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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
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
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
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
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
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
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)
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
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
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
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
Programming language
theorems is standard in Dafny with recursion employed for induction (typically, structural induction). Case analysis is performed using match statements and
Dafny
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
philosophical foundations of analytic induction originate from Aristotle and his concept of induction. Analytic induction can be thought of as a phenomenological
Humanistic_sociology
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
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)
Awareness of facts, or competency
Gettier problem Induction Internalism and externalism Justification Knowledge Meta Objectivity Privileged access Problem of induction Problem of other
Knowledge
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
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
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
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
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
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
Concept in epistemology
posteriori Belief Credence Certainty Data Experience Information Justification Induction Knowledge Meaning Rationality Reason Truth Wisdom Domains Applied epistemology
Justification_(epistemology)
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
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
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
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
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
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
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
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
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
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
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
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
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
specific receptors. Human hormones are commonly grouped into four major structural classes: Spelling conventions vary between regions. For example, current
List_of_human_hormones
Philanthropy conception of meaning
Perspectivism Pragmatism Rationalism Relativism Skepticism Solipsism Structuralism Concepts Action Analytic–synthetic distinction A priori and a posteriori
Meaning_(philosophy)
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
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
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
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
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
Branch of biology that studies biological systems at the molecular level
Nature of the Substance Inducing Transformation of Pneumococcal Types: Induction of Transformation by a Desoxyribonucleic Acid Fraction Isolated from Pneumococcus
Molecular_biology
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
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
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
American philosophy – Individualism – Indonesian philosophy – Inductionism – Induction / Informal logic – Information, philosophy of – Innatism – Instrumentalism
List_of_philosophies
Thai structural engineer
Worsak Kanok-Nukulchai is a Thai structural engineer, university administrator and Professor Emeritus at the Asian Institute of Technology. He is currently
Worsak_Kanok-Nukulchai
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
One or more words used to refer to something
Interactional linguistics Prague circle Systemic functional Usage-based Structuralism Topics Autonomy of syntax Compositionality Conservative and innovative
Name
Enzyme
site cysteine of caspase-3 thus inactivating caspase-3 and preventing induction of apoptosis. As might be expected of an enzyme involved in regulating
Formaldehyde_dehydrogenase
(July 2001). "Identification of CpG oligonucleotide sequences with high induction of IFN-alpha/beta in plasmacytoid dendritic cells". European Journal of
CpG_oligodeoxynucleotide
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 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
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
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
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
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
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
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
Production of milk in male mammals
PMID 29372185. Wamboldt, R; Shuster, S; Bikrampal, B. S. (2021). "Lactation Induction in a Transgender Woman Wanting to Breastfeed: Case Report". The Journal
Male_lactation
President of the United States from 1993 to 2001
action kept the future governor in school says he was not told of 1969 induction letter. Draft board insists none was sent". Los Angeles Times. Retrieved
Bill_Clinton
Dried, fermented, and smoked skipjack tuna
ISBN 9780387283913. Arcos, Joseph C.; Argus, Mary F. (2013-10-22). Chemical Induction of Cancer: Structural Bases and Biological Mechanisms. Elsevier. ISBN 9781483263731
Katsuobushi
German philosopher, sociologist, and theorist (1903–1969)
positivism Russian Machism Principal concepts Consilience Demarcation Evidence Induction Justification Pseudoscience Critique of metaphysics Unity of science Verificationism
Theodor_W._Adorno
Idea that knowledge comes only/mainly from sensory experience
important slant to the debate about scientific method—that of the problem of induction. Hume argued that it requires inductive reasoning to arrive at the premises
Empiricism
He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. Arnaud Denjoy (1884–1974): French mathematician
List of atheists in science and technology
List_of_atheists_in_science_and_technology
Process of generalization
making any generalizations. Bacon used and promoted induction as an abstraction tool; his induction complemented but was distinct from the ancient deductive-thinking
Abstraction
Artificial intelligence field of study
Jared; McCandlish, Sam; Olah, Chris (2022). "In-context learning and induction heads". Transformer Circuits Thread. arXiv:2209.11895. Olah, Christopher
AI_safety
Brazilian biologist
properties distinct from traditionally studied forms, especially regarding the induction of axonal growth and modulation of the inflammatory environment after
Tatiana_Sampaio
Bearer of truth values
bypass difficulties of the structured proposition view. It asserts that structural analysis is a useful theoretical tool for understanding certain aspects
Proposition
Philosophical traditions from mainland Europe
the thought of Kierkegaard and Nietzsche), hermeneutics, structuralism, post-structuralism, deconstruction, French feminism, psychoanalytic theory, posthumanism
Continental_philosophy
Biological cell regulation mechanisms
ribozymes and self-regulate their expression.[citation needed] Enzyme induction is a process in which a molecule (e.g., a drug) induces (i.e., initiates
Regulation_of_gene_expression
Material supporting an assertion
come first to possess the evidence and later form the hypothesis through induction. But this temporal order is not always reflected in scientific practice
Evidence
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
Girl/Female
Indian
Shape, Structure
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
Tamil
Shape, Structure
Girl/Female
Indian
Structure
Girl/Female
Indian, Kashmiri
Body Structure
Girl/Female
Indian
Shape, Structure
Boy/Male
Indian
Solid structure
Boy/Male
Afghan, Arabic, Gujarati, Indian, Muslim
Solid Structure; Lifetime
Girl/Female
Hindu, Indian, Telugu
The Structure of God
Boy/Male
Indian
Good Structure
Boy/Male
Muslim
Solid structure
Girl/Female
Tamil
Shape, Structure
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
Girl/Female
Hindu, Indian, Marathi
Sunny
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Telugu
Young Girl
Female
Greek
(Εἰλείθυια) Greek name derived from the word eleluthyia, EILEITHYIA means "relieve." In mythology, this is the name of a goddess of childbirth. Her Latin name is Lucina.
Girl/Female
Biblical
Covered with foam.
Boy/Male
Tamil
Lord Vishnu
Boy/Male
English French
Lives in the valley.
Boy/Male
Gujarati, Hindu, Indian, Kannada
Brother of Prahlad
Boy/Male
Gujarati, Hindu, Indian, Marathi, Sanskrit
Water
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Goddess Durga
Surname or Lastname
English
English : variant of Edmondson.
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
STRUCTURAL INDUCTION
a.
Pertaining to an edifice; structural.
a.
Having a definite organic structure; showing differentiation of parts.
n.
Having the color spots, or structural parts, arranged spirally.
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.
n.
Manner of building; form; make; construction.
n.
Organic structure; organization.
a.
Resembling shale in structure.
a.
Of lofty structure; tall.
n.
Framework; structure; edifice; building.
n.
That which is built; a building; esp., a building of some size or magnificence; an edifice.
n.
The act of building; the practice of erecting buildings; construction.
n.
Composition, or structure.
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.
a.
A typical, structural unit; a type.
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.
n.
Union of parts; structure.
a.
Of or pertaining to structure; affecting structure; as, a structural error.
a.
Bearing teeth or toothlike structures.
a.
Of or pertaining to organit structure; as, a structural element or cell; the structural peculiarities of an animal or a plant.
v. t.
To determine the homologies or structural relations of.