Checking Tests for Read-Once Functions over Arbitrary Bases
This provides a theoretical foundation for testing read-once functions, which is incremental but addresses a specific bottleneck in Boolean function analysis.
The paper tackles the problem of constructing checking tests for read-once Boolean functions over arbitrary bases, showing that every such function has a checking test with O(n^l) vectors, where n is the number of variables and l is the maximum arity in the basis, and this bound is tight up to a constant factor for some functions.
A Boolean function is called read-once over a basis B if it can be expressed by a formula over B where no variable appears more than once. A checking test for a read-once function f over B depending on all its variables is a set of input vectors distinguishing f from all other read-once functions of the same variables. We show that every read-once function f over B has a checking test containing O(n^l) vectors, where n is the number of relevant variables of f and l is the largest arity of functions in B. For some functions, this bound cannot be improved by more than a constant factor. The employed technique involves reconstructing f from its l-variable projections and provides a stronger form of Kuznetsov's classic theorem on read-once representations.