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.
ExamplePreprocessed ColumnsTrace ColumnsLogUp Columns
Permutation argument check• unordered list
• ordered list		1 / unordered list - 1 / ordered list
Range Check (0 ≤ a < 2^bits)[0,2^bits) rows• lookup columns
• multiplicities column		• 1 / lookup
• multiplicity / preprocessed
Comparator check (a > b)[0,2^bits) rows• a
• b
• multiplicities column		• 1 / (a - b)
• multiplicity / preprocessed
IsZero check (a == 0)• a
• a_inv
XOR operationsvalid XOR operations• lookup columns
• multiplicities column
Selectorscolumns of 0s and 1s
Checking whether current row is first row or notsingle column (first row = 1, other rows = 0)
Connecting multiple components (output of Component A is input of Component B)• 1 / output
• 1 / input * (-1)
Public Input/Output1 / input + 1 / output