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Topic: Recursively enumerable set

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	CBofN - Glossary
		Sets that are incomputable may be recursively enumerable (like the halting set), co-recursively enumerable (e.g., the halting set's complement), or not recursively enumerable (which, if also not CO-RE, is a random set).
		All of the Julia sets are related to the Mandelbrot set.
		Recursively Enumerable (RE) A potentially infinite set whose members can be enumerated by a universal computer; however, a universal computer may not be able to determine that something is not a member of a recursively enumerable set.
	mitpress.mit.edu /books/FLAOH/cbnhtml/glossary.html (8347 words)

	Recursively enumerable set - Wikipedia, the free encyclopedia
		A recursively enumerable language is a recursively enumerable subset of a formal language.
		Matiyasevich's theorem states that every recursively enumerable set is a Diophantine set (the converse is trivially true).
		The preimage of a recursively enumerable set under a partial recursive function is a recursively enumerable set.
	en.wikipedia.org /wiki/Recursively_enumerable_set (791 words)

	Twenty Problems in the Theory of Cellular Automata (1985)
		The set of all excluded configurations is recursively enumerable, since each of its elements is found by a finite computation.
		Thus the limit sets for cellular automata are always the complements of recursively enumerable (co-r.e.) sets, and are therefore countable in number.
		set is the limit set for a cellular automaton: one additional condition is that they must be translationally invariant.
	www.stephenwolfram.com /publications/articles/ca/85-twenty/15/text.html (649 words)

	Springer Online Reference Works
		Recursive function) that examines and classifies subsets of natural numbers from the point of view of algorithms, and also studies the structures arising as a result of such a classification.
		The mathematical posing of problems of this kind, and the development of recursive set theory, only became possible in the 1940's after the successful formalization of the intuitive concept of an (algorithmically) computable function.
		The definition of these classes of sets is not given in lattice-theoretic terms; in fact it has been shown that "being hyper-simple" is not a lattice-theoretic property.
	eom.springer.de /R/r080340.htm (1450 words)

	Recursively enumerable set - Education - Information - Educational Resources - Encyclopedia - Music
		A set A is a recursive set iff both A and the complement of A are recursively enumerable sets.
		The preimage of a recursively enumerable set under a computable function is a recursively enumerable set.
		a recursively enumerable language is a recursive enumerable set in the set of all possible words over the alphabet of the language.
	education.music.us /R/Recursively-enumerable-set.htm (529 words)

	Recursively Enumerable Set -- from Wolfram MathWorld
		Recursively undecidable problems give examples of recursively enumerable sets that are not recursive.
		In particular, the set of all Gödel numbers of total Turing machines is an example of a set which is not recursively enumerable.
		The complements of recursively enumerable but not recursive sets are all not recursively enumerable, although complements of sets that are not recursively enumerable are not necessarily recursively enumerable.
	mathworld.wolfram.com /RecursivelyEnumerableSet.html (226 words)

	PlanetMath: recursively enumerable
		The set of encodings of Turing machines which halt when given no input.
		The set of encodings of theorems of Peano arithmetic.
		This is version 3 of recursively enumerable, born on 2002-06-05, modified 2002-06-05.
	www.planetmath.org /encyclopedia/RecursivelyEnumerable.html (108 words)

	PlanetMath: recursively enumerable
		The set of encodings of Turing machines which halt when given no input.
		The set of encodings of theorems of Peano arithmetic.
		This is version 3 of recursively enumerable, born on 2002-06-05, modified 2002-06-05.
	planetmath.org /encyclopedia/RecursivelyEnumerable.html (109 words)

	Peter Suber, "Glossary of First-Order Logic"
		The set {a, b, c} is enumerated by the sequence , but also by the sequence ; it is not enumerated by the sequence .
		A property possessed by all the wffs in a set is logically hereditary iff the accepted rules of inference pass it on (transmit it) to all the conclusions derivable from that set by those rules.
		The set of all the subsets of a set.
	www.earlham.edu /~peters/courses/logsys/glossary.htm (9715 words)

	Richard A. Shore: Publications
		The recursively enumerable alpha-degrees are dense, Annals of Mathematical Logic 9 (1976), 123-155.
		The elementary theory of the recursively enumerable degrees is not aleph
		The theory of the recursively enumerable weak truth table degrees is undecidable, Journal of Symbolic Logic 57 (1992), 864-874 (with Ambos-Spies and Nies).
	www.math.cornell.edu /~shore/publications.html (2448 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		A recursively enumerable language is a recursively enumerable subset of a formal language.
		If A and B are recursively enumerable sets then A â© B, A âª B and A Ã— B (with the ordered pair of natural numbers mapped to a single natural number with the Cantor pairing function) are recursively enumerable sets.
		The preimage of a recursively enumerable set under a partial recursive function is a recursively enumerable set.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=recursively_enumerable (791 words)

	Springer Online Reference Works
		Immune set), although the productive and the immune sets do not exhaust the family of sets that are not recursively enumerable.
		the set of all Gödel numbers of general recursive functions in some Gödel enumeration of all partial recursive functions, as are the sets of all numbers of true or false formulas in elementary arithmetic in a natural enumeration of all formulas in it).
		Recursively-enumerable sets whose complements (in the number series) are productive are called creative sets; they form an important class of recursively-enumerable sets.
	eom.springer.de /p/p075040.htm (212 words)

	9. Recursively Enumerable Sets (Site not responding. Last check: 2007-10-26)
		Definition 9.1 A set X is recursively enumerable (or r.e.
		Thus, we see that the class of sets generated by partial recursive functions is identical to the class of sets accepted by partial recursive functions.
		Since the recursive sets are closed under complementation, it suffices to show that every non-empty recursive set is recursively enumerable.
	www.cs.pitt.edu /~daley/cs2110/notes/cs2110w_node36.html (644 words)

	Recursively Enumerable (Site not responding. Last check: 2007-10-26)
		Sets with a cardinality greater than that of the natural numbers (such as the set of real numbers) are not recursively enumerable.
		There are infinite sets which are recursively enumerable that have subsets which aren't; the trivial example is the set of all strings over an alphabet A, which is fully decideable; for example, the language described by the regular expression [ab]* is decideable (and thus RecursivelyEnumerable).
		Any infinite set whose inverse is finite is RecursivelyEnumerable (and decideable); as comparing any element with the finite set will always terminate.
	c2.com /cgi/wiki?RecursivelyEnumerable (517 words)

	An extension of the recursively enumerable Turing degrees
		An extension of the recursively enumerable Turing degrees
		The set of all recursively enumerable Turing degrees is denoted
		Recursively enumerable sets of positive integers and their decision problems.
	www.math.psu.edu /simpson/papers/extre/extre.html (1702 words)

	Diophantine set Diophantine equation mathematics integer recursively enumerable set polynomial (Site not responding. Last check: 2007-10-26)
		Matiyasevich's theorem, published in 1970, states that a set of integers is Diophantine if and only if it is recursively enumerable.
		A set S is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of S and otherwise runs forever.
		Hence, if the set of all primes or the set of all powers of 2 had Diophantine representations, then these sets could be represented as sets of positive values of appropriate polynomials.
	en.powerwissen.com /DLXg126YHrvnL6q05e6Wsg==_Diophantine_set.html (252 words)

	Transactions of the American Mathematical Society
		K. Ambos-Spies, On the Structure of the Polynomial-time Degrees of Recursive Sets, Habilitationschrift, Universität Dortmund, 1984.
		K. Ambos-Spies and A. Nies ``The theory of the polynomial many-one degrees of recursive sets is undecidable,'' STACS 92, Lecture Notes in Computer Science 577 (1992) 209-218.
		A. Nerode and J. Remmel, A Survey of Lattices of Recursively Enumerable Substructures, Recursion Theory, Proceedings of Symposia in Pure Mathematics 42 (1985), 322-375.
	www.ams.org /tran/2000-352-11/S0002-9947-00-02652-0/home.html (485 words)

	Thomas Sturm: Activities: ACA02: talks: Matiyasevich (Site not responding. Last check: 2007-10-26)
		but not recursive set implies that in general it is impossible to eliminate all the existential quantifiers.
		set implies that if m in is large enough, then some existential quantifiers can be eliminated.
		It was shown in [MatiyasevichR75] that m=13 variables under the existential quantifier are sufficient for constructing a Diophantine representation of an arbitrary r.e.
	www.fmi.uni-passau.de /~sturm/activities/ACA02/talks/Matiyasevich (1305 words)

	Length Functions of a Finitely Generated Group
		It is clear that not every function satisfying (D1)-(D3) can be a length function of the cyclic group in a finitely presented group: the cardinality of the set of O-equivalence classes of functions satisfying (D1)-(D3) is the continuum, and the set of embeddings of the infinite cyclic group into finitely presented groups is countable.
		Although Theorem 9 shows that all ``reasonable" functions are length functions of a given finitely generated recursively presented group inside finitely presented groups, it does not give a characterization of these functions.
		By the proper choice of a universal group H it is not difficult to sharpen the formulation of Theorems 9 and 11.
	math.vanderbilt.edu /~msapir/Talk1/node5.html (924 words)

	Review of Rodriguez-Consuegra
		But this presupposes that the set of informally provable mathematical truths is of the same character as the set of theorems of a formal system, a recursively enumerable set, an assumption cast into doubt by Lucas 1961's attempt to use Gödel's results to refute mechanism.
		At best it shows that the enumeration of a set of sentences by a Turing machine is the wrong mechanistic model for human mathematical activity.
		Since, by Tarski's theorem, the set of truth sentences of the language of first-order arithmetic is not even arithmètic, it seems likely that incompleteness theorems hold for any plausible mechanistic model, though the undecidable sentences may be of greater logical complexity than those Gödel and Jeroslow provide for formal systems and experimental logics respectively.
	www.philosophy.unimelb.edu.au /handouts/161042/gibbs.html (1599 words)

	Logical foundations and formal verification - TYPES AND SPECIFICATIONS
		The classical complement of a recursively enumerable set is not in general recursively enumerable.
		The recursively enumerable set whose characteristic function is obtained by applying Notp to some characteristic function is not uniquely determined by the recursively enumerable set determined by the characteristic function.
		If these spaces are enumerable, and if the operation of forming a representative combinator for a (dependent) function space from representative of the domain and codomains is computable, then we need only to determine one of the combinators which represents this computation and we have the basis for a maximally expressive type system.
	www.rbjones.com /rbjpub/rbjcv/papers/dtc114.htm (2156 words)

	Example showing how to find a grammar from a Turing machine
		The language of M is the set of all strings accepted by M. If a string is not accepted, then either it halts in some state other than H, or else halts in H with something other than Y on the tape, or else does not halt at all.
		DEF.An enumeration Turing machine for a set of strings, S, is a machine, such that if started on a blank string, in state 1, it prints the members of S, one after the other on the tape, separated by some special symbol such as *.
		The set of strings for which it is defined is called the domain of a function.
	www.runet.edu /~jhelm/classes/420/notes.html (3799 words)

	Logic, 8
		Therefore the valuation which sets all Ki to true will satisfy all the elements of G. Another proof is as follows: we use Konig’s lemma, which claims that if T is a binary branching tree, such that every branch is finite, the number of nodes is finite.
		Recursively enumerable can be thought of as being able to find the positive cases, but not the negative ones – i.e., one cannot falsify some inclusion in the set, since the program may not converge.
		Eventually n is enumerated in A or in cA, (since positivity/recognizing/accepting is guaranteed for RE sets), and thus we have a prf which accepts all elements.
	www.media.mit.edu /physics/pedagogy/babbage/texts/rt.html (8591 words)

	Formal Literatures (Site not responding. Last check: 2007-10-26)
		Namely, let e be a computable literature and let c be the representation of the Gödel number of the recursively enumerable set of e as a string of elements of A.
		It may be more convenient to describe natural languages as formal literatures than as formal languages: if we allow the definition of new terms and require that new terms be used in accordance with their definitions, then we have restrictions on sentences that depend on what sentences have previously been uttered.
		Any natural language may be regarded as universal with respect to the set of natural languages in the approximate sense that we might define French in terms of English and then say `From now on we shall speak only French'.
	www-formal.stanford.edu /pub/jmc/mcchay69/node18.html (680 words)

	[No title]
		Secondly, we show that the sets in a universal Martin-L\"of test for randomness have random measure, and every recursively enumerable random number is the sum of the measures represented in a universal Martin-L\"of test.
		We apply it to prove the Harrington-Slaman theorem that the theory of the recursively enumerable degrees is recursively isomorphic to the theory of first-order arithmetic.
		We prove several results about when these degrees are trivial, and when the degrees are omniscient (i.e., the set of recursive functions is learnable).
	math.berkeley.edu /~slaman/bibserver.bib (1784 words)

	Is a deterministic universe logically consistent with a probabilistic Quantum Theory?
		Lemma 6: There is no experiment, or finite set of experiments, that can completely determine the state of a finite number of, arbitrarily selected, properties of a particle P simultaneously, even though, by our premise, given any property k, there is always some experiment that will completely determine its state m at instant t.
		We define a language L as recursively enumerable if there exists a Turing machine that accepts every string of the language, and does not accept strings that are not in the language.
		Classically (3, p120, 121, 214), a partial function f of n arguments is called partial recursive if, and only if, f can be obtained from the initial functions (zero function), projection functions, and successor function (of classical recursive function theory) by means of substitution, recursion and the classical, unrestricted, μ-operator.
	alixcomsi.com /CTG_08_Quantum_consistency1.htm (2936 words)

	GATE Study Material
		Here is a triplet, whose first component, M is an encoding of a turing machine, second component M1 is the encoding of a nondeterministic linear bounded automaton, and third component I is a bit.
		Here is a triplet, whose first component, M is an encoding of a turing machine, second component M1 is the encoding of a 100 tape nondeterministic turing machine that halts on all inputs, and third component I is a bit.
		Here is a triplet, whose first component, M is an encoding of a nondeterministic 789 pushdown tape machine, second component w, is a state and third component I is a bit.
	www.onestopgate.com /question-bank/question/q18.asp (1423 words)

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