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mathematics, computable measure theory is the part of computable analysis that deals with effective versions of measure theory. As with measure theory, this
Computable_measure_theory
Study of computable functions and Turing degrees
Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function. Although initially skeptical, by 1946 Gödel
Computability_theory
Ability to solve a problem by an effective procedure
to solve the problem. The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all
Computability
Real number that can be computed within arbitrary precision
the recursive numbers, effective numbers, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel
Computable_number
Cosmological theory
preferred over other theories-of-everything by Occam's Razor. Tegmark also considers augmenting the MUH with a second assumption, the computable universe hypothesis
Mathematical universe hypothesis
Mathematical_universe_hypothesis
Process of assigning numbers to objects or events
Netherlands Stevens, S.S. On the theory of scales and measurement 1946. Science. 103, 677–80. Douglas Hubbard: "How to Measure Anything", Wiley (2007), p.
Measurement
Left-invariant (or right-invariant) measure on locally compact topological group
measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.
Haar_measure
cube System F Introduction to topos theory LF (logical framework) Computability logic Computable measure theory Finitism Ultraintuitionism Luitzen Egbertus
List of mathematical logic topics
List_of_mathematical_logic_topics
Subfield of information theory and computer science
theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably
Algorithmic information theory
Algorithmic_information_theory
Measure of algorithmic complexity
2^{*}} be a computable function mapping finite binary strings to binary strings. It is a universal function if, and only if, for any computable f : 2 ∗ →
Kolmogorov_complexity
Inherent difficulty of computational problems
the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what
Computational complexity theory
Computational_complexity_theory
Concept in theoretical computer science
become larger than any computable function. This has implications in computability theory, the halting problem, and complexity theory. The concept of a busy
Busy_beaver
Mathematical theory
probability to any computable theory. Solomonoff proved that this induction is incomputable (or more precisely, lower semi-computable), but noted that "this
Solomonoff's theory of inductive inference
Solomonoff's_theory_of_inductive_inference
Academic subfield of computer science
Information Theory. 2 (3): 113–124. Bibcode:1956IRTIT...2..113C. doi:10.1109/TIT.1956.1056813. S2CID 19519474. Alan Turing (1937). "On computable numbers
Theory_of_computation
Axioms in computational complexity theory
complexity theory the Blum axioms or Blum complexity axioms are axioms that specify desirable properties of complexity measures on the set of computable functions
Blum_axioms
Problem in computer science
verification that g is computable relies on the following constructs (or their equivalents): computable subprograms (the program that computes f is a subprogram
Halting_problem
Limit of a uniformly computable sequence of functions
computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in
Computation_in_the_limit
Yes/no problem in computer science
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a
Decision_problem
Rules out assigning to arbitrary functions their computational complexity
two parameters, then there exists a total computable predicate g {\displaystyle g} (a boolean valued computable function) so that for every program i {\displaystyle
Blum's_speedup_theorem
Halting probability of a random computer program
recognize. The domain of any universal computable function is a computably enumerable set but never a computable set. The domain is always Turing equivalent
Chaitin's_constant
Philosphical view that existence proofs must be constructive
Constructive set theory Constructive type theory Constructive analysis Constructive non-standard analysis Computability theory – Study of computable functions
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
polynomial-time computable, then we obtain a definition of p-measure: a set of sequences has p-measure 0 if there is a polynomial-time computable martingale
Resource-bounded_measure
Subfield of computer science and mathematics
to optimize some measure of performance such as minimizing the number of mistakes made on new samples. Computational number theory, also known as algorithmic
Theoretical_computer_science
British computer scientist
Computable rings and fields, in E Griffor (ed.), Handbook of Computability Theory, Elsevier (1999), pp363–447. J V Tucker and J I Zucker, Computable functions
John_V._Tucker
Generalization of Turing computability
J. F. Knight, 2000. Computable Structures and the Hyperarithmetical Hierarchy, Elsevier. ISBN 0-444-50072-3 Computability Theory of Hyperarithmetical
Hyperarithmetical_theory
Computational geometry problem
Klee's measure problem is the problem of determining how efficiently the measure of a union of (multidimensional) rectangular ranges can be computed. Here
Klee's_measure_problem
Average uncertainty in variable's states
{\displaystyle Y} . Entropy can be formally defined in the language of measure theory as follows: Let ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} be a
Entropy_(information_theory)
Mathematical method of assigning a prior probability to a given observation
not a probability and it is not computable. It is only "lower semi-computable" and a "semi-measure". By "semi-measure", it means that 0 ≤ ∑ x P ( x )
Algorithmic_probability
Topics referred to by the same term
multiverse A problem in measure theory–see Solovay model Klee's measure problem, problem of determining how efficiently the measure of a union of rectangular
Measure_problem
Near-Optimal Computable Predictions. In J. Kivinen and R. H. Sloan, editors, Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT
Speed_prior
Computer hardware technology that uses quantum mechanics
in quantum computing Quantum cognition – Application of quantum theory mathematics to cognitive phenomena Quantum sensor – Device measuring quantum mechanical
Quantum_computing
Binary sequence
other computable p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} . (Here, "Church" refers to Alonzo Church, whose 1940 paper proposed using Turing-computable rules
Algorithmically random sequence
Algorithmically_random_sequence
Measure of quantum entanglement in quantum mechanics
information theory (Thesis). University of Potsdam. arXiv:quant-ph/0610253. Bibcode:2006PhDT........59E. G. Vidal; R. F. Werner (2002). "A computable measure of
Negativity (quantum mechanics)
Negativity_(quantum_mechanics)
Yang–Mills theory in two dimensions with a well-defined measure
limit of two-dimensional Yang–Mills theory has connections to string theory. Interest in the Yang–Mills measure comes from a statistical mechanical or
Two-dimensional Yang–Mills theory
Two-dimensional_Yang–Mills_theory
There are arbitrarily large computable gaps in the hierarchy of complexity classes
states that there are arbitrarily large computable gaps in the hierarchy of complexity classes. For any computable function that represents an increase in
Gap_theorem
There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theorem, whose exact values
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Property holding for typical examples
every natural number as an element; No 1-generic is computable (or even bounded by a computable function); All 1-generics f {\displaystyle f} are generalised
Generic_property
Subject of study in ergodic theory
measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving
Measure-preserving dynamical system
Measure-preserving_dynamical_system
Method of mathematical integration
general spaces, measure spaces, such as those that arise in probability theory. The term Lebesgue integration can mean either the general theory of integration
Lebesgue_integral
Theory within consciousness research
individuals. The theory has found practical application in the development of the Perturbational Complexity Index (PCI), an empirical measure used in clinical
Integrated_information_theory
Random measure in probability theory
In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables
Empirical_measure
System with multiple networked computers
the network. Let D be the diameter of the network. On the one hand, any computable problem can be solved trivially in a synchronous distributed system in
Distributed_computing
Scientific study of digital information
field theory Information geometry Information theory and measure theory Kolmogorov complexity List of unsolved problems in information theory Logic of
Information_theory
Branch of mathematics
and infinitely large numbers. Computable analysis, the study of which parts of analysis can be carried out in a computable manner. Set-valued analysis –
Mathematical_analysis
Chinese mathematician (born 1991)
mathematician known for her research in Fourier analysis and geometric measure theory. She was awarded the Fields Medal in 2026 for her contributions to Fourier
Hong_Wang
Measure of unsolvability
natural numbers measures the level of algorithmic unsolvability of the set. The concept of Turing degree is fundamental in computability theory, where sets
Turing_degree
Infinite cardinal number
sense), the set of all algebraic numbers, the set of all computable numbers, the set of all computable functions, the set of all binary strings of finite length
Aleph_number
In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers, named after Donald A. Martin. Under
Martin_measure
class, with computable boundary, which contains all computable functions. Given a Gödel numbering φ {\displaystyle \varphi } of the computable functions
Compression_theorem
Theory of equilibrium between supply and demand
Computable General Equilibrium Modelling (2 volumes). North-Holland. ——, with Jerie, Michael; Rimmer, Maureen T. (2018). Trade Theory in Computable General
General_equilibrium_theory
American mathematician (1916–2001)
any logical numerical relationship, thereby establishing the theory behind digital computing and digital circuits. Called by some the most important master's
Claude_Shannon
Concept of moral fairness and administration of the law
Casanovas, Pompeu (2008), "Concepts and Fields of Relational Justice", Computable Models of the Law, Lecture Notes in Computer Science, vol. 4884, Springer
Justice
Topics referred to by the same term
function, a statistical tool to measure spatial correlation Polymer-clad fiber, a type of optical fiber Programming Computable Functions, a functional programming
PCF
Measure in information theory
Logical depth is a measure of complexity for individual strings devised by Charles H. Bennett based on the computational complexity of an algorithm that
Logical_depth
Branch of probability theory
deviation theory was developed in 1966, in a paper by Varadhan. Large deviations theory formalizes the heuristic ideas of concentration of measures and widely
Large_deviations_theory
Type of set in mathematics
possible, close to that of a computable set. Solovay proved in 1975 that a set can be K-trivial without being computable. The Schnorr–Levin theorem says
K-trivial_set
Hypothetical physical concept
Gödel's theorems are irrelevant for computable physics. In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose pseudo-randomness
Theory_of_everything
Branch of mathematical logic
where "recursive" means "computable", as in computable function. This name is used because RCA0 corresponds informally to "computable mathematics". In particular
Reverse_mathematics
Conditional independence of exchangeable observations
Daniel Roy (2009) "Computable exchangeable sequences have computable de Finetti measures", Proceedings of the 5th Conference on Computability in Europe: Mathematical
De_Finetti's_theorem
Basic concept of graph theory
subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. In
Connectivity_(graph_theory)
Expressing a measure as an integral of another
is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function
Radon–Nikodym_theorem
Framework for analyzing machine learning algorithms
learner to fail on data sequences with probability measure 0 [citation needed]. Algorithmic learning theory investigates the learning power of Turing machines
Algorithmic_learning_theory
Quickly growing function
examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann
Ackermann_function
Mathematics of real numbers and real functions
often include measure theory, Lebesgue integration, and function spaces. Real analysis is also known, especially in older books, as the theory of functions
Real_analysis
Feature of systems that defy description
future and measures the entropy of the probability distribution of the states within this model. It is a computable and observer-independent measure based
Complexity
Concept in cosmology
The measure problem in cosmology concerns how to compute the ratios of universes of different types within a multiverse. It typically arises in the context
Measure_problem_(cosmology)
Game where groups of players may enforce cooperative behaviour
definition of a computable simple game. In particular, all finite games are computable. Kumabe, M.; Mihara, H. R. (2011). "Computability of simple games:
Cooperative_game_theory
Models of computation
a Turing machine. Hypercomputers compute functions that a Turing machine cannot and which are, hence, not computable in the Church–Turing sense. Technically
Hypercomputation
Theorem that any three objects in space can be simultaneously bisected by a plane
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean
Ham_sandwich_theorem
Axiomatic set theories based on the principles of mathematical constructivism
are computable trees K {\displaystyle K} for which no computable such path through it exists. To prove this, one enumerates the partial computable sequences
Constructive_set_theory
Unit of measure for digital data
A unit of information is any unit of measure of digital data size. In digital computing, a unit of information is used to describe the capacity of a digital
Units_of_information
Mathematical set containing no elements
set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Common notations
Empty_set
Foundations of probability theory
subjective measure of the probability of events. The third axiom, σ-additivity, is relatively modern, and originates with Lebesgue's measure theory. Some authors
Probability_axioms
Concept in software engineering and computer science
2019-05-27. Kang, Byeong-Ho (January 2007). "Ubiquitous Computing Environment Threats and Defensive Measures". International Journal of Multimedia and Ubiquitous
Ubiquitous_computing
Hypothetical group of multiple universes
A New Simplicity Measure Yielding Near-Optimal Computable Predictions. Proc. 15th Annual Conference on Computational Learning Theory (COLT 2002), Sydney
Multiverse
require the gales to be either computable or close to computable: A gale d is called constructive, c.e., or lower semi-computable if the numbers d ( σ ) {\displaystyle
Effective_dimension
Branch of mathematical logic
recursive or polynomial-time computable functions. Functional interpretations have also been used to provide ordinal analyses of theories and classify their provably
Proof_theory
Mathematical framework to model epistemic uncertainty
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty
Dempster–Shafer_theory
Sequence of random variables
pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability. This definition is often
Random_sequence
Attraction of masses and energy
weaker as objects get farther away. Gravity is described by the general theory of relativity, proposed by Albert Einstein in 1915, which describes gravity
Gravity
Ordinals in mathematics and set theory
normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations (see ordinal analysis). However, it is not
Large_countable_ordinal
Mathematical models of strategic interactions
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively
Game_theory
Theorem in computability theory
In computability theory, Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees. The
Post's_theorem
Study of abstract machines and automata
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical
Automata_theory
discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Theory of a quantum origin of consciousness
leaves the question of the physical basis of non-computable behavior open. Most physical laws are computable, and thus algorithmic. However, Penrose determined
Orchestrated objective reduction
Orchestrated_objective_reduction
Matrix-valued random variable
spectral theory of random matrices studies the distribution of the eigenvalues as the size of the matrix goes to infinity. The empirical spectral measure μ H
Random_matrix
Study of algorithms in strategic environments
Algorithmic game theory (AGT) is an interdisciplinary field at the intersection of game theory and computer science, focused on understanding and designing
Algorithmic_game_theory
Algorithm characteristic in computations
allowed are: Polynomial-time computable distributions (P-computable): these are distributions for which it is possible to compute the cumulative density of
Average-case_complexity
Area of mathematical analysis
results of ergodic theory. The maximal ergodic theorem is analogous to the Hardy–Littlewood maximal theorem: it bounds the measure of the set on which
Harmonic_analysis
and Hermann Hankel made great use of the theory, the former in studying equations, the latter in his theory of complex numbers. Niels Henrik Abel seems
History_of_calculus
Description of physical properties at the atomic and subatomic scale
will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will
Quantum_mechanics
Collection of random variables
calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and
Stochastic_process
Class of economic models
Computable general equilibrium (CGE) models are a class of economic models that use actual economic data to estimate how an economy might react to changes
Computable general equilibrium
Computable_general_equilibrium
Study of optimal transportation and allocation of resources
somewhat different because of the development of Riemannian geometry and measure theory. The mines-factories example, simple as it is, is a useful reference
Transportation theory (mathematics)
Transportation_theory_(mathematics)
American mathematician
set theory and philosophy of set theory (particularly the idea of the set-theoretic multiverse), in computability theory, and in group theory. After
Joel_David_Hamkins
Set of marks along a ruler such that no two pairs of marks are the same distance apart
true otherwise. There is no requirement that a Golomb ruler be able to measure all distances up to its length, but if it does, it is called a perfect
Golomb_ruler
carried out in a computable manner. It is closely related to constructive analysis. Computable model theory a branch of model theory dealing with the
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Maximal proper filter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P {\textstyle P} is a certain subset of P , {\displaystyle
Ultrafilter
Measure of complexity regarding algorithmic entropy
In algorithmic information theory, sophistication is a measure of complexity related to algorithmic entropy. When K is the Kolmogorov complexity and c
Sophistication (complexity theory)
Sophistication_(complexity_theory)
COMPUTABLE MEASURE-THEORY
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COMPUTABLE MEASURE-THEORY