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Update BLOCK_WORLD.md

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- ## 🧮 Complexity Analysis
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- ### Solution Length Formula
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- For N blocks across K stacks, the **worst-case optimal solution length** is:
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- ```
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- L(N) = O(N)
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- ```
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- **Explanation**: In the worst case, each block must be moved exactly once from its initial position to its goal position. When K ≥ 3 stacks are available, there is always at least one empty stack (or a stack whose top block is already correctly positioned) that can serve as a temporary buffer.
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- ### Transition Verification
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- Checking whether a proposed move is legal requires:
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- ```
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- C_verify = O(1)
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- ```
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- **Why O(1)?** To validate a move `[block, source, dest]`:
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- 1. Check that `source` stack is non-empty (the block to be moved exists)
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- 2. Check that `dest` stack either exists or is an empty position
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- Both checks involve reading only the **top element** of two stacks, independent of N.
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- ### Action Space
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- The branching factor (number of possible moves per state):
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- ```
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- b = O(K²)
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- ```
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- With K=3 stacks, at most 3×2 = 6 moves per state (from each non-empty stack to the other two stacks).
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- ---
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- ## 📈 Benchmark Results
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- ### Model Performance (from paper)
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- | Model | Condition | Val Accuracy (N=1-7) | OOD Accuracy (N=8-10) |
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- |-------|-----------|---------------------|---------------------|
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- | **T5-small (60M)** | Pre-trained + Fine-tuned | **97.27%** | **81.00%** ✅ |
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- | GPT-2 (124M) | Pre-trained + Fine-tuned | 24.55% | 0.00% |
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- | T5-small | Trained from scratch | 0.00% | 0.00% |
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- | GPT-2 | Trained from scratch | 21.82% | 0.00% |
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- ### Key Findings
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- 1. **Architecture > Scale**: T5 (60M parameters) dramatically outperforms GPT-2 (124M parameters)
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- - T5's bidirectional encoder provides full goal access at every decoding step
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- - GPT-2's causal mask creates a planning bottleneck
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- 2. **Pre-training Helps**: Pre-trained T5 gains +97.27 percentage points over training from scratch
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- - Pre-training transfers because Block World has **O(1) local transition structure**
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- 3. **Gradual OOD Degradation**: T5 maintains strong performance even on harder instances
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- - N=8: 84% accuracy
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- - N=9: 84% accuracy
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- - N=10: 75% accuracy
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- - This gradual decay (not abrupt collapse) indicates genuine rule learning
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- ### Failure Mode Analysis
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- **T5 Fine-tuned failures** (on the 3% validation set that failed):
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- - **Loops**: 67% - Model generates valid moves but revisits previous states
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- - **Premature Stop**: 33% - Model halts before reaching goal
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- **GPT-2 Fine-tuned failures**:
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- - **Validation**: 81% loops (model learns legal moves but no goal direction)
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- - **OOD**: 91% invalid moves (systematic shift! Constraint generalization fails)
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- ---
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  ## 💡 Usage Tips
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  ### For Model Training
 
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  ## 💡 Usage Tips
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  ### For Model Training