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 for sorting. The number of unique IDs may be relevant. The total mass of certain IDs is sometimes preserved in the output.
```prolog
% Example 1 input grid_width29_height3
object(input1_idP33ffe7_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input1_idPfe7a8k_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 1 output grid_width29_height3
object(output1_idP33ffe7_t2_l29_b2_r29_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output1_idP33ffe7_t2_l16_b2_r16_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output1_idP33ffe7_t2_l7_b2_r7_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output1_idP33ffe7_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output1_idPfe7a8k_t2_l22_b2_r22_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output1_idPfe7a8k_t2_l11_b2_r11_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output1_idPfe7a8k_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output1_idPfe7a8k_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 2 input grid_width19_height3
object(input2_idPz7ea0g_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input2_idP48kmo7_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input2_idP3d53ef_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 2 output grid_width19_height3
object(output2_idPz7ea0g_t2_l11_b2_r11_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output2_idPz7ea0g_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output2_idP48kmo7_t2_l7_b2_r7_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output2_idP48kmo7_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output2_idP3d53ef_t2_l16_b2_r16_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output2_idP3d53ef_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 3 input grid_width19_height3
object(input3_idPfe7a8k_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input3_idP3d53ef_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 3 output grid_width19_height3
object(output3_idPfe7a8k_t2_l16_b2_r16_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output3_idPfe7a8k_t2_l7_b2_r7_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output3_idPfe7a8k_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output3_idP3d53ef_t2_l11_b2_r11_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output3_idP3d53ef_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output3_idP3d53ef_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_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. Sometimes it's the occluded object that gets transformed.
Consider both euclidian distance and manhatten distance between objects.
Don't invent new IDs. Check how much an object moves relative x, y.


Objects that stay stationary may be a useful landmark. A landmark may be a starting point for inserting a new object next to. Check that the examples have landmarks.


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.
Check if objects are aligned to a certain edge. Check if the mass is preserved. Count the number of object groups.
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_width29_height3
object(input4_idPz7ea0g_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input4_idPfe7a8k_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input4_idP3d53ef_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 4 output grid_width29_height3
PREDICT the text that starts with object(output4_
```
Repeat the previous example prolog code, with PREDICT replaced with your predictions.

---

Let's first try to understand the transformations that are occurring in the previous examples.

- All the objects in the input are rectangles with a width and height of 1 (they are squares).

- The number of output objects for each ID in the output is directly proportional to the position of the object with the same ID in the input.

- The IDs in the input are preserved in the output, i.e., no new IDs are introduced.

- The position of the object in the output seems to be a sequence of positions starting from its original position and moving further away. The distance it moves away increases as the number of duplicates increase.

Let's apply these transformations to the fourth example:

```prolog
% Example 4 input grid_width29_height3
object(input4_idPz7ea0g_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input4_idPfe7a8k_t2_l1_b2_r1_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(input4_idP3d53ef_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).

% Example 4 output grid_width29_height3
object(output4_idPz7ea0g_t2_l16_b2_r16_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output4_idPz7ea0g_t2_l7_b2_r7_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output4_idPz7ea0g_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output4_idPfe7a8k_t2_l22_b2_r22_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output4_idPfe7a8k_t2_l11_b2_r11_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output4_idPfe7a8k_t2_l4_b2_r4_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output4_idP3d53ef_t2_l11_b2_r11_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
object(output4_idP3d53ef_t2_l2_b2_r2_w1_h1_m1_shapeRectangle_scalex1_scaley1, transform(all)).
```

This prediction follows the rules observed from the previous examples. It duplicates the objects based on their initial position and places them further away from their initial position. The distance from the initial position for the duplicates is equal to their initial position. Objects that are occluded have been transformed according to the observed transformation rules.

