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Showing 31–60 of 1709 reference entries

Physics · MeasurementCaesium-133 defining frequency

ΔνCs denotes the unperturbed ground-state hyperfine transition frequency of caesium-133. Its numerical value is fixed exactly in SI units.

DefinitionMathematics & logic · InformationClassical propositions and connectives

Classical logic combines truth-valued propositions using defined connectives.

DefinitionMathematics & logicAxioms, deduction and mathematical proof

A mathematical proof derives a conclusion within stated assumptions and rules.

MethodMathematics & logic · InformationSets, membership and basic operations

A set specifies membership without counting duplicate listings.

DefinitionMathematics & logic · InformationFunctions, domain and codomain

A function assigns exactly one output to each input in its domain.

DefinitionMathematics & logic · InformationMathematical induction

A base case and general successor step establish a predicate on all nonnegative integers.

MethodMathematics & logic · MeasurementEuclidean right triangles and the Pythagorean theorem

In a Euclidean right triangle, squared leg lengths sum to the squared hypotenuse length.

Mathematical statementMathematics & logic · PhysicsSine, cosine and the unit circle

Sine and cosine are unit-circle coordinates whose squared values sum to one.

Mathematical statementMathematics & logic · InformationComplex numbers

A complex number has real coordinates a and b and an imaginary unit satisfying i²=−1.

DefinitionMathematics & logic · Information · PhysicsLinear maps and matrix representation

A linear map preserves linear combinations and has a matrix once bases are chosen.

DefinitionMathematics & logicLimit of a real function

A finite limit controls output error through sufficiently small input distance.

DefinitionMathematics & logic · PhysicsDerivative at a point

A derivative is a finite limit of local difference quotients.

DefinitionMathematics & logic · PhysicsDefinite integral and signed accumulation

A definite integral is the limiting signed accumulation of small contributions.

DefinitionMathematics & logic · PhysicsFundamental theorem of calculus

Continuity connects differentiation with integration.

Mathematical statementMathematics & logic · InformationProbability space and axioms

A probability assigns nonnegative, normalised, countably additive weights to events.

DefinitionMathematics & logic · InformationConditional probability and independence

Conditioning restricts attention to an event of positive probability.

DefinitionMathematics & logic · InformationExpected value of a discrete random variable

Expected value weights possible values by their probabilities.

DefinitionMathematics & logic · MeasurementConfidence interval for a population mean

A confidence level describes repeated-sampling coverage of an interval procedure.

MethodInformation · Mathematics & logicBits and unsigned binary encoding

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

DefinitionInformation · Mathematics & logicShannon entropy of a discrete distribution

Shannon entropy measures uncertainty in a specified probability distribution.

DefinitionInformationAlgorithm and decision procedure

An algorithm specifies effective steps; a decision procedure must terminate on every permitted input.

MethodInformation · Mathematics & logicUndecidability of the halting problem

No total algorithm decides halting for every program and input in a general computation model.

Mathematical statementInformation · Mathematics & logicAsymptotic upper bounds and big O

Big O bounds eventual growth up to a fixed constant factor.

DefinitionMeasurementSI base units

Seven named base units organize the SI quantity system.

DefinitionMeasurementSI decimal prefixes

Prefix symbols specify decimal scale factors.

DefinitionMeasurementMetrological traceability

Connect a result to a reference through documented calibrations.

DefinitionMeasurement · Mathematics & logicFirst-order uncertainty propagation

Combine input uncertainty through a measurement model.

MethodMeasurement · Physics · Mathematics & logicDimensional consistency

Use dimensions to detect impossible quantity equations.

MethodPhysics · Earth & cosmosNewtonian gravitation

An inverse-square force models attraction between point masses.

ModelPhysicsFrequency, wavelength and phase speed

Relate a periodic wave's spatial and temporal repetition.

Definition
Mathematical statementSource checked

Undecidability of the halting problem

InformationMathematics & logic

No total algorithm decides halting for every program and input in a general computation model.

Key knowledge

There is no algorithm that always terminates and correctly decides, for every program-input pair in a Turing-complete computation model, whether that program eventually halts on that input.

Conditions and scope

  • The theorem concerns unrestricted programs in a general model with unbounded potential memory and running time.
  • It does not rule out deciding specific programs or restricted classes, nor checking execution within a fixed resource bound.
Sources · 4

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. University of Cambridge2 sources
    1. Computation Theory (2022), Chapter 5: Halting problem, Definition 6 and Theorem 7Andrew Pitts and Tobias Kohn, University of CambridgeCourse notes · PDFcl.cam.ac.ukRead source
    2. Computation Theory, version 28 April 2022Andrew Pitts and Tobias Kohn / University of CambridgeCourse notes · PDFcl.cam.ac.ukRead source
  2. 15-112, summer 2021: The halting problemCarnegie Mellon UniversityCourse notescs.cmu.eduRead source
  3. Lectures on Philosophy and ComputationCristian S. Calude, University of AucklandTeaching resource · PDFcs.auckland.ac.nzRead source

Evidence and review

General undecidability theorem and diagonal argument.

2026-10-10 · Compared formal register-machine statement with an independent program-level argument; no claim about all restricted languages.

Foundational review

2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.

Sources from other places

Agreement within this scope

No general algorithm decides halting for every program/input pair.

The program model incorporates inputs as natural numbers; bounded or restricted cases are outside the theorem.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–2 of 2 relationships

Scientific relationships · 2 / 2

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • There is no algorithm that always terminates and correctly decides, for every program-input pair in a Turing-complete computation model, whether that program eventually halts on that input.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • A recognizer accepts every member of its language but may run forever on a nonmember. A decider halts on every input with the correct answer. A language is decidable exactly when it and its complement are both Turing-recognizable.
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Record identity
info-halting-problem
Revision
core.1
Structured reference record
Essential core · first collection