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	Structural induction - Open Encyclopedia (Site not responding. Last check: 2007-10-08)
		Structural induction is a proof method that is used in mathematical logic (e.g., the proof of Los's theorem), computer science, graph theory, and some other mathematical fields.
		The structural induction proof then consists of proving that the proposition holds for all the minimal structures, and that if it holds for the substructures of a certain structure S, then it must hold for S also.
		Structural recursion bears the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction.
	www.open-encyclopedia.com /Structural_induction (828 words)

	Induction (disambiguation) - Wikipedia, the free encyclopedia
		Induction in the fields of philosophy and logic, and used in science and the scientific method.
		Mathematical induction is a method of proof in the field of mathematics.
		Structural induction is a generalization of mathematical induction.
	www.wikipedia.org /wiki/Induction (186 words)

	Encyclopedia: Mathematical induction (Site not responding. Last check: 2007-10-08)
		Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence.
		Structural induction is a proof method that is used in mathematical logic (e.
		Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals.
	www.nationmaster.com /encyclopedia/Mathematical-induction (1793 words)

	Structural induction (Site not responding. Last check: 2007-10-08)
		Mathematical Induction Lecture notes by Peter Suber, explaining the difference between inductive inference and mathematical induction (which is a species of deductive inference).
		New Induction The new model of electromagnetic induction is claimed as superior to Faraday's law in every respect.
		Induction Induction is the conscious mental process by which we pass from the perception of particular phenomena (things and events) to the knowledge of general truths.
	www.serebella.com /encyclopedia/article-Structural_induction.html (896 words)

	Structural Induction
		Structural induction is an extension of one of the classical proof techniques in mathematics known as mathematical induction.
		The idea of structural induction is not always immediately accessible so we use the example of the stack to ease the analysis and give the reader confidence with the approach.
		With structural induction, the base case which must be established is to demonstrate that P is true for all nullary constructor operations.
	scom.hud.ac.uk /staff/scomtlm/book/node332.html (611 words)

	Structural induction -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The structural induction proof then consists of proving that the proposition holds for all the (Click link for more info and facts about minimal) minimal structures, and that if it holds for the substructures of a certain structure S, then it must hold for S also.
		Just as standard (Click link for more info and facts about mathematical induction) mathematical induction is equivalent to the (Click link for more info and facts about well-ordering principle) well-ordering principle, structural induction is also equivalent to a well-ordering principle.
		Structural recursion bears the same relationship to structural induction as ordinary ((mathematics) an expression such that each term is generated by repeating a particular mathematical operation) recursion bears to ordinary (Click link for more info and facts about mathematical induction) mathematical induction.
	www.absoluteastronomy.com /encyclopedia/s/st/structural_induction.htm (865 words)

	Handout - Induction Review (Site not responding. Last check: 2007-10-08)
		In all inductive proofs, it is good practice to state the induction hypothesis explicitly and indicate explicitly where you use it in your proof that P holds for n.
		In a proof by induction, we first show that the statement is true for the "smallest" values of our inductively defined set, which for the natural numbers is just 0.
		Induction is often used to show that a recursive procedure computes the correct answer.
	www.cs.dartmouth.edu /~brd/Teaching/AI/Handouts/Induction/induction.html (1596 words)

	Mathematical induction - Wikipedia, the free encyclopedia
		The first known proof by mathematical induction appears in Francesco Maurolico's Arithmeticorum libri fuo (1575).
		This form of mathematical induction is actually a special case of the previous form because if the statement that we intend to prove is P(n) then proving it with these two rules is equivalent with proving P(n + b) for all natural numbers n with the first two steps.
		Another generalization, called complete induction (or strong induction), allows that in the second step we assume not only that the statement holds for n = m but also that it is true for n smaller than or equal to m.
	en.wikipedia.org /wiki/mathematical_induction (1020 words)

	Citations: Proving properties of programs by structural induction - Burstall (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		contributed structured recursion, a generalised form of primitive recursion, to analytic syntax, with an associated principle of structural induction.
		This notion of induction on the structure of a term introduces the concept of a set of constructors for E. Informally, a set of constructors for E is a set of formulae such that we can....
		Essentially, structural induction shows that inductive proofs over the constructors of objects such as trees can be conducted directly without the need to recast them as numerical induction on nesting....
	citeseer.ist.psu.edu /context/53789/0 (2672 words)

	Structural induction - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-08)
		Chemical Induction of Cancer: Structural Bases and Biological Mechanisms, Part C (Chemical Induction of Cancer - Natural, Metal, Fiber & Macro)
		Structural Reliabilism : Inductive Logic as a Theory of Justification (Trends in Logic)
		Structural induction on partial algebras (Introduction to theory and application of partial algebras)
	encyclopedia.worldsearch.com /structural_recursion.htm (869 words)

	Talk:Three forms of mathematical induction - Information (Site not responding. Last check: 2007-10-08)
		This must be possible, since otherwise (by another induction I won't write out here) the tree has an infinite sequence of descendents, meaning that it is either infinite (which it isn't because it only has k+1 nodes with any children) or else cyclic (but then it wouldn't be a tree by definition of tree).
		Oh yes, and the thing that I understand by "structural induction" can usually (my guess is always, but I don't want to have to prove it) be recast as strong induction on the size of the structure.
		That's not necessarily to say that structural induction isn't sufficiently useful a concept to be added to the article (which would then have to be retitled), just that the challenge as stated can be met...
	www.book-spot.co.uk /index.php/Talk:Three_forms_of_mathematical_induction (1023 words)

	Structural Induction
		The pattern of reasoning follows exactly the structure of the inductive definition --- the base case is handled outright, and the inductive step is handled by assuming the property for the predecessor and show that it holds for the numbers.
		It is interesting to observe that the pattern of structural recursion may be directly codified in ML as a higher-order function.
		Summarizing, the principle of structural induction over a recursive datatype is naturally codified in ML using pattern matching and higher-order functions.
	www.cs.cmu.edu /People/rwh/introsml/techniques/structur.htm (1521 words)

	Mathematical induction (Site not responding. Last check: 2007-10-08)
		Mathematical induction is a method of mathematical proof typically used to establish that a statement is true of all natural numbers or otherwise is true of all of an infinite sequence.
		Note that this form mathematical induction is actually a special case the previous form because if the statement we intend to prove is P (n) then proving it with these two is equivalent with proving P (n + b) for all natural numbers n with the first two steps.
		The principle of mathematical induction is usually as an axiom of natural numbers see Peano axioms.
	www.freeglossary.com /Mathematical_induction (1035 words)

	Practical Foundations of Mathematics
		Structural recursion over free algebras is crucial for foundations because, from the outside, the mathematical world is just a string of symbols: to handle it at the most basic level we need concatenation and parsing operations.
		The last is the induction scheme, which we have already mentioned as an example of well founded induction in Example 2.5.5(b).
		This is induction for the reflexive-transitive closure of any binary relation, and is discussed in Sections 3.8 and 6.4.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s27.html (1256 words)

	CS212 F98 Structural Induction
		Recall from the handout on induction that induction is a method of proving statements about inductively defined sets.
		The induction step assumes that the fact we are trying to prove is true for all lists of length less than n, then uses that assumption to establish that it is true for lists of length n.
		In that handout, we gave a proof by induction on length that this procedure was correct.
	www.cs.cornell.edu /html/cs212-fall98/handouts/structural-induction.html (976 words)

	Mathematical Induction (Site not responding. Last check: 2007-10-08)
		Although he was unable to attend the induction ceremony due to obligations in...
		The simplest and most common form of mathematical induction proves that a statement holds for all natural numbers n and consists of two steps: # Showing that the statement holds when n = 0.
		Proofs by transfinite induction typically need to distinguish three cases: # m is a minimal element, i.e.
	www.wikiverse.org /mathematical-induction (976 words)

	CSC 236, Fall 2004, day section
		Structural induction is basically induction on the number of structure-creating rules invoked.
		Some of the earlier induction material involving induction on the height of trees could have been cast as structural induction on the recursive definition of a tree data structure.
		The midterm covers assignments 1 and 2, and chapters 1 to 4 of the notes, except for structural induction, and except for section 3.2.2 (general solution of a certain form of divide-and-conquer recurrence).
	www.dgp.toronto.edu /people/ajr/236/day (621 words)

	aether architecture (Site not responding. Last check: 2007-10-08)
		Aether induction house is an architecture prototype looking into ways of treating digital media as physical matter.
		The surface of a computer projection is unfolded onto a translucent structure, becoming a spatial experience for the visitors.
		Induction house (version1) was first exhibited at Mücsarnok (Kunsthalle Budapest) within the Közben show about contemporary Hungarian Architecture in October 2003.
	www.aether.hu /induction.htm (294 words)

	[No title]
		Furthermore, the compiler is given by a compositional (syntax-directed) translation, so properties of the compiler can be proved by structural induction on the grammar of the source language.
		Because the algorithms were tightly determined by the mathematics and we used some simple coding standards for data structures, the system has been remarkably reliable: only two errors have been discovered since the system passed its initial tests.
		Another possible development is the application of the theorem-prover to other kinds of inductions over lambda terms, such as those typically found in fixed-point reasoning or in reasoning about sequences.
	www.ccs.neu.edu /home/wand/research/mstp/techreport/paper.7-21 (3385 words)

	CS 212 Lecture: Structural Induction
		Notice that I explicitly stated the induction hypothesis and where I used it to justify the proof that P(cons i x) holds.
		Then we can prove that the P holds for data structures x such that f(x) = 0, and show that, whenever P holds for data structures such that f(x) = n, then P holds for data structures x' such that f(x) = n+1.
		Then we can argue by induction on the natural numbers that for each n, P holds for data structures x such that f(x)=n.
	www.cs.cornell.edu /courses/cs212/1998sp/lectures/lec_2_24.html (1309 words)

	Induction (disambiguation) (Site not responding. Last check: 2007-10-08)
		When we take into consideration what this book is supposed to be, that is, a guide to deeper and more productive inductions, the Handbook fills this niche very well.Those who purchase this book expecting to pour over case after case of imperical data prov...
		The Goodman Paradox and the New Riddle of I...
		Elementary induction on abstract structures (Studies in logic and the foundations of mathematics)
	www.freeglossary.com /Induction_(disambiguation) (367 words)

	Taxonomy of Proof: structural induction
		However, forms of induction can be appropriate when trying to prove things about structures defined recursively.
		This generalized induction is known as structural induction.
		A classic use of structural induction is to prove that any legal expression has the same number of left parentheses and right parentheses:
	theory.stanford.edu /~csilvers/proof/node5.html (505 words)

	Structural Induction Example - Binary Trees
		As a further example of structural induction, we consider an example worked out in detail on the domain of full binary trees.
		Whenever we consider a proof by structural induction, it is based on an inductive definition of the data domain.
		This example shows the mechanics of proofs by structural induction for recursive functional programs in all their gory detail.
	www.cs.sfu.ca /~cameron/Teaching/384/99-3/tree-induction.html (474 words)

	Exercise 13: Hoare Logic
		Structural induction breaks the proof down into several cases, one for each constructor of the type.
		This tactic takes two arguments, the first is a theorem which justifies the validity of performing structural induction on the type in question, and the second is another tactic that defines what is done with the inductive hypothesis in the recursive cases.
		There is a function supplied in HOL that takes the defining axiom of a type and proves from it a theorem that directly asserts the validity of performing structural induction on that type.
	cs.anu.edu.au /student/comp8033/ex13.html (1249 words)

	[No title]
		Attempting to prove level 3 subgoal 3 (induction step) for proof by induction on z Level 3 subgoal 3 (induction step) for proof by induction on z [] Proved by normalization.
		Level 2 subgoal 2 (induction step) for proof by induction on y: level(L(yc)) = level(z) /\ gr(L(yc)) /\ gr(z) => L(yc) = z [] Proved by structural induction on `z'.
		Level 2 subgoal 3 (induction step) for proof by induction on y: level(R(yc)) = level(z) /\ gr(R(yc)) /\ gr(z) => R(yc) = z [] Proved by structural induction on `z'.
	www.cs.technion.ac.il /~myoeli/pub/nmutex.lplog (2060 words)

	The Nuclear Pregnane X Receptor: A Key Regulator of Xenobiotic Metabolism -- Kliewer et al. 23 (5): 687 -- Endocrine ...
		that characterized induction of the CYP3A23 gene, the dexamethasone-dependent
		The three-dimensional structure of the human PXR LBD is presented as a ribbon diagram.
		Characterization of its mRNA, gene, and induction response.
	edrv.endojournals.org /cgi/content/full/23/5/687 (8582 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		NOTES ON CONSTRUCTIVE INDUCTION Constructive induction -- also known as structural induction -- is best learned by example.
		Here is a typical one: Suppose we wish to prove that SUMi=1_to_n i is a quadratic expression in n, but we don't know exactly which expression.
		The base case is n=1: SUM = a + b + c which becomes 1 = a + b + c since for n=1, SUM = 1.
	www.cs.umd.edu /class/spring2000/cmsc251/notes/induction.text (242 words)

	CS312 Mathematical Induction
		The assumption that the statement holds for all values less than n is called the induction hypothesis.
		Thus our induction hypothesis is n = 3u + 5v for some u and v, and we wish to prove under this assumption that n+1 = 3u' + 5v' for some u' and v'.
		The automatic verification of programs (checking that they meet a given specification) is an active area of research.
	www.cs.cornell.edu /courses/cs312/2002sp/handouts/induction/induction.html (1496 words)

	Roles of Shared Relations in Induction: Examination of Structural Alignment Theory (Site not responding. Last check: 2007-10-08)
		A structural alignment view of induction was generalized to account for these phenomena.
		According to the structural alignment view proposed in this paper, (1) insufficiency of the number of shared relations caused the dissociation between shared relations and inductive strength, and (2) structural alignment during similarity judgment made shared relations so salient as to increase similarity.
		The participants who rated similarity between categories of arguments prior to judgment of inductive strength judged arguments having a shared relation to be stronger, whereas the participants who only judged inductive strength did not judge so.
	www.nime.ac.jp /~ohnishi/pub/tl02abst.html (207 words)

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