By Bauml, Eggers, Tzschoppe, Huber

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A r i n g is called Noetherian if for a n y a s c e n d i n g chain of ideals -To C / 1 C " " t h e r e is a n r such t h a t I~ = I~+1 . . 2, p a g e 16, [8]). N o t e t h a t a n y finite ring is t r i v i a l l y A r t i n i a n . I n a d d i t i o n to R b e i n g a c o m m u t a t i v e ring w i t h i d e n t i t y we will also a s s u m e t h a t it A r t i n i a n . W e now f o r m a l l y define t h e r a n k of an o r d e r e d set of e l e m e n t s from an R - m o d u l e . Definition 1.

2. Bounded degree multivariate polynomials over Z~ can be interpolated using substitution queries. This paper generalizes the learning algorithm for a u t o m a t a over fields given in [4]. 1 Introduction T h e l e a r n a b i l i t y of m u l t i p l i c i t y a u t o m a t a over t h e field of r a t i o n a l s using equivalence a n d s u b s t i t u t i o n queries was shown to b e p o s s i b l e by B e r g a d a n o a n d Varricchio [7]. T h e i r a l g o r i t h m was a d i r e c t g e n e r a l i z a t i o n of A n g l u i n ' s algor i t h m for l e a r n i n g d e t e r m i n i s t i c finite a u t o m a t a [1].

L e m m a 4. The procedure described above outputs a set of at most n + 2 candidate minterms, and that set is guaranteed to contain at least one minterm of the target function. Proof sketch. The existence of the minterm was discussed in the development of the algorithm. 45 The number n + 2 is derived as follows. As noted previously, the largest possible number of splits of a graph with [V[ vertices is IV] + 1. The largest possible number of l ' s in v is n, so the split graph construction could lead to at most n + 1 candidates for the minterm.