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		be incorporated into the identification paradigm by viewing a scientist as computing a mapping from N × SEG into N, where the first argument is the size bound. We let M, with or without decorations, range over such scientists; it will be clear from context if we mean a scientist with or without additional information. Thus, M(x,  s ) denotes the hypothesis conjectured by scientist M on evidential state  s  and additional information x. The following definition formalizes the notion of convergence of a scientist on a function in the presence of additional information.

		10.1 Definition (read: M on f with additional information x converges) just in case there exists an i such that for all but finitely many n, M(x, f[n]) = i. If , then M(x, f) is defined = the unique i such that for all but finitely many n, M(x, f[n]) = i; otherwise, M(x, f) is said to be undefined.

		The next definition extends the Ex paradigm to model situations where a scientist, in addition to being fed a graph of the function, is also presented with an upper bound on the minimal program index for the function. The definition below also models cases in which an upper bound may only be available for the minimal program index for a finite variant of the function being identified. To this end, for ), we let MinProgc(f) denote the minimal program index in the  j -programming system that computes f with at most c errors, i.e., .

		10.2 Definition Let a, .

		(a) M Bexa,c-identifies f (written: ) just in case, for all , and  j M(x,f) =a f.

		(b) .

		Thus a scientist M Bexa,c-identifies f just in case M, fed the graph of f and an upper bound on MinProgc(f), converges to an index for a program that computes f with at most a errors. The notion of Bex0,0-identification was first studied by Freivalds and Wiehagen [67]. In the Bexa,c paradigm, a is the bound on the number of errors allowed in the converged program and c is the bound on the number of errors allowed in the additional information. The reader should note that in this paradigm a scientist may converge to different indexes for different upper bounds. However, if we further require that the final converged index be the same for any upper bound, we get a new identification paradigm defined below.