Recursive function theory studies functions that can be evaluated by an algorithm based on some computing model such as a Turing machine or computer program. These functions are called partial recursive (p.r.) and are defined on subsets of N, where N = {0, 1,... }, with values in N. (A good introduction to the theory may be found in [2].) This paper explains the Parameter Theorem, illustrates some unusual computer programs, and explains their connection to recursive function theory. Basing our computing model on a modern high level language, we can construct programs whose existence is known through more formal means. Our functions will operate on strings rather than numbers, however. There is no loss in generality since strings can be viewed as members of N, and vice-versa. From a practical aspect, this approach is somewhat more desirable since conventional programming languages often allow making use of long strings, but place more restrictive limits on the size of integers. We first pose a problem: Does there exist a program (written in some conventional programming language) which halts if and only if its input is identical to the program's text (i.e., source code)? For the remainder of this note X is the set of ASCII characters 'a,' 'b, +,' space, line feed, and so forth, needed to write programs in most programming languages. We denote by X * the set of all strings (finite sequences) over X. Let P1 be the set of all one variable partial recursive functions on X *. A function I belongs to P1 if and only if there is a subset of S of X* and some algorithm which computes 't by associating with each element of S a well-defined element of X *. Similarly, 't belongs to P2 if and only if there is a set S c X * X X *, and an algorithm for 't which associates with each element of S a well-defined element of X *. A partial recursive function I in P1 (or in P2) is recursive if the algorithm is defined on all of X * (or X * X X *). This is equivalent to saying that the algorithm which evaluates I always halts. Our computing model will be based on Adat; its high-level string-handling properties simplify later examples. For readability, however, we will express algorithms in a more familiar Pascal-like style. Readers familiar with either programming language should find forthcoming algorithms sufficiently clear. A formal presentation in Ada is given in the Appendix. A string of ASCII characters, such as in FIGURE 1, represents a function in our language which takes strings to strings. Of course, strings of ASCII characters such as The quick brown fox and %*!! represent nothing at all in our programming language. Let A1 be the set of all strings in X * representing functions in our language of one input STRING which outputs a STRING. Similarly let A2 be the set of all strings in
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