On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers
@article{Chaitin1969OnTS,
title={On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers},
author={Gregory J. Chaitin},
journal={Journal of the ACM (JACM)},
year={1969},
volume={16},
pages={407 - 422}
}It is suggested that there are infinite computable sets of natural numbers with the property that no infinite subset can be computed more simply or more quickly than the whole set. Attempts to establish this without restricting in any way the computer involved in the calculations are not
94 Citations
On the difficulty of computations
- Computer ScienceIEEE Trans. Inf. Theory
- 1970
The study of the running time of programs for computing infinite sets of natural numbers leads to an arithmetic of computers, which is a distributive lattice.
On the algorithmic foundation of information theory
- Computer ScienceIEEE Trans. Inf. Theory
- 1979
The equivalence of the complexity approach and the martingale approach after restriction to effective random tests is used to establish generalized source coding theorems and converses.
The Complexity of Computations
- Computer Science
- 2013
This chapter provides a survey of the main concepts and results of the computational complexity theory, starting with basic concepts such as time and space complexities, complexity classes P and NP, and Boolean circuits and two related ways to make computations more efficient.
Structural Complexity II
- MathematicsEATCS
- 1990
This is the second volume of a systematic two-volume presentation of the various areas of research in the field of structural complexity, addressed to graduate students and researchers and assumes knowledge of the topics treated in the first volume but is otherwise nearly self-contained.
An Introduction to Kolmogorov Complexity and Its Applications
- Computer ScienceTexts and Monographs in Computer Science
- 1993
The book presents a thorough treatment of the central ideas and their applications of Kolmogorov complexity with a wide range of illustrative applications, and will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics.
Why the Concept of Computational Complexity is Hard for Verifiable Mathematics
- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2015
This work proves Rice’s theorem for unprovability, claiming that each nontrivial semantic problem about programs is not almost everywhere solvable in AV-mathematics, and proves that P = NP implies the existence of algorithms X for which the claim “X solves SATISFIABILITY in polynomial time” is not provable in AV.
Language-theoretic Complexity of Disjunctive Sequences
- Mathematics, Computer ScienceDiscret. Appl. Math.
- 1997
The Computation of Finite-State Complexity by Tania K . Roblot
- Computer Science
- 2011
This thesis proposes a new variant to Algorithmic Information Theory constructed around finite-state complexity, a computable counterpart to Kolmogorov complexity based on finite transducers rather than Turing machines, and presents a first attempt at applying the finite- state complexity in a practical setting.
References
SHOWING 1-10 OF 44 REFERENCES
On the Length of Programs for Computing Finite Binary Sequences
- Computer ScienceJACM
- 1966
An application to the problem of defining a patternless sequence is proposed in terms of the concepts here developed to study the use of Turing machines for calculating finite binary sequences.
On the Length of Programs for Computing Finite Binary Sequences: statistical considerations
- Computer ScienceJACM
- 1969
An attempt is made to carry out a program for defining the concept of a random or patternless, finite binary sequence, and for subsequently defining arandom or patterned, infinite binary sequence to be a sequence whose initial segments are all random orpatternless finite binary sequences.
On the difficulty of computations
- Computer ScienceIEEE Trans. Inf. Theory
- 1970
The study of the running time of programs for computing infinite sets of natural numbers leads to an arithmetic of computers, which is a distributive lattice.
A Machine-Independent Theory of the Complexity of Recursive Functions
- Computer ScienceJACM
- 1967
The number of steps required to compute a function depends on the type of computer that is used, on the choice of computer program, and on the input-output code, but the results obtained in this paper are nearly independent of these considerations.
Retraceable Sets
- MathematicsCanadian Journal of Mathematics
- 1958
Let us compare two properties of sets of non-negative integers: (1) the set a has property Γ, if there exists an effective procedure which when applied to any element of a different from its maximum…
Machine dependence of degrees of difficulty.
- Computer Science
- 1965
Machine dependence of degree-of-difficulty relation for recursive functions, considering Turing machine as programmed computer and its implications for knowledge of Turing machines.
Abstract Set Theory
- GeographyNature
- 1954
Abstract Set TheoryBy Prof. Abraham A. Fraenkel. (Studies in Logic and the Foundations of Mathematics.) Pp. xii + 480. (Amsterdam: North-Holland Publishing Co., 1953.) 38f.; 76s.
Number: the Language of Science
- ArtNature
- 1941
Number: the Language of Science by Tobias Dantzig shows why science is inevitably and progressively doomed to use mathematics as its language.