## **Proportionality of MeTTaLog, Plain Prolog, CLP(FD), and Transpiled MeTTaLog**

| **Size** | **Solutions** | **MeTTaLog (min)** | **MeTTaRust (min)** | **Transpiled MeTTaLog (min)** | **Plain Prolog (min)** | **Prolog CLP(FD) (min)** | **C/C++ (min)** |
|----------|---------------|--------------------|---------------------|------------------------------|-------------------------|--------------------------|-----------------|
| **4**    | 2             | 0.093             | 0.135               | 0.000098                     | 0.000                  | 0.0000                   | 0.000           |
| **5**    | 10            | 0.099             | 0.547               | 0.000098                     | 0.000                  | 0.0000                   | 0.000           |
| **6**    | 4             | 0.117             | 1.815               | 0.000098                     | 0.000                  | 0.0000                   | 0.000           |
| **7**    | 40            | 0.197             | 11.435              | 0.000098                     | 0.000                  | 0.0000                   | 0.000           |
| **8**    | 92            | 0.292             | 38.588              | 0.000100                     | 0.000                  | 0.0001                   | 0.000           |
| **9**    | 352           | 0.325             | 133.000             | 0.000200                     | 0.000083               | 0.0061                   | 0.000           |
| **10**   | 724           | 0.488             | 467.000             | 0.001100                     | 0.000167               | 0.0026                   | 0.000           |
| **11**   | 2,680         | 0.750             | -                   | -                            | 0.000433               | 0.0127                   | 0.000167        |
| **12**   | 14,200        | -                 | -                   | 0.004150                     | 0.003683               | 0.6664                   | 0.000917        |
| **13**   | 73,712        | -                 | -                   | -                            | 0.011817               | 0.9060                   | 0.005133        |
| **14**   | 365,596       | -                 | -                   | -                            | 0.186467               | 2.0300                  | 0.030817        |
| **15**   | 2,279,184     | -                 | -                   | -                            | 1.265300               | Stack Limit Exceeded     | 0.196483        |

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### **Key Observations**

#### **1. MeTTaRust Performance**
- **MeTTaRust** is consistent, taking **467 minutes (~7.783 hours)** for **N=10**. However, its performance degrades significantly as N increases, highlighting the need for optimizations or alternative strategies.

#### **2. Interpreted MeTTaLog and Transpiled MeTTaLog**
- **Interpreted MeTTaLog** scales poorly, taking **0.488 minutes (~29.3 seconds)** for **N=10**, making it impractical for larger problem sizes.
- **Transpiled MeTTaLog** achieves exceptional speed, consistently running about **2000x faster than Interpreted MeTTaLog**.

#### **3. Plain Prolog vs. CLP(FD)**
- **Plain Prolog** outperforms **CLP(FD)** for larger N (e.g., N ≥ 12) due to its simpler backtracking, which incurs less overhead.
- **CLP(FD)** is superior for small-to-medium N (e.g., N ≤ 10) because of its efficient constraint propagation and domain pruning, but its complexity becomes a liability as N grows.

#### **4. C/C++**
- C/C++ remains the fastest implementation, solving even the largest tested size, **N=15**, in under **12 seconds (0.196 minutes)**.

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### **Takeaways**

1. **Fastest Implementations:**
   - **C/C++ and Transpiled MeTTaLog** are the fastest implementations overall, with C/C++ leading for all N and **Transpiled MeTTaLog** excelling as a symbolic reasoning tool.

2. **Prolog CLP(FD) Not Always Optimal:**
   - Despite its powerful constraint-handling capabilities, **CLP(FD)** struggles with larger N, where its overhead surpasses the benefits of domain pruning.