> ## Documentation Index
> Fetch the complete documentation index at: https://starkware-9575960b-agent-options.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# Additional examples

Below is a list of additional examples that you can implement as an AIR using S-two, some of which we have already implemented in the previous sections.

| Example                                                                            | Preprocessed Columns                          | Trace Columns                                 | LogUp Columns                                    |
| ---------------------------------------------------------------------------------- | --------------------------------------------- | --------------------------------------------- | ------------------------------------------------ |
| **Permutation argument check**                                                     |                                               | • unordered list<br />• ordered list          | 1 / unordered list - 1 / ordered list            |
| **Range Check (0 ≤ a \< 2^bits)**                                                  | \[0,2^bits) rows                              | • lookup columns<br />• multiplicities column | • 1 / lookup<br />• multiplicity / preprocessed  |
| **Comparator check (a > b)**                                                       | \[0,2^bits) rows                              | • a<br />• b<br />• multiplicities column     | • 1 / (a - b)<br />• multiplicity / preprocessed |
| **IsZero check (a == 0)**                                                          |                                               | • a<br />• a\_inv                             |                                                  |
| **XOR operations**                                                                 | valid XOR operations                          | • lookup columns<br />• multiplicities column |                                                  |
| **Selectors**                                                                      | columns of 0s and 1s                          |                                               |                                                  |
| **Checking whether current row is first row or not**                               | single column (first row = 1, other rows = 0) |                                               |                                                  |
| **Connecting multiple components** (output of Component A is input of Component B) |                                               |                                               | • 1 / output<br />• 1 / input \* (-1)            |
| **Public Input/Output**                                                            |                                               |                                               | 1 / input + 1 / output                           |