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Proof method in mathematical logic
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some
Structural_induction
Form of mathematical proof
this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely
Mathematical_induction
Topics referred to by the same term
Strong induction Structural induction Transfinite induction Epsilon-induction Parabolic induction Inductive reasoning, in logic Electromagnetic induction Electrostatic
Induction
Type of binary relation
is called structural induction. When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those
Well-founded_relation
Question of whether inductive reasoning leads to definitive knowledge
The problem of induction is a philosophical problem that questions the rationality of predictions about unobserved things based on previous observations
Problem_of_induction
Mathematical constructs and creation rules
{W}}_{x:A}B(x)}}.} The elimination rule for W-types works similarly to structural induction on trees. If, whenever a property (under the propositions-as-types
Inductive_type
Method of logical reasoning
degree of probability. Unlike deductive reasoning (such as mathematical induction), where the conclusion is certain, given the premises are correct, inductive
Inductive_reasoning
Heuristics in automated theorem proving
1017/CBO9780511543326. ISBN 0-521-83449-X. Aubin, Raymond (1976), Mechanizing Structural Induction, EDI-INF-PHD, vol. 76–002, University of Edinburgh, hdl:1842/6649
Rippling
Proof method in mathematical logic
concurrent interacting objects. Coinduction is the mathematical dual to structural induction.[citation needed] Coinductively defined data types are known as codata
Coinduction
Design guideline for software development
which are easier to reuse, and easier correct.ness proofs using structural induction. Lieberherr, Karl; Orleans, Doug; Ovlinger, Johan (October 2001)
Law_of_Demeter
Total order in computer science
f(...) using a lower-precedence root symbol g. In particular, by structural induction, a term f(...) is bigger than any term containing only symbols smaller
Path ordering (term rewriting)
Path_ordering_(term_rewriting)
Process of repeating items in a self-similar way
as in the preceding sections, yields structural induction — a powerful generalization of mathematical induction widely used to derive proofs in mathematical
Recursion
Defining elements of a set in terms of other elements in the set
Mathematical induction Recursive data types Recursion Recursion (computer science) Structural induction Henkin, Leon (1960). "On Mathematical Induction". The
Recursive_definition
Intellectual current and methodological approach in the social science
Structuralism is an intellectual current and methodological approach, primarily in the social sciences, that interprets elements of human culture by way
Structuralism
Algorithm to transform a regular expression into a finite automaton
expression and symbol rules as base cases, it is possible to prove with structural induction that any regular expression may be converted into an equivalent NFA
Thompson's_construction
theory of algorithms. Peano axioms Giuseppe Peano Mathematical induction Structural induction Recursive definition Naive set theory Element (mathematics)
List of mathematical logic topics
List_of_mathematical_logic_topics
Programming language
theorems is standard in Dafny with recursion employed for induction (typically, structural induction). Case analysis is performed using match statements and
Dafny
Limited form of tree data structure
Chengjiang (October 26, 2018), Lecture 22: Recursive Definitions and Structural Induction (PDF) Kenneth Rosen (2011). Discrete Mathematics and Its Applications
Binary_tree
Quickly growing function
{\displaystyle \operatorname {FGH} _{k}} . This can be shown by structural induction on the definitions of primitive recursive functions. However, the
Ackermann_function
Artificial stimulation of childbirth
Labor induction is the procedure where a medical professional starts the process of labor (giving birth) instead of letting it start on its own. Labor
Labor_induction
Class of languages studied in formal language theory in computer science
constructive. Using the closure properties of Büchi automata and structural induction over the definition of ω-regular language, it can be easily shown
Omega-regular_language
Combinatorial game theory theorem
G'} . We prove that all positions are equivalent to a nimber by structural induction. The more specific result, that the given game's initial position
Sprague–Grundy_theorem
Sociological theory of society
Structural functionalism, or simply functionalism, is "a framework for building theory that sees society as a complex system whose parts work together
Structural_functionalism
Task to construct a program meeting a formal specification
An example is shown in lines 11 to 13 of the toy example below. Structural induction. This rule allows for synthesis of recursive functions. For a given
Program_synthesis
Mathematical term; concerning axioms used to derive theorems
to describe it, led to model theory based on truth definition via structural induction over terms in first-order logic. 1935 Garrett Birkhoff HSP theorem
Axiomatic_system
Assistive listening technology
An audio induction loop system (also called an audio-frequency induction loop or AFIL, or a hearing loop) is an assistive listening device for individuals
Audio_induction_loop
Argument in the philosophy of science
In the philosophy of science, the pessimistic induction, also known as the pessimistic meta-induction, is an argument that seeks to rebut scientific realism
Pessimistic_induction
Use of functions that call themselves
All structurally recursive functions on finite (inductively defined) data structures can easily be shown to terminate, via structural induction: intuitively
Recursion_(computer_science)
Loops of electric current induced within conductors by a changing magnetic field
changing magnetic field in the conductor according to Faraday's law of induction or by the relative motion of a conductor in a magnetic field. Eddy currents
Eddy_current
Style of computer programming
special construct in which such polytypic functions can be defined via structural induction over the structure of the pattern functor of a regular datatype.
Generic_programming
Philosophical view that events are determined by prior events
biological, psychological, social, and cultural determinism, as well as structural determinism, which highlights systemic constraints. Historically, determinism
Determinism
Type of wireless power transfer
charging, is a type of wireless power transfer. It uses electromagnetic induction to provide electricity to portable devices. Inductive charging is also
Inductive_charging
Arrangement of interrelated elements in an object/system, or the object/system itself
structures are determined through structural analysis, which is one of the tasks of structural engineering. The structural elements can be classified as one-dimensional
Structure
Problem-solving method
reasoning is often based on induction, or on analogy ... Induction is the process of discovering general laws ... Induction tries to find regularity and
Heuristic
Algebraic structure
Partial Algebras. CiteSeerX 10.1.1.92.6134. Horst Reichel (1984). Structural induction on partial algebras. Akademie-Verlag. Horst Reichel (1987). Initial
Partial_algebra
Property of a statement that can be logically contradicted
proposed falsifiability as the cornerstone solution to both the problem of induction and the problem of demarcation. Popper also emphasized the related asymmetry
Falsifiability
Theory of science, reconstructing empirical theories
replaced Fresnel's. Because structure is retained, structural realism both (a) avoids pessimistic meta-induction and (b) does not make the success of science
Structuralism (philosophy of science)
Structuralism_(philosophy_of_science)
philosophical foundations of analytic induction originate from Aristotle and his concept of induction. Analytic induction can be thought of as a phenomenological
Humanistic_sociology
Extension of first-order logic with atoms expressing variable dependencies
relations of the team semantics for dependence logic are defined by structural induction over the formulas of the language. Since the negation operator interchanges
Dependence_logic
English theoretical physicist (1942–2018)
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Stephen_Hawking
Awareness of facts, or competency
Gettier problem Induction Internalism and externalism Justification Knowledge Meta Objectivity Privileged access Problem of induction Problem of other
Knowledge
Leader of the Soviet Union from 1922 to 1924
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Vladimir_Lenin
Form of pattern recognition
will yield the optimal correspondence. Grammar induction String matching Hopcroft–Karp algorithm Structural information theory Schalkoff, Robert (1992).
Syntactic_pattern_recognition
In computational linguistics, word-sense induction (WSI) or discrimination is an open problem of natural language processing, which concerns the automatic
Word-sense_induction
Statistical model to calculate the value of multiple quantities as they change over time
require as much knowledge about the forces influencing a variable as do structural models with simultaneous equations. The only prior knowledge required
Vector_autoregression
Reactions Lester I. Bockstahler c-13m January 2, 1963 Electromagnetic Induction Lester I. Bockstahler c-13m January 2, 1963 Electromagnets and Their Uses
List_of_Coronet_Films_films
Type of audible sound
terms electromagnetic vibrations or magnetic vibrations, focusing on the structural phenomenon are less ambiguous. Acoustic noise and vibrations due to electromagnetic
Electromagnetically induced acoustic noise
Electromagnetically_induced_acoustic_noise
Compound that increases the activity of a protein
Janne; Turpeinen, Miia; Pelkonen, Olavi (May 5, 2025). "Inhibition and induction of CYP enzymes in humans: an update". Archives of Toxicology. 94 (11):
Enzyme_inducer
Pharmaceutical compound
almost 10-fold preference for induction of dopamine release over serotonin release and around 1.5-fold preference for induction of dopamine release over norepinephrine
3-Chlorocathinone
Chemical compound in the ergot family of alkaloids
the ergot family of alkaloids; it is structurally and biochemically closely related to ergoline. It is structurally similar to several neurotransmitters
Ergotamine
Philosophical and social theory critical of traditional humanism
symbolically prestructured reality cannot be gained by observation alone." Structuralism was developed in post-war Paris as a response to the perceived contradiction
Antihumanism
Quality of being toxic to cells
including inhibition of cell growth, induction of apoptotic or necrotic cell death, and disruption of metabolic or structural cellular integrity. Assessing cytotoxicity
Cytotoxicity
Music museum in Cleveland, Ohio, US
thousands traveled to Ohio during induction week to participate in the events. The economic impact of the 2009 induction week activities was more than $13
Rock_and_Roll_Hall_of_Fame
Social theory proposed by Giddens that attempts to resolve the structure-agent debate
stages: structural conditioning, social interaction, its immediate outcome and structural elaboration. Thus her analysis considered embedded "structural conditions
Structuration_theory
Electric pickup truck
variants. These include a single design each for the permanent magnet motor, induction motor, motor inverter, and the 15:1 reduction gear set. All powertrain
Tesla_Cybertruck
Chinese philosophical concept
posteriori Belief Credence Certainty Data Experience Information Justification Induction Knowledge Meaning Rationality Reason Truth Wisdom Domains Applied epistemology
Unity_of_knowledge_and_action
Concept in epistemology
posteriori Belief Credence Certainty Data Experience Information Justification Induction Knowledge Meaning Rationality Reason Truth Wisdom Domains Applied epistemology
Justification_(epistemology)
Generalization of "n-th" to infinite cases
nesting) or ω ω {\displaystyle \omega ^{\omega }} (induction over the depth of nested induction). The validity of inductive counting rests on the property
Ordinal_number
Social structure made up of a set of social actors
sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group
Social_network
Different forms of the element iron
cooling process. β-Fe and the A2 critical temperature are important in induction heating of steel, such as for surface-hardening heat treatments. Steel
Allotropes_of_iron
Enzyme
1016/S0021-9258(18)64973-7. PMID 13405910. Marcus PI; Talalay P (1956). "Induction and purification of alpha- and beta-hydroxysteroid dehydrogenases". J
3alpha-hydroxysteroid dehydrogenase (B-specific)
3alpha-hydroxysteroid_dehydrogenase_(B-specific)
German social philosopher (1929–2026)
contemporary politics, particularly German politics. His major works include The Structural Transformation of the Public Sphere (1962), a social history of the emergence
Jürgen_Habermas
Text for clarification; one of four rhetorical modes
Marxism Neurophilosophy Ordinary language Pragmatism Quietism Scientific structuralism Sense data Analytic theology Analytical Thomism Concepts Analysis (paradox
Description
Two geometries based on axioms closely related to those specifying Euclidean geometry
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Non-Euclidean_geometry
American economist (born 1942)
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Deirdre_McCloskey
Process of using data analysis for predicting population data from sample data
propounded by such statisticians as Seymour Geisser. Algorithmic inference Induction (philosophy) Informal inferential reasoning Information field theory Population
Statistical_inference
The weld access hole or rat hole is a structural engineering technique in which a part of the web of an I-beam or T-beam is cut out at the end or ends
Weld_access_hole
Two types of knowledge, justification, or argument
posteriori Belief Credence Certainty Data Experience Information Justification Induction Knowledge Meaning Rationality Reason Truth Wisdom Domains Applied epistemology
A_priori_and_a_posteriori
Class of psychoactive drugs that produce empathic experiences
However, entactogens also frequently have additional actions, such as induction of dopamine and norepinephrine and serotonin 5-HT2 receptor agonism, which
Entactogen
Machine learning algorithm
splitting no longer adds value to the predictions. This process of top-down induction of decision trees (TDIDT) is an example of a greedy algorithm, and it
Decision_tree_learning
German philosopher (1833–1911)
attempts to explicate how different mental processes converge in the "structural nexus of consciousness." The distinction is based on the more general
Wilhelm_Dilthey
American philosophy – Individualism – Indonesian philosophy – Inductionism – Induction / Informal logic – Information, philosophy of – Innatism – Instrumentalism
List_of_philosophies
critics and moviegoers (destroying his career), PoP! reunites for their induction into the Rock and Roll Hall of Fame, after which their lead singer Colin
List of films with post-credits scenes
List_of_films_with_post-credits_scenes
Statement supporting a conclusion
usage. The term induction is sometimes used in a broader sense for any type of non-deductive argument. A slightly different form of induction draws a conclusion
Premise
American Automotive Engineer and professor
experience). This involved developing the architecture of the Miata’s structural elements, suspension, engine and powerplant, human occupants, etc. so
Norman Garrett (automotive engineer)
Norman_Garrett_(automotive_engineer)
In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language
Induction of regular languages
Induction_of_regular_languages
(July 2001). "Identification of CpG oligonucleotide sequences with high induction of IFN-alpha/beta in plasmacytoid dendritic cells". European Journal of
CpG_oligodeoxynucleotide
specific receptors. Human hormones are commonly grouped into four major structural classes: Spelling conventions vary between regions. For example, current
List_of_human_hormones
Set genes encoding proteins and enzymes for lactose metabolism
Bettenbrock K (January 2012). "lac operon induction in Escherichia coli: Systematic comparison of IPTG and TMG induction and influence of the transacetylase
Lac_operon
Austrian physicist, philosopher and university educator (1838–1916)
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Ernst_Mach
Conformity to reality
theorems, such as direct proof, proof by contradiction, and mathematical induction. Formal logic studies the nature of deductive reasoning and the rules
Truth
Branch of science that studies society and its relationships
three modes of social thought in particular: Durkheimian positivism and structural functionalism; Marxist historical materialism and conflict theory; and
Social_science
Epistemological philosophy advanced by Karl Popper
notably by induction, and the demarcation of science. He adopted a fallibilist approach to these problems, especially that of induction, without falling
Critical_rationalism
Philosophical idea that only one's own mind is sure to exist
that knowledge must be certain). It still entertains the points that any induction is fallible. Methodological solipsism sometimes goes even further to say
Solipsism
Medically induced loss of consciousness
principles remain. Stage 1 Stage 1, also known as induction, is the period between the administration of induction agents and loss of consciousness. During this
General_anaesthesia
as the blastopore. It is particularly important for its role in neural induction through the default model, where signaling from the dorsal lip protects
Dorsal_lip
Annual entrance test held in India
Electricity Magnetic effect of current and magnetism Electromagnetic Induction Alternating Current Electromagnetic Waves Optics Dual Nature of Radiation
IISER_Aptitude_Test
Cognition about knowledge and knowing
posteriori Belief Credence Certainty Data Experience Information Justification Induction Knowledge Meaning Rationality Reason Truth Wisdom Domains Applied epistemology
Epistemic_cognition
Critical protein in embryonic development
development. One of the most characterized functions of SHH is its role in the induction of the floor plate and diverse ventral cell types within the neural tube
Sonic_hedgehog_protein
Middlesbrough Development Corporation (Functions) Order 2023 448 The Education (Induction Arrangements for School Teachers) (England) (Amendment) Regulations 2023
List of statutory instruments of the United Kingdom, 2023
List_of_statutory_instruments_of_the_United_Kingdom,_2023
Process of aging in plants
root cells that do not acquire sugar and gases whether they are of a structural, reproductive, immature, or just plain, root nature. The idea that a plant
Plant_senescence
Position in the philosophy of science
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Instrumentalism
Weather condition in which water droplets freeze onto objects they come in contact with
Therefore, estimation of maximum potential ice load is crucial in the structural design of power line systems to withstand ice loads, and this can be done
Atmospheric_icing
Austrian–British philosopher of science (1902–1994)
writes, I approached the problem of induction through Hume. Hume, I felt, was perfectly right in pointing out that induction cannot be logically justified.
Karl_Popper
American philosopher (1891–1972)
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Stephen_Pepper
High school football stadium
district's athletic director. His career achievements at Allen resulted in his induction into the Texas High School Athletic Director Association’s 2020 Hall of
Eagle_Stadium_(Allen,_Texas)
Class of enzymes
PMC 1163832. PMID 786279. MacDonald IA, Roach PD (August 1981). "Bile induction of 7 alpha- and 7 beta-hydroxysteroid dehydrogenases in Clostridium absonum"
7alpha-hydroxysteroid dehydrogenase
7alpha-hydroxysteroid_dehydrogenase
Gene that regulates circadian rhythm in plants
the morning feedback loop and GI and ELF4 in the evening loop. Discrete induction of TOC1 gene expression results in reduced CCA1 and PRR9 expression, indicating
TOC1_(gene)
Short-acting anaesthetic and sedative drug
Amidate, is a short-acting intravenous anaesthetic agent used for the induction of general anaesthesia and sedation for short procedures such as reduction
Etomidate
1959 book by Karl Popper
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
The Logic of Scientific Discovery
The_Logic_of_Scientific_Discovery
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
Girl/Female
Indian
Shape, Structure
Girl/Female
Tamil
Shape, Structure
Boy/Male
Indian
Good Structure
Girl/Female
Indian, Kashmiri
Body Structure
Girl/Female
Indian
Structure
Boy/Male
Indian
Solid structure
Girl/Female
Indian
Shape, Structure
Girl/Female
Tamil
Shape, Structure
Boy/Male
Muslim
Solid structure
Boy/Male
Afghan, Arabic, Gujarati, Indian, Muslim
Solid Structure; Lifetime
Surname or Lastname
English
English : occupational name for a wattler, Middle English watelere, i.e. someone who made the panels of interwoven twigs that were used to fill the spaces between the structural timbers of a timber frame building. See also Dauber.
Girl/Female
Hindu, Indian, Telugu
The Structure of God
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
Boy/Male
Hindu, Indian, Marathi
Raft; Boat; A Competent Person; The Ocean
Boy/Male
Tamil
Friend
Boy/Male
Muslim
Famous
Biblical
gates; valuation; hairs
Boy/Male
Indian, Punjabi, Sikh
Hopeful
Boy/Male
Sikh
Pass through worldly cares
Surname or Lastname
English
English : habitational name from a place in Bedfordshire, so named from Old English sand ‘sand’ + ēg ‘island’, ‘dry land in a fen or marsh’.English : from the Old Norse personal name Sand(i), a short form of the various compound names with the first element sandr ‘sand’.
Girl/Female
Bengali, Hindu, Indian, Marathi, Tamil
Nectar
Boy/Male
Scottish
From the pasture.
Surname or Lastname
English
English : variant of Hickling.
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
a.
Having a definite organic structure; showing differentiation of parts.
n.
Framework; structure; edifice; building.
a.
Resembling shale in structure.
a.
Of or pertaining to structure; affecting structure; as, a structural error.
n.
Union of parts; structure.
a.
Bearing teeth or toothlike structures.
a.
Of lofty structure; tall.
n.
The act of building; the practice of erecting buildings; construction.
n.
Organic structure; organization.
n.
Manner of building; form; make; construction.
v. t.
To determine the homologies or structural relations of.
n.
Manner of organization; the arrangement of the different tissues or parts of animal and vegetable organisms; as, organic structure, or the structure of animals and plants; cellular structure.
a.
Being of the same typical structure; having like relations to a fundamental type to structure; as, those bones in the hand of man and the fore foot of a horse are homologous that correspond in their structural relations, that is, in their relations to the type structure of the fore limb in vertebrates.
a.
Pertaining to an edifice; structural.
n.
Arrangement of parts, of organs, or of constituent particles, in a substance or body; as, the structure of a rock or a mineral; the structure of a sentence.
n.
That which is built; a building; esp., a building of some size or magnificence; an edifice.
n.
Composition, or structure.
a.
A typical, structural unit; a type.
a.
Of or pertaining to organit structure; as, a structural element or cell; the structural peculiarities of an animal or a plant.
n.
Having the color spots, or structural parts, arranged spirally.