 
Summary: A Separation Theorem in Property Testing
Noga Alon
Asaf Shapira
Abstract
Consider the following seemingly rhetorical question: Is it crucial for a propertytester to know
the error parameter in advance? Previous papers dealing with various testing problems, suggest
that the answer may be no, as in these papers there was no loss of generality in assuming that is
given as part of the input, and is not known in advance. Our main result in this paper, however,
is that it is possible to separate a natural model of property testing in which is given as part
of the input from the model in which is known in advance (without making any computational
hardness assumptions). To this end, we construct a graph property P, which has the following
two properties:
(i) There is no tester for P accepting as part of the input, whose number of queries depends
only on .
(ii) For any fixed , there is a tester for P (that works only for that specific ), which makes a
constant number of queries.
Interestingly, we manage to construct a separating property P, which is combinatorially natural
as it can be expressed in terms of forbidden subgraphs and also computationally natural as it can
be shown to belong to coNP.
The main tools in this paper are efficiently constructible graphs of high girth and high chro
