Volume 26, Issue 3

We study finite automata representations of numerical rings. Such representations correspond to the class of linear p -adic automata that compute homogeneous linear functions with rational coefficients in the ring of p -adic integers. Finite automata act both as ring elements and as operations. We also study properties of transition diagrams of automata that compute a function f ( x )= cx of one variable. In particular we obtain precise values for the number of states of such automata and show that for c > 0 transition diagrams are self-dual (this property generalises self-duality of Boolean functions). We also obtain the criterion for an automaton computing a function f ( x )= cx to be a permutation automaton, and fully describe groups that are transition semigroups of such automata.

Let all vertices of a complete binary tree of finite height be independently and equiprobably labeled by the elements of some finite alphabet. We consider the numbers of pairs of identical tuples of labels on chains of subsequent vertices in the tree. Exact formulae for the expectations of these numbers are obtained. Convergence to the compound Poisson distribution is proved.

The operation of bounded prefix concatenation (BPC) is introduced on the set of word functions in the alphabet {1, 2}. The class BPC of polynomially computable functions is defined on the basis of this operation and the superposition operation. The class BPC is shown to contain a number of word functions and to be closed with respect to certain known operations. A certain type of two-tape nonerasing Turing machines is introduced, functions from the class BPC are shown to be computable on machines of this type in polynomial time.

This paper is a continuation of the paper ‘Functions without short implicents. Part I: lower estimates of weights’. In Part II we propose various methods of construction of n -place Boolean functions not admitting implicents of k variables. The first of the methods proposed is based on the gradient algorithm, the second and the third ones depend on a certain combinatorial principle of construction, while the fourth method is based on a random choice of elements in the function support. The above methods have different efficiency depending on the value of k . We give upper estimates for the minimal value w ( n, k ) of weights of the so-constructed functions. Together with the lower estimates of w ( n, k ) from the first part of the paper this allows us to obtain an asymptotically sharp estimate w ( n, k ) = Θ (ln n ) as n → ∞.

Let N${\cal N}$ be a set of N elements and F1,G1,F2,G2,…$\left( {{F_1},{G_1}} \right),\left( {{F_2},{G_2}} \right), \ldots $ be a sequence of independent pairs of random dependent mappings N→N${\cal N} \to {\cal N}$ such that F k and G k are random equiprobable mappings and P{Fk(x)=Gk(x)}=α${\bf{P}}\{ {F_k}(x) = {G_k}(x)\} = \alpha $ for all x∈N$x \in {\cal N}$ and k = 1, 2, … For a subset S0⊂N,|S0|=n${S_0} \subset {\cal N},{\mkern 1mu} |{S_0}| = n$, we consider a sequences of its images Sk=Fk(…F2(F1(S0))…)${S_k} = {F_k}( \ldots {F_2}({F_1}({S_0})) \ldots )$, Tk=Gk(…G2(G1(S0))…)${T_k} = {G_k}( \ldots {G_2}({G_1}({S_0})) \ldots )$, k = 1, 2 …, and a sequences of their unions Sk∪Tk${S_k} \cup {T_k}$ and intersections Sk∩Tk${S_k} \cap {T_k}$, k = 1, 2 … We obtain two-sided inequalities for M|Sk∪Tk|${\bf{M}}|{S_k} \cup {T_k}|$ and M|Sk∩Tk|${\bf{M}}|{S_k} \cap {T_k}|$ such that upper and lower bounds are asymptotically equivalent if N,n,k→∞$N, n, k \to \infty $, nk=o(N)$nk = o(N)$ and α=O1N$\alpha = O\left( {{\textstyle{1 \over N}}} \right)$.

The asymptotic behavior, as n → ∞, of the conditional distribution of the number of particles in a decomposable critical branching process Z ( m ) = ( Z 1 ( m ), …, Z N ( m )) with N types of particles at moment m = n − k , k = 0( n ), is investigated given that the extinction moment of the process equals to n .