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TR11-041 | 24th March 2011 15:09
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#### Testing Computability by Width-Two OBDDs

**Abstract:**
Property testing is concerned with deciding whether an object

(e.g. a graph or a function) has a certain property or is ``far''

(for a prespecified distance measure) from every object with

that property. In this work we consider the property of being

computable by a read-once width-$2$ {\em Ordered Binary Decision Diagram}

(OBDD), also known as a {\em branching program\/}, in two settings.

In the first setting the order of the variables is fixed and given

to the algorithm, while in the second setting it is not fixed.

That is, while in the first setting we should accept a function

$f$ if it is computable by a width-$2$ OBDD with a {\em given\/}

order of the variables, in the second setting we should accept a

function $f$ if there {\em exists\/} an order of the variables

according to which a width-$2$ OBDD can compute $f$.

Width-$2$ OBDDs generalize two classes of functions that have been

studied in the context of property testing - linear functions

(over $GF(2)^n$) and monomials. In both these cases membership can

be tested by performing a number of queries that is {\em independent

of the number of variables, $n$\/} (and is linear in $1/\epsilon$,

where $\epsilon$ is the distance parameter). In contrast, we show

that testing computability by width-$2$ OBDDs when the order of variables

is fixed and known requires a number of queries that grows logarithmically

with $n$ (for a constant $\epsilon$), and we provide an algorithm that performs

$\tilde{O}(\log n/\epsilon)$ queries. For the case where the order is

not fixed, we show that there is {\em no\/} testing algorithm that performs

a number of queries that is sublinear in $n$.