Topic
Ackermann function
About: Ackermann function is a research topic. Over the lifetime, 430 publications have been published within this topic receiving 7091 citations.
Papers published on a yearly basis
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Papers
TL;DR: A formulation of the simple theory oftypes which incorporates certain features of the calculus of λ-conversion into the theory of types and is offered as being of interest on this basis.
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.
2,195 citations
TL;DR: A modified version of the dynamic trees of Sleator and Tarjan is developed that is suitable for efficient recursive algorithms, and used to reduce the running time of the algorithms for both problems toO(mα(m,n), where α is a functional inverse of Ackermann's function.
Abstract: We consider the twin problems of maintaining the bridge-connected components and the biconnected components of a dynamic undirected graph. The allowed changes to the graph are vertex and edge insertions. We give an algorithm for each problem. With simple data structures, each algorithm runs inO(n logn +m) time, wheren is the number of vertices andm is the number of operations. We develop a modified version of the dynamic trees of Sleator and Tarjan that is suitable for efficient recursive algorithms, and use it to reduce the running time of the algorithms for both problems toO(mα(m,n)), where α is a functional inverse of Ackermann's function. This time bound is optimal. All of the algorithms useO(n) space.
152 citations
TL;DR: In this article, the authors present a natural definition of computational complexity of real functions and study the relationship between complexity and analytical properties of real real functions, and establish basic continuity results, such as a polynomially bounded modulus of continuity.
Abstract: Publisher Summary The aim of this chapter is to present a natural definition of computational complexity of real functions and to study the relationship between complexity and analytical properties of real functions. The field of computational complexity has been largely concerned with discrete problems. The chapter defines the notion of computable real function and establishes basic continuity results. The chapter also defines computational complexity of recursive real numbers and functions and relates this to continuity, proving, for example, that a polynomial time computable function has a polynomially bounded modulus of continuity. A real number is considered a sequence of dyadic rational numbers that converges to it. Computable real functions are to be defined not only on the computable real numbers so a natural approach is via oracle Turing machines (OTM). A root of a recursive real function must be recursive, but a root of a polynomial time computable function need not be polynomial time computable. A polynomial time computable function need not be differentiable; a standard example of an “everywhere continuous nowhere differentiable” function has been shown to be polynomial time computable. Because all polynomial time computable functions are continuous, they are also Riemann integrable. The maximum value of a recursive real function is known (e.g., Lacombe to be a recursive real number). Step functions are a useful tool in analysis and if they have recursive jump points and the jump points are ignored, they are partial recursive.
146 citations
TL;DR: One of the main problems in sound synthesis is that the composer's idea or concept of a sound does not necessarily correspond directly to the physical parameters of synthesis algorithms.
Abstract: One of the main problems in sound synthesis is that the composer's idea or concept of a sound does not necessarily correspond directly to the physical parameters of synthesis algorithms. In regard to FM syntheseis, Ackermann (1991) mentions that the transition from idea to synthesis requires "patience, skill and a little bit of luck." Even computer-based sound-generating systems like Cmusic do not have a user-interface that allows the intuitive mapping from sound idea to soundgenerating method in a musically satisfactory
126 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 14 |
| 2024 | 24 |
| 2023 | 39 |
| 2022 | 73 |
| 2021 | 21 |
| 2020 | 19 |