Testing Computability by Width Two OBDDs

Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far” (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by a read-once width-2

Ordered Binary Decision Diagram

(OBDD), also known as a

branching program

, where the order of the variables is known. Width-2 OBDDs generalize two classes of functions that have been studied in the context of property testing - linear functions (over

GF

(2)) and monomials. In both these cases membership can be tested in time that is linear in 1/

ε

. Interestingly, unlike either of these classes, in which the query complexity of the testing algorithm does not depend on the number,

n

, of variables in the tested function, we show that (one-sided error) testing for computability by a width-2 OBDD requires Ω(log(

n

)) queries, and give an algorithm (with one-sided error) that tests for this property and performs

$\tilde{O}(\log(n)/\epsilon)$

queries.