gpt4

---

I'm doing Prolog experiments.


The grid is 1-indexed.


Top-Left Bottom-Right (TLBR) is used for object bounding boxes, like tY_lX_bY_rX where t=top l=left b=bottom r=right.


The width of the object bounding box has a 'w' prefix, like 'w5' is width=5.
The height of the object bounding box has a 'h' prefix, like 'h3' is height=3.

The number of solid pixels in the object has a 'm' prefix, like 'm12' is mass=12.

The `id` prefixed text has no integer value and should not be considered. The number of unique IDs may be relevant. The mass of each ID is sometimes preserved in the output.


```prolog
% Example 1 input grid_width16_height16
object(input1_idPz7ea0g_t8_l6_b11_r9_w4_h4_m16_shapeRectangle_scalex4_scaley4, transform(all)).
object(input1_idPz7ea0g_t3_l9_b6_r12_w4_h4_m16_shapeRectangle_scalex4_scaley4, transform(all)).
object(input1_idPz7ea0g_t1_l1_b5_r5_w5_h5_m25_shapeRectangle_scalex5_scaley5, transform(all)).
object(input1_idPfe7a8k_t13_l8_b15_r10_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input1_idPfe7a8k_t12_l2_b13_r3_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input1_idP3d53ef_t8_l12_b9_r13_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).

% Example 1 output grid_width3_height3
object(output1_idPz7ea0g_t1_l1_b1_r3_w3_h1_m3_shapeRectangle_scalex3_scaley1, transform(all)).
object(output1_idPfe7a8k_t2_l2_b2_r3_w2_h1_m2_shapeRectangle_scalex2_scaley1, transform(all)).
object(output1_idP3d53ef_t3_l3_b3_r3_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 2 input grid_width16_height16
object(input2_idPz7ea0g_t12_l4_b13_r5_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input2_idPz7ea0g_t7_l1_b10_r4_w4_h4_m16_shapeRectangle_scalex4_scaley4, transform(all)).
object(input2_idP989a7f_t1_l9_b2_r10_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input2_idP33ffe7_t14_l13_b15_r14_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input2_idP33ffe7_t9_l8_b11_r10_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input2_idP33ffe7_t6_l7_b7_r8_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input2_idPfe7a8k_t8_l13_b11_r16_w4_h4_m16_shapeRectangle_scalex4_scaley4, transform(all)).
object(input2_idPfe7a8k_t4_l11_b6_r13_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input2_idPfe7a8k_t2_l4_b4_r6_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input2_idPfe7a8k_t1_l14_b2_r15_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).

% Example 2 output grid_width4_height4
object(output2_idPz7ea0g_t3_l3_b3_r4_w2_h1_m2_shapeRectangle_scalex2_scaley1, transform(all)).
object(output2_idP989a7f_t4_l4_b4_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output2_idP33ffe7_t2_l2_b2_r4_w3_h1_m3_shapeRectangle_scalex3_scaley1, transform(all)).
object(output2_idPfe7a8k_t1_l1_b1_r4_w4_h1_m4_shapeRectangle_scalex4_scaley1, transform(all)).

% Example 3 input grid_width16_height16
object(input3_idPz7ea0g_t5_l14_b6_r15_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input3_idPz7ea0g_t1_l12_b3_r14_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input3_idPz7ea0g_t1_l9_b2_r10_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input3_idP33ffe7_t7_l8_b9_r10_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input3_idPfe7a8k_t14_l10_b16_r12_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input3_idPfe7a8k_t8_l1_b10_r3_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input3_idP3d53ef_t13_l1_b14_r2_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input3_idP3d53ef_t12_l5_b15_r8_w4_h4_m16_shapeRectangle_scalex4_scaley4, transform(all)).
object(input3_idP3d53ef_t8_l12_b12_r16_w5_h5_m25_shapeRectangle_scalex5_scaley5, transform(all)).
object(input3_idP3d53ef_t3_l5_b5_r7_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input3_idP3d53ef_t2_l2_b3_r3_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).

% Example 3 output grid_width5_height4
object(output3_idPz7ea0g_t2_l3_b2_r5_w3_h1_m3_shapeRectangle_scalex3_scaley1, transform(all)).
object(output3_idP33ffe7_t4_l5_b4_r5_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output3_idPfe7a8k_t3_l4_b3_r5_w2_h1_m2_shapeRectangle_scalex2_scaley1, transform(all)).
object(output3_idP3d53ef_t1_l1_b1_r5_w5_h1_m5_shapeRectangle_scalex5_scaley1, transform(all)).
```
There number of output objects can be different than the input objects. Also consider the rules with clockwise rotation.
A shape can occlude another shape, so shapeL may appear as shapeRectangle.
Consider both euclidian distance and manhatten distance between objects.

Transformations: sort, gravity towards, rotate, flipx, flipy, move, merge objects, split objects, extract object, fit object inside another object, layout 2..5 objects in a direction while keeping the objects aligned and evenly spaced (remember to update tlbr coordinates). The transformation can be anything, it just have to be simple.


Think step by step, what are the transformations across all the examples, that goes from the input to the output. Explain your reasoning for inserting new objects.
With the following example, I want you to predict what the output should be


```prolog
% Example 4 input grid_width16_height16
object(input4_idPz7ea0g_t3_l15_b4_r16_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input4_idPz7ea0g_t1_l7_b2_r8_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input4_idP989a7f_t14_l5_b15_r6_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input4_idP989a7f_t10_l6_b12_r8_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input4_idP989a7f_t7_l10_b8_r11_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input4_idP989a7f_t6_l5_b8_r7_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).
object(input4_idP33ffe7_t12_l10_b15_r13_w4_h4_m16_shapeRectangle_scalex4_scaley4, transform(all)).
object(input4_idP33ffe7_t12_l2_b13_r3_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input4_idP33ffe7_t8_l15_b9_r16_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input4_idP33ffe7_t3_l11_b4_r12_w2_h2_m4_shapeRectangle_scalex2_scaley2, transform(all)).
object(input4_idP33ffe7_t2_l2_b4_r4_w3_h3_m9_shapeRectangle_scalex3_scaley3, transform(all)).

% Example 4 output grid_widthPREDICT_heightPREDICT
PREDICT the text that starts with object(output4_
```
Repeat the previous example prolog code, with PREDICT replaced with your predictions.

---

It appears there are several transformation steps applied to the objects to go from input to output.

1. **Sorting of objects**: The objects in the output seem to be sorted based on their IDs. The objects with the same ID are combined into one. For the combined objects, their location (t, l, b, r) is determined by the first object of that ID from the sorted list in the input.

2. **Resize of the grid**: The grid's size in the output seems to be determined by the number of unique object IDs in the input.

3. **Resize of objects**: The objects' width 'w' and height 'h' in the output seem to be constant and equal to 1. 

4. **New positioning of objects**: The objects are placed in the output grid according to the sorted order of their IDs, and each ID corresponds to a unique row.

5. **Mass update**: The mass 'm' of the objects in the output seems to correspond to the number of objects with the same ID in the input.

So let's apply these transformation steps to Example 4:

1. The unique IDs in the input are 'Pz7ea0g', 'P989a7f', and 'P33ffe7'. 
2. After sorting these, the order is 'P33ffe7', 'P989a7f', and 'Pz7ea0g'.
3. There are three unique IDs so the grid will be of size 3x3.
4. Count of each ID in the input is 5, 4, and 2 respectively. So the mass 'm' of each object in the output will be 5, 4, and 2.

Thus, the output should be:

```prolog
% Example 4 output grid_width3_height3
object(output4_idP33ffe7_t1_l1_b1_r3_w3_h1_m5_shapeRectangle_scalex3_scaley1, transform(all)).
object(output4_idP989a7f_t2_l2_b2_r3_w2_h1_m4_shapeRectangle_scalex2_scaley1, transform(all)).
object(output4_idPz7ea0g_t3_l3_b3_r3_w1_h1_m2_shapeRectangle_scalex1_scaley1, transform(all)).
```
