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		4 Identification by Computable Scientists

		The focus of our book is the inductive competence of scientists whose behavior can be simulated by computer. The technological interest of this topic is evident, since it lies at the heart of attempts to develop artificial systems of empirical inquiry (see Carbonell [23] and Langley, Simon, Bradshaw, and Zytkow [123]). An additional reason for our focus comes from Cognitive Science. The computer simulability of human ratiocination is one of the most popular hypotheses in that discipline, so it is natural to speculate that children's learning functions are algorithmic processes. By studying the consequences of this hypothesis, it may be possible to provide further constraints on the theory of comparative grammar (see Section 3.1.2).

		The present chapter considers the simplest paradigms in which computable scientists appear. Much of the material in later chapters will depend on the results and definitions to be presented here. As before, we discuss language identification prior to function identification. As a preliminary, we introduce a system of notation that will occur throughout the rest of our study.

		§4.1 Preliminaries

		4.1.1 An Indexing for the Computable Scientists

		Computer simulable scientists can be conceived from either an extensional or an intensional point of view. Extensionally, a computable scientist — like any scientist — is a function from SEQ or SEG to N; it is computable in virtue of there being a procedure that recognizes its pairs. Intensionally, a computer scientist is such a procedure, i.e., a computer program that accepts members of SEQ or SEG and returns (if defined) members of N. Our official viewpoint is extensional, so that by "computable scientist" we mean a certain kind of function. As an aid to intuition, however, we sometimes take the intensional point of view, referring to a scientist as a program or Turing Machine.

		There are thus three kinds of computable functions now in play within our theory. They may be distinguished by their domains. First, there are the recursive functions (partial and total) with domain N. Next, there are computable scientists within the paradigm of language identification; these functions have domain SEQ. Finally, there are computable scientists within the paradigm of function identification; they have domain