But almost all results in inductive inference are independent of the .... to the torso T and then step by step to to the synthesized program resp. program t.orso ...
All'l!Cll!A!tIC,SYNl'IIESIS 01' PROGRAMS AJlll INDUCT:IVE Illl'ERENCE 01' 1'UNCTI0l1S Klaus P. Jantke l)
Since the early deys of programming hes been the dream of a system cap able ot constructing. exact programs (in a fixed programming language) by s,U:ficiently extensive specifications of the underlying problem given by the user. In recent years in, reply to the enonnous increase of pro
gramming problems there has been an, increasing activity' in efforts to
automatic programming. Many papers described ingenious techniques for automatic program synthesis. But in many cases there is no assertion
about the capability of the developed method. The automatic synthesis of programs seems to be� interpretation of
the mathematical theory of inductive inference of recursive functions.
But almost all results in inductive inference are independent of the used definition of computability, that means of the underlying GODEL numbering. On the other hand, papers which dealt with automatic syn
thesis of programs are written �rom the point of view of a certain programming langu�ge, for instance LISP. The quest.ion is, whether. every GODEL numbering with an associated com
plexity measure in the sense of BLUl,I /4/ can be interpreted to be an
encoded simple programmin� language and vice versa. Let us refer to
a recursion theoretic result of BUCHBERGER /5/: Theorem 1
Suppose that 'I' is any GODEL numbering and cf, is a complexity
measure associated with f• Then there are four functibns
c, d, f and t with (1) For all natural numbers 1 and x it holds that q>1Cx) =rn< t(f"(c(i,x))) = o). (2) For all natural numbers i and x it holds that 'fi(x) = d(�(cl:i,x))), where m =i (x).
1) Department of Mathematics, Humboldt University Berlin,· 1066 Berlin, PSF 1297, GDR
220 rt is easy to show that there are f and
'1> such
that it is impossible
to find· four primitive recursive functions c,d,f,t to decompose f resp. (p in the sense of theorem 1. Since our main concern will be abou.t. programming languages with very simple statements and tests, we formulate a variation of the preceding theorem: Theorem 2.
Ir' 'f is any Gt5DEL numbering obtained in the usual way by enco ding a programming language wi.th only primitive recursive tests an4 statements (in the sense of register machines, for instance) and if
