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Showing 1–30 of 204 reference entries

Physics · Information · MeasurementScientific records

Build an inspectable physics record, test and revise a bounded cart model, preserve processing history, and plan findable data with appropriate sharing conditions.

MethodMathematics & logic · InformationClassical propositions and connectives

Classical logic combines truth-valued propositions using defined connectives.

DefinitionMathematics & 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 · 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 & 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.

DefinitionInformation · 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.

DefinitionInformationFinite-alphabet mutual information

Mutual information measures statistical dependence in bits.

DefinitionInformationNear-lossless source coding threshold

Entropy bounds asymptotic compression for an iid source.

Mathematical statementInformationDiscrete memoryless channel capacity

Capacity is the maximal asymptotically reliable rate.

Mathematical statementInformationHamming distance and error correction

Separated codewords permit bounded symbol-error recovery.

Mathematical statementInformationInternet protocol layers

Protocol layers divide communication responsibilities.

DefinitionInformationTCP reliable byte-stream transport

TCP orders bytes and retransmits detected losses.

MethodInformationEncryption, hashes and message authentication

Cryptographic primitives serve distinct security properties.

DefinitionInformationPublic-key digital signatures

A signing key and a verification key have different roles.

DefinitionInformationComputable reductions and undecidability

Reduction direction determines which impossibility transfers.

Mathematical statementInformationDecision classes P and NP

Polynomial solving and polynomial verification differ.

DefinitionInformationNP-hardness and NP-completeness

NP-completeness combines verification and reduction hardness.

DefinitionInformationBounded-error randomized computation

Random choices permit a stated probability of error.

DefinitionInformationFinite graphs and trees

Graphs represent pairwise connections; trees impose acyclicity.

DefinitionInformationData-structure interfaces and operation costs

An interface defines operations; an implementation determines costs.

MethodInformationLoop invariants and total correctness

An invariant establishes correctness only with its proof obligations.

Method
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