Article history: Received 22 June 2010 Revised 21 November 2010 Accepted 12 February 2011 Available online xxxx Communicated by Zhe-Xian Wan MSC: 11T35 11T06 20G40 15B05

The Ackermann function is a famous total recursive binary function on the natural numbers. It is the archetypal example of such a function that is not primitive recursive, in the sense of classical recursion theory. However, and in seeming contradiction, there are generalized notions of total recursion, for which the Ackermann function is in fact primitive recursive, and often featured as a wit...

We investigate the computational power of C++ compilers. In particular, it is known that any partial recursive function can be computed at compile time, using the template mechanism to define primitive recursion, composition, and minimalization. We show that polynomial time computable functions can be computed at compile-time using the same mechanism, together with template specialization.

Thrs paper develops a transformational paradrgm by which nonnumertcal algorrthms are treated as fixed pomt computatrons derived from very htgh level problem specrficatrons We begin by presenting an abstract functronal problem specrficatron language SQ’, which 1s shown to express any partial recursive function m a fixed pomt normal form Next, we gtce a nondetermuusttc Iterative schema that rn th...

the re is no uniform bound on the loss of time incurred when some universal automaton globally simulates another automaton. More precisely, define an automaton to be a partial recursive function with decidable domain. Say that an automaton U locally simulates an automaton f via recursive, injective encoding cr with recursive time bound b iff for all z, z ' C IN the following holds: f ( z ) = z'...

1. Let [qo, qi,q2, ■ • ■ J be a Kleene enumeration of partial recursive functions. If / is such a function, denote by 9f its index set, 9f= {ra|g„^/}. Insofar as the indices of a partial recursive function correspond to the different sets of "instructions" for computing its values, it is natural to ask how much of the "complexity" of the function is reflected by its index set; for example, one ...

Kechris, A.S., Boundedness theorems for dilators and ptykes, Annals of Pure and Applied Logic 52 (1991) 79-92. The main theorem of this paper is: If f is a partial function from K, to X, which is Pi-bounded, then there is a weakly finite primitive recursive dilator D such that for all infinite a E domu), f(a) s D(a). The proof involves only elementary combinatorial constructions of trees. A gen...

A subset A of the Baire space ω satisfies the polarized partition property if there is an infinite sequence 〈Hi | i ∈ ω〉 of finite subsets of ω, with |Hi| ≥ 2, such that ∏ iHi ⊆ A or ∏ iHi ∩ A = ∅. It satisfies the bounded polarized partition property if, in addition, the Hi are bounded by some pre-determined recursive function. DiPrisco and Todorčević [6] proved that both partition properties ...

We study the value-distribution of Dirichlet L-functions L(s, χ) in the half-plane σ = 1/2. The main result is that a certain average of the logarithm of L(s, χ) with respect to χ, or of the Riemann zeta-function ζ(s) with respect to =s, can be expressed as an integral involving a density function, which depends only on σ and can be explicitly constructed. Several mean-value estimates on L...