SIGACT News Complexity Theory Column 52
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Time Hypothesis on a Quantum Computer the Implications Of
The implications of breaking the strong exponential time hypothesis on a quantum computer Jorg Van Renterghem Student number: 01410124 Supervisor: Prof. Andris Ambainis Master's dissertation submitted in order to obtain the academic degree of Master of Science in de informatica Academic year 2018-2019 i Samenvatting In recent onderzoek worden reducties van de sterke exponenti¨eletijd hypothese (SETH) gebruikt om ondergrenzen op de complexiteit van problemen te bewij- zen [51]. Dit zorgt voor een interessante onderzoeksopportuniteit omdat SETH kan worden weerlegd in het kwantum computationeel model door gebruik te maken van Grover's zoekalgoritme [7]. In de klassieke context is SETH wel nog geldig. We hebben dus een groep van problemen waarvoor er een klassieke onder- grens bekend is, maar waarvoor geen ondergrens bestaat in het kwantum com- putationeel model. Dit cre¨eerthet potentieel om kwantum algoritmes te vinden die sneller zijn dan het best mogelijke klassieke algoritme. In deze thesis beschrijven we dergelijke algoritmen. Hierbij maken we gebruik van Grover's zoekalgoritme om een voordeel te halen ten opzichte van klassieke algoritmen. Grover's zoekalgoritme lost het volgende probleem op in O(pN) queries: gegeven een input x ; :::; x 0; 1 gespecifieerd door een zwarte doos 1 N 2 f g die queries beantwoordt, zoek een i zodat xi = 1 [32]. We beschrijven een kwantum algoritme voor k-Orthogonale vectoren, Graaf diameter, Dichtste paar in een d-Hamming ruimte, Alle paren maximale stroom, Enkele bron bereikbaarheid telling, 2 sterke componenten, Geconnecteerde deel- graaf en S; T -bereikbaarheid. We geven ook nieuwe ondergrenzen door gebruik te maken van reducties en de sensitiviteit methode. -
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/. -
Input-Oblivious Proof Systems and a Uniform Complexity Perspective on P/Poly
Electronic Colloquium on Computational Complexity, Report No. 23 (2011) Input-Oblivious Proof Systems and a Uniform Complexity Perspective on P/poly Oded Goldreich∗ and Or Meir† Department of Computer Science Weizmann Institute of Science Rehovot, Israel. February 16, 2011 Abstract We initiate a study of input-oblivious proof systems, and present a few preliminary results regarding such systems. Our results offer a perspective on the intersection of the non-uniform complexity class P/poly with uniform complexity classes such as NP and IP. In particular, we provide a uniform complexity formulation of the conjecture N P 6⊂ P/poly and a uniform com- plexity characterization of the class IP∩P/poly. These (and similar) results offer a perspective on the attempt to prove circuit lower bounds for complexity classes such as NP, PSPACE, EXP, and NEXP. Keywords: NP, IP, PCP, ZK, P/poly, MA, BPP, RP, E, NE, EXP, NEXP. Contents 1 Introduction 1 1.1 The case of NP ....................................... 1 1.2 Connection to circuit lower bounds ............................ 2 1.3 Organization and a piece of notation ........................... 3 2 Input-Oblivious NP-Proof Systems (ONP) 3 3 Input-Oblivious Interactive Proof Systems (OIP) 5 4 Input-Oblivious Versions of PCP and ZK 6 4.1 Input-Oblivious PCP .................................... 6 4.2 Input-Oblivious ZK ..................................... 7 Bibliography 10 ∗Partially supported by the Israel Science Foundation (grant No. 1041/08). †Research supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. ISSN 1433-8092 1 Introduction Various types of proof systems play a central role in the theory of computation. -
Notes on Hierarchy Theorems 1 Proof of the Space
U.C. Berkeley | CS172: Automata, Computability and Complexity Handout 8 Professor Luca Trevisan 4/21/2015 Notes on Hierarchy Theorems These notes discuss the proofs of the time and space hierarchy theorems. It shows why one has to work with the rather counter-intuitive diagonal problem defined in Sipser's book, and it describes an alternative proof of the time hierarchy theorem. When we say \Turing machine" we mean one-tape Turing machine if we discuss time. If we discuss space, then we mean a Turing machine with one read-only tape and one work tape. All the languages discussed in these notes are over the alphabet Σ = f0; 1g. Definition 1 (Constructibility) A function t : N ! N is time constructible if there is a Turing machine M that given an input of length n runs in O(t(n)) time and writes t(n) in binary on the tape. A function s : N ! N is space constructible if there is a Turing machine M that given an input of length n uses O(s(n)) space and writes s(n) on the (work) tape. Theorem 2 (Space Hierarchy Theorem) If s(n) ≥ log n is a space-constructible function then SPACE(o(s(n)) 6= SPACE(O(s(n)). Theorem 3 (Time Hierarchy Theorem) If t(n) is a time-constructible function such that t(n) ≥ n and n2 = o(t(n)), then TIME(o(t(n)) 6= TIME(O(t(n) log t(n))). 1 Proof of The Space Hierarchy Theorem Let s : N ! N be a space constructible function such that s(n) ≥ log n. -
The Structure of Logarithmic Advice Complexity Classes
The Structure of Logarithmic Advice Complexity Classes Jose L Balcazar Department of Software LSI Universitat Politecnicade Catalunya Pau Gargallo E Barcelona Spain balquilsiupces Montserrat Hermo Department of Software LSI Universidad del Pas Vasco PO Box E San Sebastian Spain jiphehumsiehues May Abstract A nonuniform class called here FullPlog due to Ko is studied It corresp onds to p olynomial time with logarithmically long advice Its imp ortance lies in the struc tural prop erties it enjoys more interesting than those of the alternative class Plog sp ecically its introduction was motivated by the need of a logarithmic advice class closed under p olynomialtime deterministic reductions Several characterizations of FullPlog are shown formulated in terms of various sorts of tally sets with very small information content A study of its inner structure is presented by considering the most usual reducibilitie s and lo oking for the relationships among the corresp onding reduction and equivalence classes dened from these sp ecial tally sets Partially supp orted by the EU through the ESPRIT Long Term Research Pro ject ALCOM IT and through the HCM Network CHRXCT COLORET by the Spanish DGICYT through pro ject PB KOALA and by Acciones Integradas HispanoAlemanas HAB and ALB Introduction Nonuniform complexity classes were essentially introduced in where the main rela tionships to uniform classes were already shown In order to capture characteristics of nonuniform mo dels of computation in which xed input lengths are compulsory in the denition -
Notes for Lecture Notes 2 1 Computational Problems
Stanford University | CS254: Computational Complexity Notes 2 Luca Trevisan January 11, 2012 Notes for Lecture Notes 2 In this lecture we define NP, we state the P versus NP problem, we prove that its formulation for search problems is equivalent to its formulation for decision problems, and we prove the time hierarchy theorem, which remains essentially the only known theorem which allows to show that certain problems are not in P. 1 Computational Problems In a computational problem, we are given an input that, without loss of generality, we assume to be encoded over the alphabet f0; 1g, and we want to return as output a solution satisfying some property: a computational problem is then described by the property that the output has to satisfy given the input. In this course we will deal with four types of computational problems: decision problems, search problems, optimization problems, and counting problems.1 For the moment, we will discuss decision and search problem. In a decision problem, given an input x 2 f0; 1g∗, we are required to give a YES/NO answer. That is, in a decision problem we are only asked to verify whether the input satisfies a certain property. An example of decision problem is the 3-coloring problem: given an undirected graph, determine whether there is a way to assign a \color" chosen from f1; 2; 3g to each vertex in such a way that no two adjacent vertices have the same color. A convenient way to specify a decision problem is to give the set L ⊆ f0; 1g∗ of inputs for which the answer is YES. -
1 Time Hierarchy Theorem
princeton university cos 522: computational complexity Lecture 3: Diagonalization Lecturer: Sanjeev Arora Scribe:scribename To separate two complexity classes we need to exhibit a machine in one class that is different (namely, gives a different answer on some input) from every machine in the other class. This lecture describes diagonalization, essentially the only general technique known for constructing such a machine. We also indicate why this technique has been unable thus far to resolve P =? NP and other interesting questions. 1 Time Hierarchy Theorem The Time Hierarchy Theorem shows that allowing Turing Machines more com- putation time strictly increases the class of languages that they can decide. Theorem 1 If f,g are running times satisfying f(n) log f(n)=o(g(n)), then DTIME(f(n)) DTIME(g(n)) (1) To showcase the essential idea of the proof of Theorem 1, we prove the sim- pler statement DTIME(n) DTIME(n2). We use diagonalization. Suppose M1,M2,M3,... is a numbering of all Turing Machines, where the description of Mi can be produced from i in time O(log i). (Such numberings exist. For example, one can order TMs according to the number of states in their tran- sition diagrams, and use lexicographic ordering among then all TMs that have the same number of states. Note that we allow machines that do not halt on all inputs.) Consider the following Turing Machine, D:“On input x,ifx =0j1k for 1.5 some j, k then construct Mk and simulate it on x for |x| steps. If Mk halts and accepts, reject. -
Non-Uniform Computation
Non-Uniform Computation Lecture 10 Non-Uniform Computational Models: Circuits 1 Non-Uniform Computation 2 Non-Uniform Computation Uniform: Same program for all (the infinitely many) inputs 2 Non-Uniform Computation Uniform: Same program for all (the infinitely many) inputs Non-uniform: A different “program” for each input size 2 Non-Uniform Computation Uniform: Same program for all (the infinitely many) inputs Non-uniform: A different “program” for each input size Then complexity of building the program and executing the program 2 Non-Uniform Computation Uniform: Same program for all (the infinitely many) inputs Non-uniform: A different “program” for each input size Then complexity of building the program and executing the program Sometimes will focus on the latter alone 2 Non-Uniform Computation Uniform: Same program for all (the infinitely many) inputs Non-uniform: A different “program” for each input size Then complexity of building the program and executing the program Sometimes will focus on the latter alone Not entirely realistic if the program family is uncomputable or very complex to compute 2 Non-uniform advice 3 Non-uniform advice Program: TM M and advice strings {An} 3 Non-uniform advice Program: TM M and advice strings {An} M given A|x| along with x 3 Non-uniform advice Program: TM M and advice strings {An} M given A|x| along with x An can be the program for inputs of size n 3 Non-uniform advice Program: TM M and advice strings {An} M given A|x| along with x An can be the program for inputs of size n n |An|=2 is sufficient 3 Non-uniform advice Program: TM M and advice strings {An} M given A|x| along with x An can be the program for inputs of size n n |An|=2 is sufficient But {An} can be uncomputable (even if just one bit long) 3 Non-uniform advice Program: TM M and advice strings {An} M given A|x| along with x An can be the program for inputs of size n n |An|=2 is sufficient But {An} can be uncomputable (even if just one bit long) e.g. -
Circuit Complexity Contents 1 Announcements 2 Relation to Other
CS221: Computational Complexity Prof. Salil Vadhan Lecture 19: Circuit Complexity 11/6 Scribe: Chris Crick Contents 1 Announcements ² Problem Set 3, problem 2: You may use ², the regular expression for the empty string. ² PS3 due tomorrow. ² Be aware that you do not necessarily have to solve all of the problems on the problem set — the point is to get you to think about them. ² Reading: Papadimitriou x4.3, x11.4 2 Relation to other complexity measures Definition: The circuit complexity or circuit size of a function f : f0; 1gn ! f0; 1g is the number of gates in the smallest boolean circuit that computes f. Circuit complexity depends our choice of a universal basis, but only up to a constant. We can therefore use any such basis to represent our circuit. Because it’s familiar and convenient, then, we will usually use AND, OR and NOT, though sometimes the set of all boolean functions of two inputs and one output is more natural. Example: To provide a sense for how one programs with circuits, we will examine the thresh- 1 if at least two x s are 1 old function Th (x ; ¢ ¢ ¢ ; x ) = f i g. Figure 1 shows a simple circuit to 2 1 n 0 otherwise compute this function. In this circuit, li = 1 if at least one of (x1; : : : ; xi) = 1. ri becomes 1 once two xis equal 1. li is just xi _ li¡1, and ri is ri¡1 _ (li¡1 ^ xi). The output node is rn. Conventions for counting circuit gates vary a little – for example, some people count the inputs in the total, while others don’t. -
Linear Advice for Randomized Logarithmic Space
Linear Advice for Randomized Logarithmic Space Lance Fortnow1 and Adam R. Klivans2? 1 Department of Computer Science University of Chicago Chicago, IL 60637 [email protected] 2 Department of Computer Science The University of Texas at Austin Austin, TX 78712 [email protected] Abstract. We show that RL ⊆ L/O(n), i.e., any language computable in randomized logarithmic space can be computed in deterministic log- arithmic space with a linear amount of non-uniform advice. To prove our result we use an ultra-low space walk on the Gabber-Galil expander graph due to Gutfreund and Viola. 1 Introduction The question of whether RL, randomized logarithmic space, can be simulated in L, deterministic logarithmic space, remains a central challenge in complexity- theory. The best known deterministic simulation of randomized logarithmic space is due to Saks and Zhou [16] who, building on seminal work due to Nisan [13], proved that BPL ⊆ L3/2. Recently in a breakthrough result, Rein- gold [14] proved that the s-t connectivity problem on undirected graphs could be solved in L; this implies SL, the symmetric analogue of NL, equals L. The possibility of extending his techniques to prove RL = L has subsequently been investigated by Reingold, Trevisan and Vadhan [15]. 1.1 Randomness and Non-Uniformity The relationship between randomness and non-uniformity is a topic of funda- mental importance in complexity theory. Derandomizing complexity classes fre- quently involves a consideration of the smallest non-uniform complexity class or circuit family which contains a particular randomized class. Adleman’s well known result on BPP [1], for example, shows that BPP can be computed by polynomial-size circuits. -
18.405J S16 Lecture 3: Circuits and Karp-Lipton
18.405J/6.841J: Advanced Complexity Theory Spring 2016 Lecture 3: Circuits and Karp-Lipton Scribe: Anonymous Student Prof. Dana Moshkovitz Scribe Date: Fall 2012 1 Overview In the last lecture we examined relativization. We found that a common method (diagonalization) for proving non-inclusions between classes of problems was, alone, insufficient to settle the P versus NP problem. This was due to the existence of two orcales: one for which P = NP and another for which P =6 NP . Therefore, in our search for a solution, we must look into the inner workings of computation. In this lecture we will examine the role of Boolean Circuits in complexity theory. We will define the class P=poly based on the circuit model of computation. We will go on to give a weak circuit lower bound and prove the Karp-Lipton Theorem. 2 Boolean Circuits Turing Machines are, strange as it may sound, a relatively high level programming language. Because we found that diagonalization alone is not sufficient to settle the P versus NP problem, we want to look at a lower level model of computation. It may, for example, be easier to work at the bit level. To this end, we examine boolean circuits. Note that any function f : f0; 1gn ! f0; 1g is expressible as a boolean circuit. 2.1 Definitions Definition 1. Let T : N ! N.A T -size circuit family is a sequence fCng such that for all n, Cn is a boolean circuit of size (in, say, gates) at most T (n). SIZE(T ) is the class of languages with T -size circuits. -
Complexity Classes
CHAPTER Complexity Classes In an ideal world, each computational problem would be classified at least approximately by its use of computational resources. Unfortunately, our ability to so classify some important prob- lems is limited. We must be content to show that such problems fall into general complexity classes, such as the polynomial-time problems P, problems whose running time on a determin- istic Turing machine is a polynomial in the length of its input, or NP, the polynomial-time problems on nondeterministic Turing machines. Many complexity classes contain “complete problems,” problems that are hardest in the class. If the complexity of one complete problem is known, that of all complete problems is known. Thus, it is very useful to know that a problem is complete for a particular complexity class. For example, the class of NP-complete problems, the hardest problems in NP, contains many hundreds of important combinatorial problems such as the Traveling Salesperson Prob- lem. It is known that each NP-complete problem can be solved in time exponential in the size of the problem, but it is not known whether they can be solved in polynomial time. Whether ? P and NP are equal or not is known as the P = NP question. Decades of research have been devoted to this question without success. As a consequence, knowing that a problem is NP- complete is good evidence that it is an exponential-time problem. On the other hand, if one such problem were shown to be in P, all such problems would be been shown to be in P,a result that would be most important.