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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

Computable reductions and undecidability

Information

Reduction direction determines which impossibility transfers.

Key knowledge

A≤ₘB means a total computable f maps every input w so that w∈A iff f(w)∈B. If B is decidable, so is A; hence undecidable A with A≤ₘB implies undecidable B.

Conditions and scope

  • A and B are languages over declared finite alphabets.
  • The transformation must halt on every input; this is computable many-one reduction, without a polynomial-time requirement.
Sources · 1

References

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

  1. Mapping Reducibility, Class 9 slidesNancy Lynch / MITCourse notes · PDFocw.mit.eduRead source

Evidence and review

Physical pages 5–6 total computable functions; pages 11 and 16–18 mapping reductions and decidability implication.

2026-10-10 · Mapping reduction and Theorem 1/Corollary 2.

Related reference entries

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

Showing 1–3 of 3 relationships

Scientific relationships · 3 / 3

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

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • A≤ₘB means a total computable f maps every input w so that w∈A iff f(w)∈B. If B is decidable, so is A; hence undecidable A with A≤ₘB implies undecidable B.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • Every nontrivial property of the language recognized by a Turing machine is undecidable from its machine description. Nontrivial means some recognizable language has the property and another does not; machine-syntax properties are outside this theorem.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • A logspace many-one reduction computes f with x∈A exactly when f(x)∈B, using a read-only input, O(log n) work tape and write-only output. Directed s–t reachability is NL-complete under these reductions: bounded path guessing uses logarithmic space, and each NL computation reduces to its polynomial-size configuration graph.
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info-reductions-undecidability
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university-reference.1
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University reference