A separation theorem in property testing

Consider the following seemingly rhetorical question: Is it crucial for a property-tester to know the error parameter 2 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 2 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 2 is given as part of the input from the model in which 2 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 2 as part of the input, whose number of queries depends only on 2. (ii) For any fixed 2, there is a tester for P (that works only for that specific 2), 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 chromatic number, a result about testing monotone graph properties, as well as basic ideas from the theory of recursive functions. Of independent interest is a precise characterization of the monotone graph properties that can be tested with 2 being part of the input, which we obtain as one of the main steps of the paper.