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Topic: Speedup theorem

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	Computational complexity theory
		In computational complexity theory Blum's speedup theorem, first stated by Manuel Blum in 1967, is an important theorem about the complexity of computable functions.
		In computational complexity theory, Cook's theorem, proved by Stephen Cook in his 1971 paper "The Complexity of Theorem Proving Procedures", states that the Boolean satisfiability problem is NP-complete.
		Fagin's theorem is a result in descriptive complexity theory which states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP.
	www.shortopedia.com /C/O/Computational_complexity_theory (1256 words)

	Linear speedup theorem - Wikipedia, the free encyclopedia
		In computational complexity theory, the linear speedup theorem for Turing machines proves that given any c > 0 and any Turing machine solving a problem in time f(n), there is another machine that solves the same problem in time cf(n).
		It should be easy to see how to generalize the proof to all values of c, and also that the proof applies as much to space as it does to time.
		The linear speedup theorem is the reason why complexity theory ignores linear factors and represents the complexity of algorithms with big O notation.
	en.wikipedia.org /wiki/Linear_speedup_theorem (320 words)

	CS 662: Speedup
		On many algorithms the fraction of the operations that must be done sequentially is not a constant f, but a function, f(N), of the size of the input of the algorithm.
		You would expect less than linear speedup since with only one machine, no messages are sent over the ethernet, they are just communicated via sockets.
		Superlinear speedup is commonly attributable to caching effects.
	www.eli.sdsu.edu /courses/spring96/cs662/notes/speedup/speedup.html (1196 words)

	CS267: Notes for Lecture 4, Jan 26, 1995
		A "speedup plot" is Speedup(p) plotted versus p.
		Speedup and efficiency are the most common ways to report the performance of a parallel algorithm
		For example, even if only s=1% of a program is serial, the speedup is limited to 100, and so it is not worth using a machine with more than 100 processors.
	www.cs.berkeley.edu /~demmel/cs267-1995/lecture04.html (1613 words)

	NOTES ON PARALLEL ALGORITHMS (Site not responding. Last check: 2007-10-20)
		Compare this to the sequential approach: we would take n-1 steps to sum all the numbers, as opposed to 3 -- which is lg 8, and not by accident: each parallel step halves the number of results, so in lg n steps we are down to 1 result.
		The speedup for addition of 2p = 8 numbers as above will be 7/3 = 2.333.
		But if we buy one million processors, and add two million numbers, the speedup is (roughly) 1,000,000/21, which is about 50,000 times faster than the sequential method.
	www.cs.umd.edu /class/spring2000/cmsc251/notes/par.html (476 words)

	Type theory: Andrews
		Most current research on automated theorem proving is concerned with proving theorems of first-order logic, the basic features of which (connectives and quantifiers) occur in most formal languages.
		Any proof of this theorem in axiomatic set theory requires using the axioms of set theory many hundreds of times, so the problem of automatically deriving the theorem from these axioms is much more complex than the corresponding problem in type theory.
		A related result by Statman ~\cite{Statman78}) establishes that the minimal length of a proof in first-order logic of a theorem of first-order logic may be extraordinarily longer than that the minimal length of a proof of the same theorem in second-order logic.
	www.cs.duke.edu /NSFwkshopAD/contributions/wkshop/andrewsE.html (615 words)

	Highbeam Encyclopedia - Search Results for speedup
		A Dictionary of Computing; 1/1/2004; JOHN DAINTITH; 100 words; speedup theorem A theorem in complexity theory that, like the gap theorem, can be expressed in terms of abstract complexity measures...
		The group, consisting of textile, fiber and apparel trade associations and labor unions, warned that a speedup in the MTN cuts would lead to adverse effects for industries already hard hit by competitive imports.
		Roundup: Iran raps European delay, pursues speedup in nuclear talks, by Chen Wendi.
	www.encyclopedia.com /SearchResults.aspx?Q=speedup (975 words)

	Jason Hickey and Aleksey Nogin: Fast tactic-based theorem proving. (Site not responding. Last check: 2007-10-20)
		Theorem provers for higher-order logics often use tactics to implement automated proof search.
		Our speedup is due to efficient data structures and modularity, which allows parts of the prover to be customized on a domain-specific basis.
		MetaPRL logical framework, with speedups of more than two orders of magnitude over traditional tactic-based proof search.
	nogin.org /papers/fast_proving.html (142 words)

	JP Lewis - Homepage
		We extend an active appearance model to employ local, positive-only reconstruction, resulting in surprisingly good extrapolation.
		Describes a frequency-domain algorithm for normalized cross-correlation (the convolution theorem gives un-normalized cross correlation).
		10-15x speedup) and has been used extensively in subsequent ILM projects.
	www.idiom.com /~zilla (1092 words)

	Fast Tactic-Based Theorem Proving, by Jason Hickey and Aleksey Nogin (Site not responding. Last check: 2007-10-20)
		Theorem provers for higher-order logics often use tactics to implement automated proof search.
		Our speedup is due to efficient data structures and modularity, which allows parts of the prover to be customized on a domain-specific basis.
		Our architecture is used in the MetaPRL logical framework, with speedups of more than two orders of magnitude over traditional tactic-based proof search.
	www.nuprl.org /documents/hickey/fast_proving.html (130 words)

	Computational Complexity: Favorite Theorems: Abstract Complexity
		Trakhtenbrot independently proved the gap theorem for abstract complexity measures.
		McCreight and Meyer also give an honesty theorem showing that computable t there is (in a weak sense) a time-constructible t' such that languages computable with resource bound t are equal to languages computable with resource bound t'.
		After the P versus NP problem was popularized by Cook and Karp in 1971, the focus of complexity went to polynomial-time (which also was machine independent) and away from abstract complexity.
	weblog.fortnow.com /2005/08/favorite-theorems-abstract-complexity.html (442 words)

	RSA Security - Appendix
		A speedup of 1.8 to 2.0 has been achieved in practice.
		In Step 2, one hopes that the order is smooth up to B1 except for a single prime factor lying between B1 and B2 for a suitably chosen value of B2.
		The second step has the effect of about an order-of-magnitude speedup in practice, although it does not affect the asymptotic complexity of the method.
	www.rsasecurity.com /rsalabs/node.asp?id=2849 (1632 words)

	hobbit - Performance of Compiled Code
		The author has so far compiled and tested a number of large programs (theorem provers for various logics and hobbit itself).
		The speedup for the provers was between 25 and 40 times for various provable formulas.
		T. Moore has reported a 16-fold speedup for a large gate-level IC optimizer.
	www-swiss.ai.mit.edu /~jaffer/hobbit_4.html (900 words)

	URCS Theory Technical Reports
		We eliminate some special cases from the proofs of two theorems in which a machine instantiating a many-query reduction to a p-selective set is made to use only one query.
		Since the Kaemper-AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of results---ranging from the study of unique solutions to issues of approximation---our results implicitly strengthen all those results.
		We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard, and we prove that every P-constructibly semi-switching counting property of circuits is PP-hard.
	www.cs.rochester.edu /trs/theory-trs.html (16130 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		He is no doubt aware that his statement "It is like trying to say something about the property of rational numbers but formulating a theorem that has the reals as its sample space.
		Date: Mon, 27 Feb 95 8:04:55 PST Subject: Super-Linear Speedup when Parallelizing GAs It is well known that genetic algorithms are amenable to parallelization in several different ways.
		From time to time, I have heard it mentioned that GAs may deliver SUPER-LINEAR SPEEDUP in the sense that the parallel GA solves (at least for some problems) faster when semi-isolated sub-populations (Sewell Wright's demes) are used.
	www.aic.nrl.navy.mil /galist/digests/v9n11 (2128 words)

	Amazon.com: Computability, Complexity, and Languages, Second Edition : Fundamentals of Theoretical Computer Science ... (Site not responding. Last check: 2007-10-20)
		Predicate calculus is then discussed, and Herbrand's theorem, which effectively reduces logical inference in predicate calculus to a problem of satisfiability of universal sentences, is proven.
		This theorem is fascinating and has important applications to automated theorem proving, as it ties together semantic and syntactical properties of a formal system.
		This is followed by a detailed discussion of the speedup theorem, which essentially states that there is a wildly complicated recursive function such that for any program computing this function, there exists another program computing the function that works a lot faster for almost every input.
	www.amazon.com /exec/obidos/tg/detail/-/0122063821?v=glance (2370 words)

	parallel
		[We usually have in mind ``good'' algorithms, i.e., ones that run as fast as possible in the worst-case, both for t(n) and for T_p(n), although in practice we may not know precisely what those are.] For instance, in the above summing example, the speedup is (n-1)/lg n, where p = n/2.
		So a parallel machine with p processors can never be more than p times faster than a suitable sequential machine solving the same problem.
		We define the EFFICIENCY of a parallel algorithm as the speedup divided by the number of processors.
	www.cs.umd.edu /class/spring2003/cmsc351/notes/parallel (1977 words)

	CS 611, v3.0: Exams Archives
		Be comfortable with the s-m-n theorem, the recursion theorem and Rice's theorem.
		For the proof project, you need to turn in revised definitions, revised theorem statements and scratchwork for your proof.
		Then it proves some useful theorems, like Rice's theorem, the SMN theorem and the recursion theorem, then it uses those theorems in proofs of other theorems.
	lal.cs.byu.edu /cs611/archives/exams (1226 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Theorem: A combined input- and output-queued switch with a speedup of 4 operating under the MUCF algorithm exactly matches cells with a FIFO output-queued switch.
		The theorem is proven by showing that when S>= 4, the same is true for input threads.
		More work may be done in exploring what happens at speedups between 1 and 2 which is exactly what most commercial switches use.
	theory.stanford.edu /~nmishra/CS361-2002/lecture19.2-scribe.doc (1270 words)

	Nat' Academies Press, Probability and Algorithms (1992)
		Please use the page image as the authoritative form to ensure accuracy.
		BPP that are not in P. Yao (1982) announced a converse result suggesting that at most a subexponential speedup is possible:
		Analysis of various primality testing algorithms led to study of the complexity classes described in §4.2.
	www.nap.edu /books/0309047765/html/42.html (517 words)

	A New Cosmogony [Free Republic] (Site not responding. Last check: 2007-10-20)
		What the Speed Up Theorem tells us is that if Finite Nature is true, and if the thing in Other was competent, then there is no way for it to get the answer any faster than by letting the Universe run its course.
		He even proved several theorems about Omega...which later turned out to be virtually identical with things the theologians had discovered about God.
		Cantor proved a similar theorem about Omega: that it cannot be defined by appeal to characteristics of any lesser order of infinity.
	www.freerepublic.com /forum/a39fb9923213c.htm (5454 words)

	Claus Bauer - Publications (Site not responding. Last check: 2007-10-20)
		Packet scheduling in input - queued switches with a speedup of less than two and scheduling algorithms for switches with a configuration overhead.
		Hua's theorem for five almost equal prime squares.
		Claus Bauer, Hua's theorem on sums of five prime squares in arithmetic progressions.
	www.clausbauer.com /pub.html (781 words)

	CS 8751 Spring 2003 Sample Exam 2 Questions (Site not responding. Last check: 2007-10-20)
		Discuss two examples showing how Bayes theorem can be used to justify approaches to learning.
		Also, discuss an example of a learning method based on Bayes theorem.
		Give two examples of ways to speedup this algorithm.
	www.d.umn.edu /~rmaclin/cs8751/spring2003/exam2-sample.html (550 words)

	Introduction
		In [7] Slaney and Lusk described a parallel algorithm for computing the closure of a set under an operation and presented several application areas.
		In this paper we describe the application to automated theorem proving, which can be viewed as the computation of the closure of a set of clauses under a set of inference rules.
		[5], currently the fastest sequential theorem proving system for large problems.
	www-fp.mcs.anl.gov /~lusk/papers/roo/node1.html (349 words)

	Parallel algorithms (Site not responding. Last check: 2007-10-20)
		According to the main Theorem of parallel computing, the maximal speedup which is possible to obtain using N processors over the best sequantial algorithm using 1 processor is N { and in our case when we use N/log(N) processors - the maximal speedup is N/log(N) }
		Further: there is a theorem in the computer science that the best sequantial sorting algorithm has a complexity of N*log(N).
		Let us now assume that there is a parallel sorting program that uses N/log(N) processors which runs with a complexity log(N).
	www.pa.uky.edu /~sorokin/stuff/cs685S/algoritm2.html (395 words)

	PHYS771 Lecture 5: Paleocomplexity
		This is a consequence of a fundamental result called the Time Hierarchy Theorem, which was proven by Hartmanis and Stearns in the mid-1960's and later rewarded with a Turing Award.
		(T-Rex might've been a dinosaur, but it still had pretty sharp teeth!) In this case, a 1967 result called the Blum Speedup Theorem says that there really are problems that admit no fastest algorithm.
		Our goal is to define a function f, from integers to {0,1}, such that if f can be computed in O(t(n)) steps, then it can also be computed in O(t(n-i)) steps for any positive integer i.
	www.scottaaronson.com /democritus/lec5.html (2554 words)

	Five big questions with pretty simple answers
		This is somewhat similar to the situation regarding mathematical proofs: There are an arbitrarily large number of correct proofs for every correct theorem, but we prefer those that are most concise and elegant.
		A variant of Noether's theorem states that for every microscopically conserved discrete quantity, there is an asymptotically continuous symmetry.
		Because the CA underlying DM is regular and simple despite being computation-universal, it is not surprising that there are mathematical shortcuts beyond conservation laws that partially escape from the dictates of the speedup theorem and computational irreducibility.
	www.research.ibm.com /journal/rd/481/fredkin.html (10873 words)

	Combinatorial Representations of Partial Information Classes and their Truth-Table Closures (Site not responding. Last check: 2007-10-20)
		As partial information classes can be represented purely combinatorially by so-called families, all of the different notions of partial information can be compared by comparing families.
		In the first part of the thesis, the combinatorial properties of families are studied and as a consequence several new results on partial information classes are obtained, including a generalisation of the Generalised Non-Speedup Theorem.
		In the second part, the stability and relative difficulty of partial information classes are analysed using truth-table reduction closures.
	www.eccc.uni-trier.de /eccc-local/ECCC-Theses/tantau.html (260 words)

	Outline of Material for Test #1 (Site not responding. Last check: 2007-10-20)
		If a parallel algorithm consists of a fully parallel part and a fully serial part such that the fraction of instructions that are fully serial is given by f, then the speedup for this parallel algorithm can be given by:
		Be able to use Brent's theorem to estimate the run time of arbitrary control circuits.
		Be able to explain why we can simulate a PRAM model on fewer processors by using virtual processors.
	www.erc.msstate.edu /~lush/fl2000/cs4163/test1_outline.html (686 words)

	Complexity course outline Spring 1999 (Site not responding. Last check: 2007-10-20)
		Time and Space Hierarchy theorems, Gap theorem (w/o proof), relations between deterministic and nondeterministic complexity classes (incomplete);
		relations between deterministic and nondeterministic complexity classes, Savitch's theorem; the configuration graph, the reachability method, Immerman's theorem.
		Bin Packing is Strongly NP-complete (redcution from 3DM); 3-Partition (w/o proof); Interval Sequencing: Pseudo polynomial reduction from Bin Packing Subgraph Isomorphism is NPC; Subforest isomorphism: pseudo-polynomial reduction from bin packing.
	www.math.tau.ac.il /~rshamir/complexity/99/outline.html (386 words)

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