Checking Tests for Read-Once Functions over Arbitrary Bases

TL;DR

This paper establishes that read-once Boolean functions over any basis have concise checking tests of size proportional to n^l, with a reconstruction technique that strengthens classical theorems in the field.

Contribution

It proves that every read-once function over a basis has a checking test of size O(n^l), improving understanding of test complexity and providing a new reconstruction method.

Findings

Checking tests for read-once functions are of size O(n^l).

The bound on test size is tight for some functions.

A new reconstruction technique from l-variable projections is introduced.

Abstract

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.