DOCSLIB.ORG

Complexity Theory Lectures 1–6

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Complexity Theory Lectures 1–6 Complexity Theory 1 Complexity Theory Lectures 1–6 Lecturer: Dr. Timothy G. Griffin Slides by Anuj Dawar Computer Laboratory University of Cambridge Easter Term 2009 http://www.cl.cam.ac.uk/teaching/0809/Complexity/ Cambridge Easter 2009 Complexity Theory 2 Texts The main texts for the course are: Computational Complexity. Christos H. Papadimitriou. Introduction to the Theory of Computation. Michael Sipser. Other useful references include: Computers and Intractability: A guide to the theory of NP-completeness. Michael R. Garey and David S. Johnson. Structural Complexity. Vols I and II. J.L. Balc´azar, J. D´ıaz and J. Gabarr´o. Computability and Complexity from a Programming Perspective. Neil Jones. Cambridge Easter 2009 Complexity Theory 3 Outline A rough lecture-by-lecture guide, with relevant sections from the text by Papadimitriou (or Sipser, where marked with an S). Algorithms and problems. 1.1–1.3. • Time and space. 2.1–2.5, 2.7. • Time Complexity classes. 7.1, S7.2. • Nondeterminism. 2.7, 9.1, S7.3. • NP-completeness. 8.1–8.2, 9.2. • Graph-theoretic problems. 9.3 • Cambridge Easter 2009 Complexity Theory 4 Outline - contd. Sets, numbers and scheduling. 9.4 • coNP. 10.1–10.2. • Cryptographic complexity. 12.1–12.2. • Space Complexity 7.1, 7.3, S8.1. • Hierarchy 7.2, S9.1. • Descriptive Complexity 5.6, 5.7. • Cambridge Easter 2009 Complexity Theory 5 Complexity Theory Complexity Theory seeks to understand what makes certain problems algorithmically difficult to solve. In Data Structures and Algorithms, we saw how to measure the complexity of specific algorithms, by asymptotic measures of number of steps. In Computation Theory, we saw that certain problems were not solvable at all, algorithmically. Both of these are prerequisites for the present course. Cambridge Easter 2009 Complexity Theory 6 Algorithms and Problems Insertion Sort runs in time O(n2), while Merge Sort is an O(n log n) algorithm. The first half of this statement is short for: If we count the number of steps performed by the Insertion Sort algorithm on an input of size n, taking the largest such number, from among all inputs of that size, then the function of n so defined is eventually bounded by a constant multiple of n2. It makes sense to compare the two algorithms, because they seek to solve the same problem. But, what is the complexity of the sorting problem? Cambridge Easter 2009 Complexity Theory 7 Lower and Upper Bounds What is the running time complexity of the fastest algorithm that sorts a list? By the analysis of the Merge Sort algorithm, we know that this is no worse than O(n log n). The complexity of a particular algorithm establishes an upper bound on the complexity of the problem. To establish a lower bound, we need to show that no possible algorithm, including those as yet undreamed of, can do better. In the case of sorting, we can establish a lower bound of Ω(n log n), showing that Merge Sort is asymptotically optimal. Sorting is a rare example where known upper and lower bounds match. Cambridge Easter 2009 Complexity Theory 8 Review The complexity of an algorithm (whether measuring number of steps, or amount of memory) is usually described asymptotically: Definition For functions f : IN IN and g : IN IN, we say that: → → f = O(g), if there is an n0 IN and a constant c such that for • ∈ all n > n0, f(n) cg(n); ≤ f = Ω(g), if there is an n0 IN and a constant c such that for • ∈ all n > n0, f(n) cg(n). ≥ f = θ(g) if f = O(g) and f = Ω(g). • Usually, O is used for upper bounds and Ω for lower bounds. Cambridge Easter 2009 Complexity Theory 9 Lower Bound on Sorting An algorithm A sorting a list of n distinct numbers a1,...,an. ai < aj ? ap < aq ? ak < al? . ar < a.s? . done done done done done To work for all permutations of the input list, the tree must have at least n! leaves and therefore height at least log2(n!) = θ(n log n). Cambridge Easter 2009 Complexity Theory 10 Travelling Salesman Given V — a set of nodes. • c : V V IN — a cost matrix. • × → Find an ordering v1,...,vn of V for which the total cost: n−1 c(v , v )+ c(v , v ) n 1 X i i+1 i=1 is the smallest possible. Cambridge Easter 2009 Complexity Theory 11 Complexity of TSP Obvious algorithm: Try all possible orderings of V and find the one with lowest cost. The worst case running time is θ(n!). Lower bound: An analysis like that for sorting shows a lower bound of Ω(n log n). Upper bound: The currently fastest known algorithm has a running time of O(n22n). Between these two is the chasm of our ignorance. Cambridge Easter 2009 Complexity Theory 12 Formalising Algorithms To prove a lower bound on the complexity of a problem, rather than a specific algorithm, we need to prove a statement about all algorithms for solving it. In order to prove facts about all algorithms, we need a mathematically precise definition of algorithm. We will use the Turing machine. The simplicity of the Turing machine means it’s not useful for actually expressing algorithms, but very well suited for proofs about all algorithms. Cambridge Easter 2009 Complexity Theory 13 Turing Machines For our purposes, a Turing Machine consists of: K — a finite set of states; • Σ — a finite set of symbols, including . • ⊔ s K — an initial state; • ∈ δ :(K Σ) (K a, r ) Σ L,R,S • × → ∪ { } × × { } A transition function that specifies, for each state and symbol a next state (or accept acc or reject rej), a symbol to overwrite the current symbol, and a direction for the tape head to move (L – left, R – right, or S - stationary) Cambridge Easter 2009 Complexity Theory 14 Configurations A complete description of the configuration of a machine can be given if we know what state it is in, what are the contents of its tape, and what is the position of its head. This can be summed up in a simple triple: Definition A configuration is a triple (q,w,u), where q K and w,u Σ⋆ ∈ ∈ The intuition is that (q,w,u) represents a machine in state q with the string wu on its tape, and the head pointing at the last symbol in w. The configuration of a machine completely determines the future behaviour of the machine. Cambridge Easter 2009 Complexity Theory 15 Computations Given a machine M =(K, Σ,s,δ) we say that a configuration (q,w,u) yields in one step (q′, w′,u′), written ′ ′ ′ (q,w,u) M (q , w ,u ) → if w = va ; • δ(q, a)=(q′,b,D); and • either D = L and w′ = v u′ = bu • or D = S and w′ = vb and u′ = u or D = R and w′ = vbc and u′ = x, where u = cx. If u is empty, then w′ = vb and u′ is empty. ⊔ Cambridge Easter 2009 Complexity Theory 16 Computations ⋆ The relation is the reflexive and transitive closure of M . →M → A sequence of configurations c1,...,cn, where for each i, ci M ci+1, is called a computation of M. → The language L(M) Σ⋆ accepted by the machine M is the set of ⊆ strings x (s, ⊲, x) ⋆ (acc,w,u) for some w and u { | →M } A machine M is said to halt on input x if for some w and u, either (s, ⊲, x) ⋆ (acc,w,u) or (s, ⊲, x) ⋆ (rej,w,u) →M →M Cambridge Easter 2009 Complexity Theory 17 Decidability A language L Σ⋆ is recursively enumerable if it is L(M) for some ⊆ M. A language L is decidable if it is L(M) for some machine M which halts on every input. A language L is semi-decidable if it is recursively enumerable. A function f : Σ⋆ Σ⋆ is computable, if there is a machine M, → such that for all x, (s, ⊲, x) ⋆ (acc, f(x), ε) →M Cambridge Easter 2009 Complexity Theory 18 Example Consider the machine with δ given by: ⊲ 0 1 ⊔ s s,⊲,R s, 0,R s, 1,R q, , L ⊔ q acc,⊲,R q, , L rej, ,R q, , L ⊔ ⊔ ⊔ This machine will accept any string that contains only 0s before the first blank (but only after replacing them all by blanks). Cambridge Easter 2009 Complexity Theory 19 Multi-Tape Machines The formalisation of Turing machines extends in a natural way to multi-tape machines. For instance a machine with k tapes is specified by: K, Σ, s; and • δ :(K Σk) K a, r (Σ L,R,S )k • × → ∪ { }× × { } Similarly, a configuration is of the form: (q, w1,u1,...,wk,uk) Cambridge Easter 2009 Complexity Theory 20 Complexity For any function f : IN IN, we say that a language L is in → TIME(f(n)) if there is a machine M =(K, Σ,s,δ), such that: L = L(M); and • The running time of M is O(f(n)). • Similarly, we define SPACE(f(n)) to be the languages accepted by a machine which uses O(f(n)) tape cells on inputs of length n. In defining space complexity, we assume a machine M, which has a read-only input tape, and a separate work tape. We only count cells on the work tape towards the complexity. Cambridge Easter 2009 Complexity Theory 21 Nondeterminism If, in the definition of a Turing machine, we relax the condition on δ being a function and instead allow an arbitrary relation, we obtain a nondeterministic Turing machine.
Load more

Recommended publications
  • Database Theory
    DATABASE THEORY Lecture 4: Complexity of FO Query Answering Markus Krotzsch¨ TU Dresden, 21 April 2016 Overview 1. Introduction | Relational data model 2. First-order queries 3. Complexity of query answering 4. Complexity of FO query answering 5. Conjunctive queries 6. Tree-like conjunctive queries 7. Query optimisation 8. Conjunctive Query Optimisation / First-Order Expressiveness 9. First-Order Expressiveness / Introduction to Datalog 10. Expressive Power and Complexity of Datalog 11. Optimisation and Evaluation of Datalog 12. Evaluation of Datalog (2) 13. Graph Databases and Path Queries 14. Outlook: database theory in practice See course homepage [) link] for more information and materials Markus Krötzsch, 21 April 2016 Database Theory slide 2 of 41 How to Measure Query Answering Complexity Query answering as decision problem { consider Boolean queries Various notions of complexity: • Combined complexity (complexity w.r.t. size of query and database instance) • Data complexity (worst case complexity for any fixed query) • Query complexity (worst case complexity for any fixed database instance) Various common complexity classes: L ⊆ NL ⊆ P ⊆ NP ⊆ PSpace ⊆ ExpTime Markus Krötzsch, 21 April 2016 Database Theory slide 3 of 41 An Algorithm for Evaluating FO Queries function Eval(', I) 01 switch (') f I 02 case p(c1, ::: , cn): return hc1, ::: , cni 2 p 03 case : : return :Eval( , I) 04 case 1 ^ 2 : return Eval( 1, I) ^ Eval( 2, I) 05 case 9x. : 06 for c 2 ∆I f 07 if Eval( [x 7! c], I) then return true 08 g 09 return false 10 g Markus Krötzsch, 21 April 2016 Database Theory slide 4 of 41 FO Algorithm Worst-Case Runtime Let m be the size of ', and let n = jIj (total table sizes) • How many recursive calls of Eval are there? { one per subexpression: at most m • Maximum depth of recursion? { bounded by total number of calls: at most m • Maximum number of iterations of for loop? { j∆Ij ≤ n per recursion level { at most nm iterations I • Checking hc1, ::: , cni 2 p can be done in linear time w.r.t.
    [Show full text]
  • CS601 DTIME and DSPACE Lecture 5 Time and Space Functions: T, S
    CS601 DTIME and DSPACE Lecture 5 Time and Space functions: t, s : N → N+ Definition 5.1 A set A ⊆ U is in DTIME[t(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M) ≡ w ∈ U M(w)=1 , and 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps. Definition 5.2 A set A ⊆ U is in DSPACE[s(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M), and 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells. (Input tape is “read-only” and not counted as space used.) Example: PALINDROMES ∈ DTIME[n], DSPACE[n]. In fact, PALINDROMES ∈ DSPACE[log n]. [Exercise] 1 CS601 F(DTIME) and F(DSPACE) Lecture 5 Definition 5.3 f : U → U is in F (DTIME[t(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. Definition 5.4 f : U → U is in F (DSPACE[s(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. (Input tape is “read-only”; Output tape is “write-only”.
    [Show full text]
  • Interactive Proof Systems and Alternating Time-Space Complexity
    Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement.
    [Show full text]
  • On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs*
    On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs* Benny Applebaum† Eyal Golombek* Abstract We study the randomness complexity of interactive proofs and zero-knowledge proofs. In particular, we ask whether it is possible to reduce the randomness complexity, R, of the verifier to be comparable with the number of bits, CV , that the verifier sends during the interaction. We show that such randomness sparsification is possible in several settings. Specifically, unconditional sparsification can be obtained in the non-uniform setting (where the verifier is modelled as a circuit), and in the uniform setting where the parties have access to a (reusable) common-random-string (CRS). We further show that constant-round uniform protocols can be sparsified without a CRS under a plausible worst-case complexity-theoretic assumption that was used previously in the context of derandomization. All the above sparsification results preserve statistical-zero knowledge provided that this property holds against a cheating verifier. We further show that randomness sparsification can be applied to honest-verifier statistical zero-knowledge (HVSZK) proofs at the expense of increasing the communica- tion from the prover by R−F bits, or, in the case of honest-verifier perfect zero-knowledge (HVPZK) by slowing down the simulation by a factor of 2R−F . Here F is a new measure of accessible bit complexity of an HVZK proof system that ranges from 0 to R, where a maximal grade of R is achieved when zero- knowledge holds against a “semi-malicious” verifier that maliciously selects its random tape and then plays honestly.
    [Show full text]
  • NP-Completeness: Reductions Tue, Nov 21, 2017
    CMSC 451 Dave Mount CMSC 451: Lecture 19 NP-Completeness: Reductions Tue, Nov 21, 2017 Reading: Chapt. 8 in KT and Chapt. 8 in DPV. Some of the reductions discussed here are not in either text. Recap: We have introduced a number of concepts on the way to defining NP-completeness: Decision Problems/Language recognition: are problems for which the answer is either yes or no. These can also be thought of as language recognition problems, assuming that the input has been encoded as a string. For example: HC = fG j G has a Hamiltonian cycleg MST = f(G; c) j G has a MST of cost at most cg: P: is the class of all decision problems which can be solved in polynomial time. While MST 2 P, we do not know whether HC 2 P (but we suspect not). Certificate: is a piece of evidence that allows us to verify in polynomial time that a string is in a given language. For example, the language HC above, a certificate could be a sequence of vertices along the cycle. (If the string is not in the language, the certificate can be anything.) NP: is defined to be the class of all languages that can be verified in polynomial time. (Formally, it stands for Nondeterministic Polynomial time.) Clearly, P ⊆ NP. It is widely believed that P 6= NP. To define NP-completeness, we need to introduce the concept of a reduction. Reductions: The class of NP-complete problems consists of a set of decision problems (languages) (a subset of the class NP) that no one knows how to solve efficiently, but if there were a polynomial time solution for even a single NP-complete problem, then every problem in NP would be solvable in polynomial time.
    [Show full text]
  • Deterministic Execution of Multithreaded Applications
    DETERMINISTIC EXECUTION OF MULTITHREADED APPLICATIONS FOR RELIABILITY OF MULTICORE SYSTEMS DETERMINISTIC EXECUTION OF MULTITHREADED APPLICATIONS FOR RELIABILITY OF MULTICORE SYSTEMS Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op vrijdag 19 juni 2015 om 10:00 uur door Hamid MUSHTAQ Master of Science in Embedded Systems geboren te Karachi, Pakistan Dit proefschrift is goedgekeurd door de promotor: Prof. dr. K. L. M. Bertels Copromotor: Dr. Z. Al-Ars Samenstelling promotiecommissie: Rector Magnificus, voorzitter Prof. dr. K. L. M. Bertels, Technische Universiteit Delft, promotor Dr. Z. Al-Ars, Technische Universiteit Delft, copromotor Independent members: Prof. dr. H. Sips, Technische Universiteit Delft Prof. dr. N. H. G. Baken, Technische Universiteit Delft Prof. dr. T. Basten, Technische Universiteit Eindhoven, Netherlands Prof. dr. L. J. M Rothkrantz, Netherlands Defence Academy Prof. dr. D. N. Pnevmatikatos, Technical University of Crete, Greece Keywords: Multicore, Fault Tolerance, Reliability, Deterministic Execution, WCET The work in this thesis was supported by Artemis through the SMECY project (grant 100230). Cover image: An Artist’s impression of a view of Saturn from its moon Titan. The image was taken from http://www.istockphoto.com and used with permission. ISBN 978-94-6186-487-1 Copyright © 2015 by H. Mushtaq All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means without the prior written permission of the copyright owner.
    [Show full text]
  • Week 1: an Overview of Circuit Complexity 1 Welcome 2
    Topics in Circuit Complexity (CS354, Fall’11) Week 1: An Overview of Circuit Complexity Lecture Notes for 9/27 and 9/29 Ryan Williams 1 Welcome The area of circuit complexity has a long history, starting in the 1940’s. It is full of open problems and frontiers that seem insurmountable, yet the literature on circuit complexity is fairly large. There is much that we do know, although it is scattered across several textbooks and academic papers. I think now is a good time to look again at circuit complexity with fresh eyes, and try to see what can be done. 2 Preliminaries An n-bit Boolean function has domain f0; 1gn and co-domain f0; 1g. At a high level, the basic question asked in circuit complexity is: given a collection of “simple functions” and a target Boolean function f, how efficiently can f be computed (on all inputs) using the simple functions? Of course, efficiency can be measured in many ways. The most natural measure is that of the “size” of computation: how many copies of these simple functions are necessary to compute f? Let B be a set of Boolean functions, which we call a basis set. The fan-in of a function g 2 B is the number of inputs that g takes. (Typical choices are fan-in 2, or unbounded fan-in, meaning that g can take any number of inputs.) We define a circuit C with n inputs and size s over a basis B, as follows. C consists of a directed acyclic graph (DAG) of s + n + 2 nodes, with n sources and one sink (the sth node in some fixed topological order on the nodes).
    [Show full text]
  • Computational Complexity: a Modern Approach
    i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist.
    [Show full text]
  • Sustained Space Complexity
    Sustained Space Complexity Jo¨elAlwen1;3, Jeremiah Blocki2, and Krzysztof Pietrzak1 1 IST Austria 2 Purdue University 3 Wickr Inc. Abstract. Memory-hard functions (MHF) are functions whose eval- uation cost is dominated by memory cost. MHFs are egalitarian, in the sense that evaluating them on dedicated hardware (like FPGAs or ASICs) is not much cheaper than on off-the-shelf hardware (like x86 CPUs). MHFs have interesting cryptographic applications, most notably to password hashing and securing blockchains. Alwen and Serbinenko [STOC'15] define the cumulative memory com- plexity (cmc) of a function as the sum (over all time-steps) of the amount of memory required to compute the function. They advocate that a good MHF must have high cmc. Unlike previous notions, cmc takes into ac- count that dedicated hardware might exploit amortization and paral- lelism. Still, cmc has been critizised as insufficient, as it fails to capture possible time-memory trade-offs; as memory cost doesn't scale linearly, functions with the same cmc could still have very different actual hard- ware cost. In this work we address this problem, and introduce the notion of sustained- memory complexity, which requires that any algorithm evaluating the function must use a large amount of memory for many steps. We con- struct functions (in the parallel random oracle model) whose sustained- memory complexity is almost optimal: our function can be evaluated using n steps and O(n= log(n)) memory, in each step making one query to the (fixed-input length) random oracle, while any algorithm that can make arbitrary many parallel queries to the random oracle, still needs Ω(n= log(n)) memory for Ω(n) steps.
    [Show full text]
  • The Complexity Zoo
    The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/.
    [Show full text]
  • Notes on Space Complexity of Integration of Computable Real
    Notes on space complexity of integration of computable real functions in Ko–Friedman model Sergey V. Yakhontov Abstract x In the present paper it is shown that real function g(x)= 0 f(t)dt is a linear-space computable real function on interval [0, 1] if f is a linear-space computable C2[0, 1] real function on interval R [0, 1], and this result does not depend on any open question in the computational complexity theory. The time complexity of computable real functions and integration of computable real functions is considered in the context of Ko–Friedman model which is based on the notion of Cauchy functions computable by Turing machines. 1 2 In addition, a real computable function f is given such that 0 f ∈ FDSPACE(n )C[a,b] but 1 f∈ / FP if FP 6= #P. 0 C[a,b] R RKeywords: Computable real functions, Cauchy function representation, polynomial-time com- putable real functions, linear-space computable real functions, C2[0, 1] real functions, integration of computable real functions. Contents 1 Introduction 1 1.1 CF computablerealnumbersandfunctions . ...... 2 1.2 Integration of FP computablerealfunctions. 2 2 Upper bound of the time complexity of integration 3 2 3 Function from FDSPACE(n )C[a,b] that not in FPC[a,b] if FP 6= #P 4 4 Conclusion 4 arXiv:1408.2364v3 [cs.CC] 17 Nov 2014 1 Introduction In the present paper, we consider computable real numbers and functions that are represented by Cauchy functions computable by Turing machines [1]. Main results regarding computable real numbers and functions can be found in [1–4]; main results regarding computational complexity of computations on Turing machines can be found in [5].
    [Show full text]
  • CUDA Toolkit 4.2 CURAND Guide
    CUDA Toolkit 4.2 CURAND Guide PG-05328-041_v01 | March 2012 Published by NVIDIA Corporation 2701 San Tomas Expressway Santa Clara, CA 95050 Notice ALL NVIDIA DESIGN SPECIFICATIONS, REFERENCE BOARDS, FILES, DRAWINGS, DIAGNOSTICS, LISTS, AND OTHER DOCUMENTS (TOGETHER AND SEPARATELY, "MATERIALS") ARE BEING PROVIDED "AS IS". NVIDIA MAKES NO WARRANTIES, EXPRESSED, IMPLIED, STATUTORY, OR OTHERWISE WITH RESPECT TO THE MATERIALS, AND EXPRESSLY DISCLAIMS ALL IMPLIED WARRANTIES OF NONINFRINGEMENT, MERCHANTABILITY, AND FITNESS FOR A PARTICULAR PURPOSE. Information furnished is believed to be accurate and reliable. However, NVIDIA Corporation assumes no responsibility for the consequences of use of such information or for any infringement of patents or other rights of third parties that may result from its use. No license is granted by implication or otherwise under any patent or patent rights of NVIDIA Corporation. Specifications mentioned in this publication are subject to change without notice. This publication supersedes and replaces all information previously supplied. NVIDIA Corporation products are not authorized for use as critical components in life support devices or systems without express written approval of NVIDIA Corporation. Trademarks NVIDIA, CUDA, and the NVIDIA logo are trademarks or registered trademarks of NVIDIA Corporation in the United States and other countries. Other company and product names may be trademarks of the respective companies with which they are associated. Copyright Copyright ©2005-2012 by NVIDIA Corporation. All rights reserved. CUDA Toolkit 4.2 CURAND Guide PG-05328-041_v01 | 1 Portions of the MTGP32 (Mersenne Twister for GPU) library routines are subject to the following copyright: Copyright ©2009, 2010 Mutsuo Saito, Makoto Matsumoto and Hiroshima University.
    [Show full text]