main__":
+    # Run the tests manually to see output
+    print("Testing duplicate PyObjects with same class instance...")
+    test_duplicate_pyobjects_same_class_instance()
+
+    print("\nTesting duplicate PyObjects with nested class...")
+    test_duplicate_pyobjects_nested_class()
+
+    print("\nTesting duplicate PyObjects with closure...")
+    test_pyobject_with_closure()