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 • 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 operations
valid XOR operations
• lookup columns • 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)
