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Topic: Space hierarchy theorem

	Time hierarchy theorem - Wikipedia, the free encyclopedia
		In computational complexity theory, the time hierarchy theorems are important statements that ensure the existence of certain "hard" problems which cannot be solved in a given amount of time.
		The analogous theorems for space are the space hierarchy theorems.
		However, the time hierarchy theorems provide no means to relate deterministic and non-deterministic complexity, or time and space complexity, so they cast no light on the great unsolved questions of complexity theory: whether P and NP, NP and PSPACE, PSPACE and EXPTIME, or EXPTIME and NEXPTIME are equal or not.
	en.wikipedia.org /wiki/Time_hierarchy_theorem (1015 words)

	[No title]
		Space and time complexity classes Deterministic and nondeterministic space and time are defined traditionally for DTMs and NDTMs using proper resource functions.
		Space Hierarchy Theorem: If r(n) and s(n) are prope and r(n) = o(s(n)), then SPACE(r(n)) is strictly contained in SPACE(s(n)).
		Theorem: P contained in NP contained in EXPTIME contained in NEXPTIME but P is strictly contained in EXPTIME.
	www.cs.brown.edu /courses/cs159/lect.04.txt (520 words)

	Time [Internet Encyclopedia of Philosophy]
		Space and time are not material substances, he added, but are like substances, in not being dependent on matter or motions or anything else except God.
		One says that both space and time emerge from some micro-substrate, although there is no agreed upon theory of what the substrate is. The second camp says that space emerges but time does not.
		Ascending another level up the hierarchy of multiverses, if there are different possible values for the physical constants and for the kinds of elementary particles in our universe, then there must be parallel universes far from us which have all those possible values.
	www.iep.utm.edu /t/time.htm (16669 words)

	Xah: The Shapes of Space
		A plane or sphere is a homogeneous space since the geometry is everywhere the same, while a torus or monkey saddle surface is not.
		Among these 2d spaces, there's the idea of orientability, which is a intriguing property: that it is possible to move about in the space and become mirror image.
		Steve Smale proved a theorem in 1958 that means it is possible, to the amazement of fellow mathematicians.
	xahlee.org /Periodic_dosage_dir/20031225_shape_space.html (2619 words)

	Comparing the Space Efficiency of PKR Formalisms (Site not responding. Last check: 2007-10-23)
		Namely, the next theorem shows how to infer the existence of theorem-preserving reductions, while the other one gives a way to prove that there is no theorem-preserving reduction from one formalism to another one.
		Theorems 4-7 show that compilability classes characterize very precisely the relative capability of PKR formalisms to represent sets of models or sets of theorems.
		For example, as a consequence of Theorems 3 and 7 there is no poly-size reduction from PL to the syntactic restriction of PL allowing only Horn clauses that preserves the theorems, unless the polynomial hierarchy collapses.
	www.cs.cmu.edu /afs/cs/project/jair/pub/volume13/cadoli00a-html/node9.html (774 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		By the Space Hierarchy theorem, Space(log^2 n) is properly contained in Space(n).
		Time Hierarchy Theorem: For any time constructible function t(n), there is a language in O(Time(t(n))) which is not in Time(o(t(n)/log t(n))).
		The proof idea is the same as in the case of space hierarchy from last time: we define a "diagonal" language that cannot be decided by any TM running in time o(t(n)/log t(n)).
	www.cs.sfu.ca /~kabanets/cmpt308/lectures/21.txt (546 words)

	Self-Organizing Systems FAQ for Usenet newsgroup comp.theory.self-org-sys
		The larger area of state space that leads to an attractor is called its basin of attraction and comprises all the pre-images of the attractor state.
		In the world of possible systems (the state space for the system) two possibilities are neighbours if a change or mutation to one parameter can change the first system into the second or vice versa.
		The mutation space of a system with 2 alleles at each node is a Boolean Hypercube of dimension N (number of neighbours).
	www.calresco.org /sos/sosfaq.htm (8398 words)

	Computational Complexity 2003 (Site not responding. Last check: 2007-10-23)
		We will present the PCP theorem in detail, and show that that certain functions that were earlier believed to be hard to compute exactly are equally hard even to estimate approximately.
		Nisan's theorem: BPLog is a subset of ZP*Log.
		The Saks-Shiyu theorem: RL is subset of DSpace((log n)^1.5).
	www.tcs.tifr.res.in /~jaikumar/Courses/Complexity (272 words)

	Research
		A dichotomy theorem for turbulence (or the latex file, or the ps file).
		There is a closely connected partition theorem for atomic models, stating that if the structure has non-trivial automorphism group then we can find a partition of some 2-type into two sets both of which are met in any elementary submodel.
		The analogue of Ladner's theorem is unprovable in ZFC for infinite time Turing machines (or the postscript version).
	www.math.ucla.edu /~greg/research.html (1658 words)

	Boaz Barak's Home Page
		Unlike other complexity measures such as deterministic and nondeterministic time and space, and non-uniform size, it is not known whether probabilistic time has a strict hierarchy.
		In this work we prove that a hierarchy theorem does hold for ``slightly non-uniform'' probabilistic machines.
		We also discuss conditions under which a hierarchy theorem can be proven for fully uniform Turing machines.
	www.math.ias.edu /~boaz/Papers/bptime.html (186 words)

	Wikinfo \| List of mathematical proofs (Site not responding. Last check: 2007-10-23)
		2 Articles devoted to theorems of which a (sketch of a) proof is given
		Theorems of which articles are exclusively devoted to proving them
		Articles devoted to theorems of which a (sketch of a) proof is given
	www.wikinfo.org /wiki.php?title=List_of_mathematical_proofs (232 words)

	CS154 Course Notes (Site not responding. Last check: 2007-10-23)
		Goedel's incompleteness theorem and the class of arithmetical languages.
		This also proves that NL is included in P. I sketched a proof of the Space Hierarchy Theorem, roughly saying that all space complexity classes that could be different are different.
		Covered material on complexity theory: Sipser Chapters 7 (all of it), 8 (except for the proof of Theorem 8.22) and 9.1 (except for the proof of Theorem 9.10, and theorem 9.15), as well as the handout on complexity classes.
	kilby.stanford.edu /~rvg/154-2001/notes.html (2216 words)

	Gödel's shock
		By asserting the existence of complex infinite sets one can indirectly define levels in the hierarchy of mathematical truth that are difficult to approach in other ways.
		In Chapters 3 and 4 we describe the structure of the mathematical hierarchy of self reflecting structures and possible approaches to extending it.
		We argue that there are more powerful approaches to exploring mathematical truth that have no need to transcend the limits of Gödel's theorem with mystical intuition.
	www.mtnmath.com /whatt72h/node11.html (227 words)

	Manifesto for the Reputation Society (Site not responding. Last check: 2007-10-23)
		From a society–wide point of view, the goal would be to move from a negative–sum arms race in which advertisers must compete just to play in the commercial world — wasting consumer time and advertiser dollars — to a positive–sum experience which provides effective quality and relevance information for the advertising dollars that are spent.
		Space for insightful yet unpopular points of view may ironically be provided by the same polarizing phenomenon that discounts minority views, since easy polarization implies that differing points of view can easily be filtered out and ignored.
		In the vector space model, documents are still represented as lists of words, but the number of times each word occurs is also stored — so the index would have an entry similar to "In this document, ‘a’ occurs 127 times, ‘aardvark’ occurs 1 time,...".
	www.firstmonday.dk /issues/issue9_7/masum (17985 words)

	The Complexity World below Logarithmic Space (ResearchIndex) (Site not responding. Last check: 2007-10-23)
		However, machines with such little space may still be quite powerful.
		10 Lower bounds for the space complexity of context free recogn..
		4 A lower bound for the nondeterministic space complexity of c..
	citeseer.ist.psu.edu /34203.html (611 words)

	Review of Stephen G. Simpson: Subsystems of Second Order Arithmetic
		Thence, if a theorem has been proven from the right (set existence) axioms, the axioms themselves can be proven from the theorem.
		Hence the theorem and the axioms used to prove it are equivalent (over a weak base theory).
		Bolzano-Weierstraß, every sequence in a compact metric space has a convergent subsequence, the Ascoli-lemma, every countable commutative ring has a maximal ideal, every countable vector space over a countable field has a basis, full König's Lemma, and a Ramsey theorem for colorings.
	www.math.psu.edu /simpson/sosoa/moellerfeld (1238 words)

	Paul C. Kainen at Georgetown University
		It is shown that for subsets A and B of a metric space X, A x B (the cartesian product) is approximatively compact (ac) when A is ac and B is compact.
		It is also shown that A + B is ac when A is ac and B is compact, where A and B are subsets of an F-space and so of a normed linear space, and A + B {a + b: a in A and b in B}.
		Papy16 (with Vera Kurkova and Marcello Sanguineti, in a normed linear space, the error functional of a compact subset is well-posed in the generalized Hadamard sense, when restricted to an approx.
	www.georgetown.edu /faculty/kainen/homepage.html (682 words)

	[No title]
		Find the time and space complexity of the RAM.
		Linear Speed-Up Theorem A T(n) time-bounded Turing machine M
		A LOOP(1) program is a LOOP program in which no nesting of do's is allowed.
	www.cse.ohio-state.edu /~gurari/theory-bk/theory-bk-fiveli1.html (773 words)

	CBofN - Glossary - C
		Chaos/Chaotic Irregular motion of a dynamical system that is deterministic, sensitive to initial conditions, and impossible to predict in the long term with anything less than an infinite and perfect representation of analog values.
		Chomsky Hierarchy Four classes of languages (or computing machines) that have increasing complexity: regular (finite-state automata), context-free (push-down automata), context-sensitive (linear bounded automata), and recursive (Turing machines).
		Because of the vast numbers involved, explicit search an entire search space is not always possible.
	mitpress.mit.edu /books/FLAOH/cbnhtml/glossary-C.html (829 words)

	The shock of Gödel's proof
		In Chapters 5 and 6 we describe the structure of the mathematical hierarchy of self reflecting structures and possible approaches to extending it.
		Chapters 5 and 6 develop in an intuitive and semi-formal way the basics of formal set theory.
		Section 5.8 contains a sketch of a proof of a limited version of Gödel's Incompleteness Theorem called the Halting Problem.
	www.mtnmath.com /whatrh/node13.html (194 words)

	ACO - Typical Program of Study (Site not responding. Last check: 2007-10-23)
		The max-flow min-cut theorem and the associated algorithm, Hoffman's circulation theorem, Hu's 2-commodity flow theorem
		Max-flow Min-cut theorem, Menger's theorem, the structure of 1-, 2-, 3- connected graphs (blocks, ear-decomposition, contractible edges, Tutte's synthesis of 3-connected graphs)
		Ramsey's theorem for graphs, upper and lower bounds, Ramsey's theorem for k-tuples
	www.math.gatech.edu /aco/syllabi.html (488 words)

	Computer Science @ UC Davis \| Course Descriptions (Site not responding. Last check: 2007-10-23)
		To provide students with a grounding in the theory of computation, and to help them develop the skills necessary to do and understanding more complex proofs in theoretical computer science.
		First order logic, completeness, second-order logic, undecidability and incompleteness, the recursion theorem.
		Complexity classes, the time and space hierarchy, Savitch's theorem, reductions, completeness.
	www.cs.ucdavis.edu /courses/exp_course_desc/220.html (106 words)

	MFCS 2001 - Contributed Talks
		Finally, we focus on cellular automata and prove a dichotomy theorem: continuous cellular automata are either equivalent to the identity or to a constant one.
		The main contribution to the well-known {\em Space Hierarchy Theorem} is that {\em (i)}~the language~$\cal L$ separating the two space classes is unary (tally), {\em (ii)}~the hierarchy is independent of whether $h(n)$ or~$\ell(n)$ are in $\Omega(\log n)$ or in $o(\log n)$, {\em (iii)}~the functions $h(n)$ or~$\ell(n)$ themselves need not be space constructible nor monotone increasing.
		Among others we will show a hierarchy theorem of $c$-mc real numbers that, for any constants $c_2>c_1\ge 1$, there is a $c_2$-mc real number which is not $c_1$-mc and that there is an $\omega$-mc real number which is not $c$-mc for any $c\in\IR$.
	www.math.cas.cz /~mfcs2001/abstrakty.html (5520 words)

	Complexity Zoo - Qwiki
		Proving a stronger time hierarchy theorem for BPTIME is a longstanding open problem; see [BH97] for details.
		The Space Hierarchy Theorem: For constructible f(n) greater than log n, DSPACE(f(n)) is strictly contained in DSPACE(f(n) log(f(n))) [HLS65].
		The Time Hierarchy Theorem: For constructible f(n) greater than n, DTIME(f(n)) is strictly contained in DTIME(f(n) log(f(n)) loglog(f(n))) [HS65].
	qwiki.caltech.edu /index.php?title=Complexity_Zoo&printable=yes (6509 words)

	CS172 Computability and Complexity
		These models of computation are thoroughly understood and there are some elegant equivalences and theorems that can be shown.However, as we will see, the models in this section are a bit too restrictive to accurately capture a modern notion of what can and can’t be done with computers.
		This arises if the time (or space) required is so large as to be intractable.
		Complexity theory gives us a handle on this by analyzing how the time (or space) required by a problem grows with the size of the input.
	www-inst.eecs.berkeley.edu /~cs172/sp05 (803 words)

	The Math Forum - Math Library - Order/Lattices
		A lattice is an infinite arrangement of points spaced with sufficient regularity that one can shift any point onto any other point by some symmetry of the arrangement.
		More formally, a lattice can be defined as a discrete subgroup of a finite-dimensional vector space (the subgroup is often required not to lie within any subspace of the vector space, which can be expressed formally by saying that the quotient of the space by the lattice is compact).
		The goals of the presentation are to explain why Polya's theorem is true and to develop techniques for applying Rayleigh's method.
	mathforum.org /library/topics/lattices (1129 words)

	Developing for Developers
		Moreover, a theorem of linear algebra tells us precisely how many input numbers we need to guarantee that a square can be found: as long as we have more columns than rows, the null space is guaranteed to be nontrivial, so that we have a nonzero solution.
		This still requires a fair amount of space; it's common to use block algorithms that work on small portions of the matrix at one time, storing the rest of the matrix on disk.
		We set a number of roughly evenly spaced bits, where the first hash establishes the initial index, the second hash establishes the spacing, and the spacing is slightly increased as we go along.
	blogs.msdn.com /devdev (16531 words)

	Transactions of the American Mathematical Society
		Our methods show that the Perron-Frobenius theorem is ``really'' about the boundedness of invariant subsets in the Hilbert projective metric.
		Spectral theorem for convex monotone homogeneous maps and ergodic control.
		Linear operators leaving invariant a cone in a Banach space.
	www.ams.org /tran/2004-356-12/S0002-9947-04-03470-1/home.html (384 words)

	CMSC 652 --- Complexity Theory (Site not responding. Last check: 2007-10-23)
		Everyone is encouraged to attend the distinguished lecture by Cynthia Dwork, being held in CSIC 1115 from 4-5.
		The relation of BPP to P/poly and the polynomial hierarchy.
		The PCP theorem: The theorem, a sketch of a proof, and applications
	www.cs.umd.edu /~jkatz/complexity (820 words)

	COMP 531
		Quick recap of Turing Machines, Strong Church--Turing hypothesis, time and space complexity measures for TMs, simulation of multi-tape TM by single-tape TM, linear speed up for time and space.
		Another story also discussing the subsequent developments including the PCP Theorem and complete with pictures of the cast of characters.
		A course on the PCP theorem and hardness of approximation
	www.cs.mcgill.ca /~navin/531/531.html (683 words)

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