We characterize iterated log depth circuit classes between AC 0 and AC 1 by Cobham-like bounded recursion schemata. We also give alternative characterizations which utilize the safe recursion method developed by Bellantoni and Cook.
Variance components of tree height (HT) and stem diameter at 1.3 m above the ground (DBH) were investigated for the eight open-pollinated families of Zelkova serrata (Thumb.) Makino planted with three different initial planting spacings in a progeny test site, Chiba, Japan. Parent–offspring correlations were also evaluated by using these families and their mother trees. The smallest values of HT and DBH were observed in the narrowest initial planting spacing (1.10 x 1.10 m) compared to those in medium (1.30 x 1.36 m) and wide (2.00 x 1.80 m) spacings, suggesting that adverse effects of competition with neighboring trees occurred on both height and radial growth. Similar to HT and DBH, the initial planting spacings also affected the genetic parameter estimates: the ratio of family variance component to total phenotypic variance showed the highest value in narrow initial planting spacing for both HT and DBH. Thus, family variance component might include competition effects, leading to biased genetic parameter estimates. In contrast, parent–offspring correlation coefficients showed the highest value in wide initial planting spacing where competition effect might be smaller. Therefore, the growth traits of Z. serrata might be inherited from the parent to the offspring when competition effect was small.
We define a fragment of Primitive Recursive Arithmetic by replacing the defining axioms for primitive recursive functions by those for functions in some specific complexity class. In this note we consider such theory for AC'. We present a model-theoretical property of this theory, by means of which we are able to characterize its provably total functions. Next we consider the problem of how strong the induction scheme can be in this theory.Mathematics Subject Classification: 03C62, 03F30, 68615.Primitive Recursive Arithmetic (PRA) is a first order theory which consists of all primitive recursive functions together with their defining axioms and induction scheme for all bounded formulae. It is known that PRA is preserved under substructures, namely if M is a model of PRA, then any substructure of M is also a model of PRA. It follows using HERBRAND'S theorem and the theorem of LO< and TARSKI that if y ( z , y) is a bounded formula such that PRA I-Vz3yy(z, y), then PRA I -Vzcp(z, f(z)) holds for some primitive recursive function f .From the point of view of computational complexity, it is interesting to find analogous systems for complexity classes of functions such as PTIME, LOGSPACE, ACO, etc.In this paper, we shall give such a system AC'CA for ACo which consists of the symbols for the functions in ACo together with their defining axioms and appropriately chosen induction schemes so that the system is preserved under substructures. The main idea for choosing the induction scheme is to be able to compute a witness for a formula of a certain complexity by an algorithm which can be carried out within the given complexity class. So it is necessary to find more elaborate technique as the complexity class in concern becomes smaller.
CLOTE and TAKEUTI [7]defined Buss-like weak first order theories corresponding to small Boolean circuit complexity classes. Their systems have a weak induction scheme together with some additional axiom schemes, while our system will have built-in functions for these complexity classes.')The author wishes to thank the anonymous referee for his helpful remarks and suggestions.
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