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Topic: Undecidable

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	Halting problem - Wikipedia, the free encyclopedia
		One such consequence of the halting problem's undecidability is that there cannot be a general algorithm that decides whether a given statement about natural numbers is true or not.
		Yet another, quite amazing, consequence of the undecidability of the halting problem is Rice's theorem which states that the truth of any non-trivial statement about the function that is defined by an algorithm is undecidable.
		The undecidability of the halting problem relies on the fact that algorithms are assumed to have potentially infinite storage: at any one time they can only store finitely many things, but they can always store more and they never run out of memory.
	en.wikipedia.org /wiki/Halting_problem (4381 words)

	Undecidable - Wikipedia, the free encyclopedia
		A decision problem is undecidable if there is no known algorithm that decides it.
		"Undecidable" is sometimes used as a synonym of "independent".
		This is a disambiguation page: a list of articles associated with the same title.
	en.wikipedia.org /wiki/Undecidable (96 words)

	Constructivity_abstracts
		We thus argue that the formal undecidability of true Arithmetical propositions is a characteristic not of relations that are Platonistically inherent in any Arithmetic of the natural numbers, but of the particular formalisation chosen to represent them.
		Hence it does not universally establish the existence of undecidable propositions that are true under interpretation.
		We conclude that the formal undecidability of Arithmetical propositions that are true under interpretation is a characteristic not of any relations that are Platonistically inherent in any Arithmetic of the natural numbers, but of the particular formalisation chosen to represent the Arithmetic.
	alixcomsi.com /Constructivity_abstracts.htm (1087 words)

	Undecidable Problems in Fractal Geometry
		This paper is motivated by a conjecture by Penrose in his book [10] that the Mandelbrot set is undecidable under a reasonable model of computation which allows you to work with real numbers.
		Lemma 1 and its generalisation to 2-D case show that there is exact correspondence between the strings and the concatenation operation, and the application of the corresponding affine transformations.
		Theorem 1 implies that it is undecidable to test if the intersection of the attractors of two given IFSs is empty.
	journal-ci.csse.monash.edu.au /ci/vol02/undecide/undecide.html (1961 words)

	Undecidability, Epistemology and Anti-Realist Iniuitionism
		Second, he argues that if certain ``undecidable'', or ``not effectively decidable'', or ``effectively undecidable'' mathematical statements are to satisfy the manifestation requirement, then there's something wrong with applying the principle of bivalence to these statements.
		The more fundamental worry is that if undecidability is relative to which axioms and rules are accepted, then it would seem that what counts as undecidable in the case of classical mathematics presupposes the acceptance of classical reasoning.
		Hence the proposed characterization of undecidability, conjoined with an intuitionistic conception of proof, has the consequence that all mathematical sentences are provably not undecidable.
	www.hf.uio.no /filosofi/njpl/vol2no2/decidable/node1.html (4174 words)

	Is Fermat's last theorem undecidable?
		Fermat claimed to have a proof which the margin was 'too small to contain.' It is likely that he had a proof for the case n=4 and thought incorrectly that a similar proof could be used for all other values of n.
		In the past there was speculation Fermat's last theorem might be undecidable, and it was realised that this would mean that there couldn't be a counterexample (which would be a proof that it was false), and hence that it was true.
		Although Godel's undecidable statement was an artificial construction (and much to long to be actually written down), in the 1980's some meaningful statements in number theory were shown to be undecidable.
	www.chronon.org /articles/fermat_undecidable.html (844 words)

	Undecidable at opensource encyclopedia (Site not responding. Last check: 2007-11-07)
		In computability theory, a decision problem is undecidable if there is no algorithm that can always give the correct answer.
		If there is an algorithm that answers YES if and only if the correct answer is YES, but that may run forever when the correct answer is NO, then the problem is partially decidable.
		A formal language is said to be undecidable if the decision problem "is a given string in this language" is undecidable.
	www.wiki.tatet.com /Undecidable.html (172 words)

	collatz problem
		Because there is a difference proving a problem to be undecidable and whether it is provable or not.
		Weel, from my understanding of it, and it appears to be wrong, and I am basing this on something I read a while ago that I didn't pay attention to, he constructed various iterative formulae with convergence conjectures akin to Collatz type conjectures that were unprovable in ZFC, or something like that.
		As far as I know (and I may also be wrong) what was proven is that there are Collatz-type problems which are undecidable, but that the Collatz problem itself may or may not be; that is still an open problem.
	www.physicsforums.com /showthread.php?p=418685 (1488 words)

	Term Rewriting and Unification (Site not responding. Last check: 2007-11-07)
		A recent breakthrough result of [Schmidt-Schauss & Schulz 97] on the upper bound on the exponent of periodicity of solutions to context equations suggests that the ultimate decidability of context unification may possibly be obtained by using this reduction and a termination argument similar to Makanin's.
		Finite linear finitely terminating rewrite systems have undecidable $\exists\forall\exists$-theories of one step rewriting, in general.
		Since the full first-order theories of one-step rewriting turn out to be undecidable, in general, it is interesting to figure out decidable cases, both for specific classes of systems and particular single systems, as well as for restricted theories of bounded quantifier depth.
	www.mpi-sb.mpg.de /~sv/index-rew.html (465 words)

	XOTeam Software - XOMines Lite 1.2
		This is especially enjoyable at the start of a new game and when you face two equivalent options at the end of a large one.
		A situation is deemed undecidable whenever the information displayed by the game board does not enable the player to make a risk-free decision.
		In this case, you know that the upper left corner cell is surrounded by exactly one mine, but you still have to pick it amongst three covered cells.
	www.xoteam.com /products/XOMines_short.html (585 words)

	MLLW2 is undecidable
		Lincoln, Scedrov, and Shankar showed the undecidability of IMLL2 and IMALL2 by embedding of LJ2 (announced in this forum, LICS '95).
		Lafont has proved the undecidability of MALL2 (announced in this forum, to appear in the Journal of Symbolic Logic).
		The undecidability of some of these logics are already known: IMLL2, IMALL2 (Lincoln, Scedrov and Shankar), MALL2 (Lafont), MALL2 (Lafont and Scedrov), N-IMLL2 (Emms), N-MLL2, LC2 (Kanovich).
	www.cis.upenn.edu /~bcpierce/types/archives/1996/msg00136.html (629 words)

	Intersection of FSA and off-line parsable DCG is undecidable
		Intersection of FSA and off-line parsable DCG is undecidable
		The question whether the intersection of a FSA and an off-line parsable DCG is empty is undecidable.
		The PCP problem however is known to be undecidable.
	odur.let.rug.nl /~vannoord/papers/acl95/node4.html (514 words)

	CSC 432 Notes (Site not responding. Last check: 2007-11-07)
		So, it is quite important in proving undecidability to have an undecidable problem to start with.
		The trick with undecidability is that we begin with the assumption that the new problem is already solved.
		Consider one example of a nontrivial property: prove undecidable the problem of deciding whether or not a language is regular.
	web.presby.edu /~wasmith/courses/432/notes/Undecidability.htm (446 words)

	The Gestalt of Determinism
		In a sense, this is trivially true, because a finite Turing machine can always be written to generate any given finite string, although the algorithm necessary to generate a very irregular string may be nearly as long as the string itself.
		Since determinism is inherently undecidable, we may try to define a more tractable notion, such as predictability, in terms of the exhibited complexity manifest in our observations.
		On this basis it might seem that we could eventually assert with certainty that the universe is inherently unpredictable (on some level of experience), i.e., that the length of the shortest Turing machine required to duplicate the results grows in proportion with the number of observations.
	www.mathpages.com /rr/s9-08/9-08.htm (1168 words)

	undecidable language (Site not responding. Last check: 2007-11-07)
		Definition: A language for which the membership cannot be decided by an algorithm --- equivalently, cannot be recognized by a Turing machine that halts for all inputs.
		See also decidable language, undecidable problem, decidable problem.
		Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "undecidable language", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/undecidableLanguage.html (126 words)

	Switch Stability is Undecidable
		Second, the simple satisfaction of applying undergraduate computer science to claim that the general version of the problem that one is studying is undecidable.
		Nothing in this short snippet is non obvious, and it is a direct consequence from undecibility theory.
		A: We try and solve problems which in general are undecidable.
	klamath.stanford.edu /~sundaes/stability.html (731 words)

	Gödel's Theorem: On Formally Undecidable Propositions
		The proposition that there are undecidable problems in the system P would therefore read, for example, as follows: There exist propositional formulae a such that neither a nor the negation of a are provable formulae.
		-consistency, mere consistency as such is assumed for c, then there follows, indeed, not the existence of an undecidable proposition, but rather the existence of a property (r) for which it is possible neither to provide a counter-example nor to prove that it holds for all numbers.
		is therefore provable in P, and hence the undecidability of the one follows from that of the other, whereby Proposition IX is proved.
	www.geier.hu /GOEDEL/Godel_orig/godel3.htm (6903 words)

	Formal Computational Models and Computability (Site not responding. Last check: 2007-11-07)
		No undecidable problem can ever be solved by a computer or computer program of any kind.
		We have not said that undecidable means we don't know of a solution today but might find one tomorrow.
		Once we've seen one problem that is undecidable, it is often easy to show that other similar problems must also be undecidable.
	www.cs.rochester.edu /u/leblanc/csc173/computability/undecidable.html (1272 words)

	Polymorphic Type Systems (Site not responding. Last check: 2007-11-07)
		System Fsub, the second-order polymorphic typed lambda-calculus with subtyping [Cardelli-Wegner, 85], [Bruce-Longo, 90], [Curien-Ghelli, 92], appeared to be undecidable because of the undecidability of its subtyping component [Pierce, 92].
		Both subtyping and typing relations in the system Fsub, the well-known second-order polymorphic typed lambda-calculus with subtyping appeared to be undecidable.
		In his POPL'92 paper B.Pierce proved the undecidability of subtyping relation in Fsub, the second-order polymorphic typed lambda-calculus with subtyping, introduced by L.Cardelli and P.Wegner in 1985, and later studied and improved by many researchers.
	www.mpi-sb.mpg.de /~sv/index-pts.html (1394 words)

	Non-commutative MLL2 is undecidable (Site not responding. Last check: 2007-11-07)
		Lincoln, Scedrov, and Shankar showed the undecidability of IMLL2 and IMALL2 by embedding of LJ2 (announced in this forum, to appear in LICS '95).
		As for non-commutative linear logic, Emms shows embedding of LJ2 into N-IMLL2 as well, Kanovich proved the undecidability of Lambek calculus enriched by the exponential !, and thereby the undecidability of N-MELL.
		We follow the pattern developed in the Lafont's paper and refined in the subsequent Lafont and Scedrov's paper.
	www.seas.upenn.edu /~sweirich/types/archive/1995/msg00078.html (406 words)

	undecidable problem (Site not responding. Last check: 2007-11-07)
		Definition: A problem that cannot be solved for all cases by any algorithm whatsoever---equivalently, whose associated language cannot be recognized by a Turing machine that halts for all inputs.
		See also decidable problem, unsolvable problem, undecidable language, intractable, busy beaver.
		Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "undecidable problem", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/undecidableProblem.html (133 words)

	Alan Turing and the Enigma of Computability [encyclopedia]
		Gödel had shown that any sufficiently rich mathematical axiom system is incomplete in that there must exist propositions whose truth can never be determined (undecidable propositions within the system).
		Turing graduated in 1934 then, in the spring of 1935, he attended Max Newman's advanced course on the foundations of mathematics.
		Turing was motivated by Gödel's work to seek an algorithmic method of determining whether any given propositions were undecidable, with the ultimate goal of eliminating them from mathematics.
	www.artzia.com /History/Biography/Turing (2014 words)

	Undecidable and Unthinkable Problems
		The problem is undecidable if the language is re, but not recursive.
		A decision is not guaranteed, hence the problem is undecidable.
		the halting problem, is often undecidable, while questions surrounding the language of a tm are unthinkable.
	www.mathreference.com /lan-tm,undec.html (792 words)

	Amazon.com: Undecidable: Basic Papers on Problems Propositions Unsolvable Problems and Computable Functions: Books: ... (Site not responding. Last check: 2007-11-07)
		This anthology of fundamental papers dealing with undecidability and unsolvability begins with del's epoch-making paper of 1931.
		Already the standard reference work on the subject, The Undecidable is also ideally suited as a text or supplementary text for courses in logic, philosophy, and foundations of mathematics.
		This is a great collection of seminal papers by Goedel, Church, Turing, Rosser, Kleene, and Post on the topic of undecidability.
	www.amazon.com /exec/obidos/tg/detail/-/0911216014?v=glance (686 words)

	Notes on Universal Turing Machine
		However the works were shown then and later were included in a London retrospective (2001) as part of a hard copy show to mark the launching of the Digital Art Museum's presence on line: www.dam.org.
		By 1929 Gödel had shown that all formal systems are incomplete by their very nature so there will always be propositions that cannot be proven within any given formal system.
		However, a properly coded "undecidable" remains "undecidable" for a universal mime.
	www.verostko.com /archive/writings/utm-notes.htm (770 words)

	The Church Project: Typability is undecidable for {F}+eta
		Typability for F+eta is the problem of determining for any term M whether there is any type tau that can be given to it using the type inference rules of F+eta.
		Typability has been proven undecidable for System F (without the eta rule), but the decidability of typability has been an open problem for F+eta.
		The proof methods are similar in outline to those used to prove the undecidability of typability for System F, but the fine details differ greatly.
	www.church-project.org /reports/Wells:TUFE-1996.html (400 words)

	undecidable - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "undecidable" is defined.
		Undecidable : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include undecidable: undecidable language, undecidable set, list of statements undecidable in zfc, totally undecidable problem, undecidable figure, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=undecidable (106 words)

	Hereditary History Preserving Bisimilarity is Undecidable (Site not responding. Last check: 2007-11-07)
		We show undecidability of hereditary history preserving bisimilarity for finite asynchronous transition systems by a reduction from the halting problem of deterministic 2-counter machines.
		To make the proof more transparent we introduce an intermediate problem of checking domino bisimilarity for origin constrained tiling systems.
		We also argue that the undecidability result holds for finite 1-safe Petri nets, which can be seen as a proper subclass of finite asynchronous transition systems
	www.brics.dk /RS/99/19 (103 words)

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