Property Testing - Definition and Variants

Definition and Variants

Formally, a property testing algorithm with query complexity q(n) and proximity parameter ε for a decision problem L is a randomized algorithm that, on input x (an instance of L) makes at most q(|x|) queries to x and behaves as follows:

• If x is in L, the algorithm accepts x with probability at least ⅔.
• If x is ε-far from L, the algorithm rejects x with probability at least ⅔.

Here, "x is ε-far from L" means that the Hamming distance between x and any string in L is at least ε|x|.

A property testing algorithm is said to have one-sided error if it satisfies the stronger condition that the accepting probability for instances x ∈ L is 1 instead of ⅔.

A property testing algorithm is said be non-adaptive if it performs all its queries before it "observes" any answers to previous queries. Such an algorithm can be viewed as operating in the following manner. First the algorithm receives its input. Before looking at the input, using its internal randomness, the algorithm decides which symbols of the input are to be queried. Next, the algorithm observes these symbols. Finally, without making any additional queries (but possibly using its randomness), the algorithm decides whether to accept or reject the input.