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Topic: Computably enumerable

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	Computable function Summary
		In computational complexity theory, the problem of determining the complexity of a computable function is known as a function problem.
		The fact that these models give equivalent classes of computable functions stems from the fact that each model is capable of reading and mimicking a procedure for any of the other models, much as a compiler is able to read instructions in one computer language and emit instructions in another langauge.
		The notion of computability of a function can be relativized to an arbitrary set of natural numbers A, or equivalently to an arbitrary function f from the naturals to the naturals, by using Turing machines (or any other model of computation) extended by an oracle for A or f.
	www.bookrags.com /Computable_function (2322 words)

	NationMaster - Encyclopedia: Recursive
		In mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of...
		In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if There is an algorithm that, when given an input â€” typically an integer or a tuple of integers or a sequence of characters â€” eventually halts if it...
		In mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of the class.
	www.nationmaster.com /encyclopedia/Recursive (599 words)

	Recursively enumerable set - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-30)
		In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if
		a recursively enumerable language is a recursively enumerable set in the set of all possible words over the alphabet of the language.
		The preimage of a recursively enumerable set under a computable function is a recursively enumerable set.
	www.sciencedaily.com /encyclopedia/recursively_enumerable_set (624 words)

	Turing degree - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-30)
		In Computer science and Mathematical logic, the Turing degree or degree of unsolvability of a set X of natural numbers is the equivalence class of all sets that are Turing equivalent to X.
		The concept of Turing degree is fundamental in computability theory.
		Sometimes, a number can be enumerated into X to satisfy one requirement but doing this would cause a previously satisfied requirement to become unsatisfied (that is, to be injured).
	www.sciencedaily.com /encyclopedia/turing_degree (1281 words)

	NationMaster - Encyclopedia: Turing degree
		A recursively enumerable Turing degree (computably enumerable Turing degree) is one containing a recursively enumerable set (computably enumerable set).
		The recursively enumerable Turing degrees under the partial order induced by Turing reducibility form an upper semi-lattice and are an object that has been much studied by the logic community.
		In computability theory, the Turing degree of a subset X of the natural numbers, ω, is the equivalence class of all subsets of ω equivalent to X under Turing reducibility.
	www.nationmaster.com /encyclopedia/Turing-degree (383 words)

	Computational Complexity: Foundations of Complexity Lesson 2: Computable and Computably Enumerable Languages
		Lesson 1 we described the Turing machine model to answer the question, "What is a computer?" The next question is "What can we compute?" First we need some definitions to describe problems to be computed.
		The class of computably enumerable languages is the set of languages L such that L = L(M) for some Turing machine M. You might see recognizable or recursively enumerable as other names for the computably enumerable languages.
		The class of computable languages consists of the set of languages L such that L = L(M) for some total Turing machine M. You might see decidable or recursive as other names for the computable languages.
	weblog.fortnow.com /2002/09/foundations-of-complexitylesson-2.html (423 words)

	Sample SRC Proposal for AMS Conferences
		Computability theory (or recursion theory) is the area of mathematical logic dealing with the theoretical bounds on, and structure of, computability and with the interplay between computability and definability in mathematical languages and structures.
		The {\it computably enumerable} (or {\it recursively enumerable}) sets are those whose members can be listed by a Turing machine or, equivalently, whose membership relation is definable by an existential formula in the language of arithmetic.
		There are many specific results establishing the interrelations among the computability properties of such notions depending on various properties of the structure in question (e.g., the dimension of the vector space or the transcendence degree of the field).
	www.ams.org /meetings/srcsample1.html (2785 words)

	Theory - Wikipedia, the free encyclopedia
		This knowledge consists of axioms, definitions, theorems and computational techniques, all related in some way by tradition or practice.
		Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood).
		Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them.
	en.wikipedia.org /wiki/Theory (2549 words)

	recursively, function, recursive, partial, natural, algorithm, total, recursion, halts, formal, texts, right, provable ...
		In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable or provable if: *There is an algorithm that, when given an input number, eventually ha
		The preimage of a recursively enumerable set under a partial recursive function is a recursively enumerable set.
		The definition of a recursively enumerable set as the domain of a partial function, rather than the range of a total recursive function, is common in contemporary texts.
	www.alphasearch.org /Recursively-enumerable-set.html (771 words)

	Research Interests of Robert I. Soare (Site not responding. Last check: 2007-10-30)
		The theory of computable function emerged during the 1930's with the primitive recursive functions used in Godel's incompleteness theorem, and then the full definitions by Godel [1934] Turing [1936] and others of computable functions which played a significant role in later development of computing devices.
		For example, the problem of deciding whether a program computes a total function is of strictly greater degree than that of the halting problem.
		The new book is intended as a replacement, updating the initial chapters, omitting many later chapters and writing ones on new results, and adding chapters on the relation of computability to model theory and to computational complexity in computer science.
	www.people.cs.uchicago.edu /~soare/CV/research.html (715 words)

	Recent papers of Geoff LaForte
		We study the relationship between computably enumerable reals and their approximations by means of sequences of prefix-free binary strings.
		We investigate algorithms which use the information in a set X to produce a set with information content that includes that of X and is relatively computably enumerable in X. These procedures, called pseudojumps, are the most natural ways (from a computational standpoint) to increase the information in a given set.
		He was the co-organizer of VIC '96, a conference on computational complexity and computability held in Wellington, New Zealand in December 1996, and the special session in computability theory at the regional AMS conference in Gainesville, Florida in March of 1999.
	www.cs.uwf.edu /~glaforte/paperlist.html (964 words)

	Recursively enumerable set (Site not responding. Last check: 2007-10-30)
		In the theory of computability (often less suggestively called recursion theory), a set ''S of natural numbers or tuples of natural numbers, or of literal string s, is recursively enumerable or computably enumerable or semi-decidable if it satisfies either (and therefore both) of the following equivalent conditions.
		There is an algorithm that, when given a natural number n (or tuple of natural numbers, or word, as the case may be) eventually halts if n is a member of S and otherwise runs forever.
		The word recursive is in this context taken to be synonymous with computable ; see recursive function.
	www.serebella.com /encyclopedia/article-Recursively_enumerable_set.html (255 words)

	Recursively enumerable set (Site not responding. Last check: 2007-10-30)
		In the theory of computability (often less called recursion theory) a set S of natural numbers or tuples of numbers or of literal strings is recursively enumerable or computably enumerable or semi-decidable if it satisfies either (and therefore of the following equivalent conditions.
		There is an algorithm that when given a natural number n (or tuple of natural numbers or as the case may be) eventually halts n is a member of S and otherwise runs forever.
		The first condition why the term semi-decidable is sometimes used; the second suggests computably enumerable is used.
	www.freeglossary.com /Recursively_enumerable (610 words)

	Russell Miller (Site not responding. Last check: 2007-10-30)
		I study computability theory, the branch of mathematical logic concerned with finite algorithms and the mathematical problems which such algorithms can or cannot solve.
		Computable model theory, one of my specialties, applies such techniques to general mathematical structures such as trees, linear orders, groups, fields, and algebraic varieties.
		Other interests of mine lie in pure computability theory; these include automorphisms of the lattice of computably enumerable sets (i.e., sets whose elements can be listed by an algorithm) and undecidability of the partial order of Turing degrees of those sets.
	qcpages.qc.edu /math/web/faculty/miller.htm (107 words)

	Nick’s Research Blog
		\nu(x) is an enumerable semimeasure: { (x,p) : p <= \nu(x) } is computably enumerable.
		Enumerable semimeasures are those where we can ask but we may not receive a reply if the answer is "no".
		If the enumerated element is (p,y) for some output y with o < y then output the tail of y not included in o and set o <- y.
	nicksresearch.wordpress.com (2232 words)

	Theodore A. Slaman: Bibliography (Site not responding. Last check: 2007-10-30)
		Greenberg, Noam and Shore, Richard A. and Slaman, Theodore A. The theory of the metarecursively enumerable degrees.
		Slaman, Theodore A. and Soare, Robert I. Extension of embeddings in the computably enumerable degrees.
		Slaman, Theodore A. The density of infima in the recursively enumerable degrees.
	math.berkeley.edu /~slaman/papers/Publications.html (1439 words)

	Blogger: Computational Complexity - Email Post to a Friend
		Machine M does not accept input }is not computably enumerable.
		If LA were computable then so wouldLD which we know is not even computably enumerable.
		We can view it as computing afunction f from Σ* to Σ* where thevalue of f is the contents of the output tape after the machine entersa specified halting state.
	www.blogger.com /email-post.g?blogID=3722233&postID=85544652 (229 words)

	Recursively enumerable
		In the theory of computability (often less suggestively called recursion theory), a set S of natural numbers or tuples of natural numbers, or of literal strings, is recursively enumerable orcomputably enumerable or semi-decidable if it satisfies either (and therefore both) of thefollowing equivalent conditions.
		There is an algorithm that, when given a natural number n (or tupleof natural numbers, or word, as the case may be) eventually halts if n is a member of S and otherwise runsforever.
		The first condition suggests why the term semi-decidable issometimes used; the second suggests why computably enumerable is used.
	www.therfcc.org /recursively-enumerable-43242.html (208 words)

	Randomness everywhere
		Moreover, although the infinite amount of information contained in Ω's digits is algorithmically incompressible, it turns out that Ω is `computably enumerable', which means that it can be calculated by an infinite process during which one can never know how close one is to the final value.
		An Ω is computably enumerable because a systematic run of all programs will produce better and better approximations (without being able to compute its digits exactly), and random because it is incompressible; there is no better way to find its digits than by tossing a fair coin.
		The existence of a computably enumerable random real number that is not an Ω became less plausible, but was not ruled out.
	www.cs.auckland.ac.nz /CDMTCS/chaitin/nature.html (1396 words)

	Department of Computer Science (Site not responding. Last check: 2007-10-30)
		In this talk we present some of the basic results about computably enumerable algebras and their properties.
		It is known, as proved by Bergstra and Tucker in the 80s, when the algebra is computable the question has a positive solution.
		However, we provide an example of a computably enumerably finitely generated algebra that has no finitely presented expansions.
	www.cs.uchicago.edu /events/249 (154 words)

	Courses in the Department of Mathematics
		Topics include the Kleene normal form theorem for representing computable functions and computably enumerable (c.e.) sets; the enumeration and s-m-n theorem, unsolvable problems, classification of c.e.
		Math 30300 develops the deeper properties of computability and the classification of relative computability on sets and (Turing) degrees.
		It begins with the finite injury priority method of Friedberg and Muchnik, continues with the infinite injury priority method of Sacks, and minimal pair of computably enumerable (c.e) degrees method by Lachlan.
	catalogs.uchicago.edu /divisions/math-courses.html (2661 words)

	Department of Computer Science
		University of Chicago, Department of Mathematics and Computer Science
		Extension Theorems and Automorphisms of the Computably Enumerable Sets
		Let E denote the structure of the computably enumerable (c.e) sets under inclusion modulo finite sets.
	www.cs.uchicago.edu /events/256 (250 words)

	Logic Colloquium '99 : Scientific Program
		Computational logic is the topic of special focus of the conference (but not at the expense of the other main topics).
		For every computably enumerable degree a, if the double jump of a is Turing equal to J then there is a computably enumerable set A in a with F(A).
		For all n > 1, the high_n (non-low_n) computably enumerable degrees are invariant.
	www.cwi.nl /lc99/scientific_program.html (2711 words)

	Computably Enumerable Reals and Uniformly Presentable Ideals - Downey, Terwijn (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		A set A presents a computably enumerable real if A is a computably enumerable pre x-free set of strings such that =.
		1 The theory of the recursively enumerable weak truth-table de..
		1 free languages and initial segments of computably enumerable..
	citeseer.ist.psu.edu /downey02computably.html (462 words)

	3130CIT course outline
		Computational complexity, tractable and intractable problems, and complexity classes for different models will be covered.
		Computable and computably enumerable languages (a language is computable iff it and its complement are computably enumerable).
		Unsolvability: Examples of languages that are computably enumerable but not computable, languages that are not computably enumerable.
	www.cit.gu.edu.au /teaching/3130CIT/outline.html (488 words)

	UMASS Boston Department of Computer Science
		If r is a reducibility between sets of numbers, a natural question to ask about the structure Cr of the r-degrees containing computably enumerable sets is whether every element not equal to the greatest one is branching.
		In this paper, we answer the question in the tt case by showing that every tt-incomplete computably enumerable truth-table degree a is branching in Ctt.
		The fact that every Turing-incomplete computably enumerable truth-table degree is branching can be shown using a technique of Ambos-Spies.
	www.cs.umb.edu /Researches/Publications/fejer/ttbranch.htm (180 words)

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