Ackermann function

About: Ackermann function is a research topic. Over the lifetime, 430 publications have been published within this topic receiving 7091 citations.

A Formulation of the Simple Theory of Types

Abstract: The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.The class of type symbols is described by the rules that i and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols i and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.

Nonlinearity of Davenport-Schinzel Sequences and of Generalized Path Compression Schemes

Abstract: Davenport-Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a Davenport-Schinzel sequence composed of n symbols is (n /spl alpha/(n)), where /spl alpha/ (n) is the functional inverse of Ackermann's function, and is thus very slow growing. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes.

Sliding mode control design based on Ackermann's formula

Abstract: The sliding mode control methods are developed to design systems which have the desired dynamic behavior and are robust with respect to perturbations. It is shown that the discontinuity plane for sliding mode control may be found in an explicit form using Ackermann's formula. Two design procedures are derived. First, static controllers are designed to enforce sliding modes with the desired dynamic properties after a finite-time interval. Then, dynamic controllers are designed that exhibit the desired dynamic properties during the entire control process.

Nonlinearity of Daveport:80Schinzel sequences and of generalized path compression schemes

Abstract: Davenport—Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a Davenport—Schinzel sequence composed ofn symbols is Θ (nα(n)), where α(n) is the functional inverse of Ackermann’s function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes.

Recursive star-tree parallel data structure

Abstract: This paper introduces a novel parallel data structure called the recursive star-tree (denoted “${}^ * $-tree”). For its definition a generalization of the $ * $ functional is used (where for a function $f * f(n) = \min \{ {i|f^{(i)} (n) \leqslant 1} \}$ and $f^{(i)} $ is the ith iterate of f). Recursive ${}^ * $-trees are derived by using recursion in the spirit of the inverse Ackermann function.The recursive ${}^ * $-tree data structure leads to a new design paradigm for parallel algorithms. This paradigm allows for extremely fast parallel computations, specifically, $O(\alpha (n))$ time (where $\alpha (n)$ is the inverse of the Ackermann function), using an optimal number of processors on the (weakest) concurrent-read, concurrent-write parallel random-access machine (CRCW PRAM).These computations need only constant time, and use an optimal number of processors if the following nonstandard assumption about the model of parallel computation is added to the CRCW PRAM: an extremely small number of processor...