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		part of (or at least connected to) the reality she. is trying to discover, she might have a certain set of ideas if the world is one way, and a, different set if the world is another. To take the starkest example, if DNA had different chemical properties than it does then geneticists (whose minds are partly determined by their own stock of DNA) might have started off with a different set of potential hypotheses about enzyme production and cell regulation. This kind of interaction suggests a richer representation of scientists, to be pursued in the present chapter. Specifically, scientists will be fitted out with a numerical parameter that communicates partial information about the reality inscribed in the text they are examining. In this way, the correct hypothesis impacts the class of potential realities for which the scientist is suited. We shall consider two such parameterization schemes, as follows.

		Upper bounds on hypotheses. First, we shall assume that scientists have prior information about the size of a correct explanation for the phenomenon under investigation. Specifically, they are given an upper bound on the minimum size program for the function they are facing. Section 10.2 considers the extent to which this kind of information facilitates inquiry.

		Approximate hypotheses. Second, we assume that scientists have approximately true theories about the language or function with which Nature confronts them. The utility of different kinds of approximations will be examined in Section 10.3.

		§10.2 Upper Bound on the Size of Hypothesis

		We begin by considering the impact on function identification of advance information about the size of the smallest, explanatory hypothesis for the incoming data. Then we consider the analogous issue for language learning.

		§10.2.1 Function Identification with Program Size Upper Bound

		Recall from the discussion on page 23 that our standard program size measure is That is, the size of program index p is simply p itself. This measure, while not completely natural, is mathematically convenient and, as discussed in Chapter 2, recursion theoretically equivalent to any other program size measure (Definition 2.8). Thus, in the following, "an upper bound on the size of the minimal sized hypothesis for " shall mean "an upper bound on the minimal index for f." Such additional information can