slightly larger than 1.8, we surmise that because OBDDs and d-DNNFs utilizes the compactness of solutions, positions of critical points for these two languages differ from those for PIs. 

![0_image_0.png](0_image_0.png) 

![0_image_1.png](0_image_1.png) 
Interestingly, we exhibit that when r (r>0.3 for 3-SAT) is fixed, the sizes of target languages increase exponentially as the number of variables n grows linearly. We take 6 values of r between 0 and rp uniformly. For each r, we generate 1000 3-SAT instances varying n from 10 to 60 at increments of 5, and convert them into d-DNNFs. Compilation results are shown in Fig. 3. Curves with different values of r are all nearly linear when the logarithmic vertical axis is used, so we believe that the size grows exponentially in the general cases. However, the slope of each curve depends on r. When r is close to 1.8, the slope grows larger, and the curve of r=1.8 has the biggest slope among the 6 curves. As a result, though the average sizes increase exponentially, the sizes around phase transition regions grow fastest. 

![0_image_2.png](0_image_2.png) 
