Additional Examples

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

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