Advanced Topics in Complexity Theory: Exercise
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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. -
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/. -
LSPACE VS NP IS AS HARD AS P VS NP 1. Introduction in Complexity
LSPACE VS NP IS AS HARD AS P VS NP FRANK VEGA Abstract. The P versus NP problem is a major unsolved problem in com- puter science. This consists in knowing the answer of the following question: Is P equal to NP? Another major complexity classes are LSPACE, PSPACE, ESPACE, E and EXP. Whether LSPACE = P is a fundamental question that it is as important as it is unresolved. We show if P = NP, then LSPACE = NP. Consequently, if LSPACE is not equal to NP, then P is not equal to NP. According to Lance Fortnow, it seems that LSPACE versus NP is easier to be proven. However, with this proof we show this problem is as hard as P versus NP. Moreover, we prove the complexity class P is not equal to PSPACE as a direct consequence of this result. Furthermore, we demonstrate if PSPACE is not equal to EXP, then P is not equal to NP. In addition, if E = ESPACE, then P is not equal to NP. 1. Introduction In complexity theory, a function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem [6]. A functional problem F is defined as a binary relation (x; y) 2 R over strings of an arbitrary alphabet Σ: R ⊂ Σ∗ × Σ∗: A Turing machine M solves F if for every input x such that there exists a y satisfying (x; y) 2 R, M produces one such y, that is M(x) = y [6]. -
The Classes FNP and TFNP
Outline The classes FNP and TFNP C. Wilson1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function Problems What are Function Problems? Function Problems FSAT Defined Total Functions TSP Defined Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function -
The Computational Complexity Column by V
The Computational Complexity Column by V. Arvind Institute of Mathematical Sciences, CIT Campus, Taramani Chennai 600113, India [email protected] http://www.imsc.res.in/~arvind The mid-1960’s to the early 1980’s marked the early epoch in the field of com- putational complexity. Several ideas and research directions which shaped the field at that time originated in recursive function theory. Some of these made a lasting impact and still have interesting open questions. The notion of lowness for complexity classes is a prominent example. Uwe Schöning was the first to study lowness for complexity classes and proved some striking results. The present article by Johannes Köbler and Jacobo Torán, written on the occasion of Uwe Schöning’s soon-to-come 60th birthday, is a nice introductory survey on this intriguing topic. It explains how lowness gives a unifying perspective on many complexity results, especially about problems that are of intermediate complexity. The survey also points to the several questions that still remain open. Lowness results: the next generation Johannes Köbler Humboldt Universität zu Berlin [email protected] Jacobo Torán Universität Ulm [email protected] Abstract Our colleague and friend Uwe Schöning, who has helped to shape the area of Complexity Theory in many decisive ways is turning 60 this year. As a little birthday present we survey in this column some of the newer results related with the concept of lowness, an idea that Uwe translated from the area of Recursion Theory in the early eighties. Originally this concept was applied to the complexity classes in the polynomial time hierarchy. -
Logic and Complexity
o| f\ Richard Lassaigne and Michel de Rougemont Logic and Complexity Springer Contents Introduction Part 1. Basic model theory and computability 3 Chapter 1. Prepositional logic 5 1.1. Propositional language 5 1.1.1. Construction of formulas 5 1.1.2. Proof by induction 7 1.1.3. Decomposition of a formula 7 1.2. Semantics 9 1.2.1. Tautologies. Equivalent formulas 10 1.2.2. Logical consequence 11 1.2.3. Value of a formula and substitution 11 1.2.4. Complete systems of connectives 15 1.3. Normal forms 15 1.3.1. Disjunctive and conjunctive normal forms 15 1.3.2. Functions associated to formulas 16 1.3.3. Transformation methods 17 1.3.4. Clausal form 19 1.3.5. OBDD: Ordered Binary Decision Diagrams 20 1.4. Exercises 23 Chapter 2. Deduction systems 25 2.1. Examples of tableaux 25 2.2. Tableaux method 27 2.2.1. Trees 28 2.2.2. Construction of tableaux 29 2.2.3. Development and closure 30 2.3. Completeness theorem 31 2.3.1. Provable formulas 31 2.3.2. Soundness 31 2.3.3. Completeness 32 2.4. Natural deduction 33 2.5. Compactness theorem 36 2.6. Exercices 38 vi CONTENTS Chapter 3. First-order logic 41 3.1. First-order languages 41 3.1.1. Construction of terms 42 3.1.2. Construction of formulas 43 3.1.3. Free and bound variables 44 3.2. Semantics 45 3.2.1. Structures and languages 45 3.2.2. Structures and satisfaction of formulas 46 3.2.3. -
Physics 212B, Ken Intriligator Lecture 6, Jan 29, 2018 • Parity P Acts As P
Physics 212b, Ken Intriligator lecture 6, Jan 29, 2018 Parity P acts as P = P † = P −1 with P~vP = ~v and P~aP = ~a for vectors (e.g. ~x • − and ~p) and axial pseudo-vectors (e.g. L~ and B~ ) respectively. Note that it is not a rotation. Parity preserves all the basic commutation relations, but it is not necessarily a symmetry of the Hamiltonian. The names scalar and pseudo scalar are often used to denote parity even vs odd: P P = for “scalar” operators (like ~x2 or L~ 2, and P ′P = ′ for pseudo O O O −O scalar operators (like S~ ~x). · Nature respects parity aside from some tiny effects at the fundamental level. E.g. • in electricity and magnetism, we can preserve Maxwell’s equations via V~ ~ V and → − V~ +V~ . axial → axial In QM, parity is represented by a unitary operator such that P ~r = ~r and • | i | − i P 2 = 1. So the eigenvectors of P have eigenvalue 1. A state is even or odd under parity ± if P ψ = ψ ; in position space, ψ( ~x) = ψ(~x). In radial coordinates, parity takes | i ±| i − ± cos θ cos θ and φ φ + π, so P α,ℓ,m =( 1)ℓ α,ℓ,m . This is clear also from the → − → | i − | i ℓ = 1 case, any getting general ℓ from tensor products. If [H,P ] = 0, then the non-degenerate energy eigenstates can be written as P eigen- states. Parity in 1d: PxP = x and P pP = p. The Hamiltonian respects parity if V ( x) = − − − V (x). -
Constructive Galois Connections
ZU064-05-FPR main 23 July 2018 12:36 1 Constructive Galois Connections DAVID DARAIS University of Vermont, USA (e-mail: [email protected]) DAVID VAN HORN University of Maryland, College Park, USA (e-mail: [email protected]) Abstract Galois connections are a foundational tool for structuring abstraction in semantics and their use lies at the heart of the theory of abstract interpretation. Yet, mechanization of Galois connections using proof assistants remains limited to restricted modes of use, preventing their general application in mechanized metatheory and certified programming. This paper presents constructive Galois connections, a variant of Galois connections that is effective both on paper and in proof assistants; is complete with respect toa large subset of classical Galois connections; and enables more general reasoning principles, including the “calculational” style advocated by Cousot. To design constructive Galois connections we identify a restricted mode of use of classical ones which is both general and amenable to mechanization in dependently-typed functional programming languages. Crucial to our metatheory is the addition of monadic structure to Galois connections to control a “specification effect.” Effectful calculations may reason classically, while pure calculations have extractable computational content. Explicitly moving between the worlds of specification and implementation is enabled by our metatheory. To validate our approach, we provide two case studies in mechanizing existing proofs from the literature: the first uses calculational abstract interpretation to design a static analyzer; the second forms a semantic basis for gradual typing. Both mechanized proofs closely follow their original paper-and-pencil counterparts, employ reasoning principles arXiv:1807.08711v1 [cs.PL] 23 Jul 2018 not captured by previous mechanization approaches, support the extraction of verified algorithms, and are novel. -
The Relative Complexity of NP Search Problems
The Relative Complexity of NP Search Problems PAUL BEAME* STEPHEN COOKT JEFF EDMONDS$ Computer Science and Engineering Computer Science Dept. I. C.S.I. University of Washington University of Toronto Berkeley, CA 94704- ”1198 beame@cs. Washington. edu sacook@cs. toronto. edu edmonds@icsi .berke,ley.edu RUSSELL IMPAGLIAZZO TONIANN PITASSI$ Computer Science Dept. Mathematics and Computer Science UCSD University of Pittsburgh russelltlcs .ucsd.edu toniflcs .pitt, edu Abstract might be that of finding a 3-coloring of the graph if it ex- ists. One can always reduce a search problem to a related Papadimitriou introduced several classes of NP search prob- decision problem and, as in the reduction of 3-coloring to lemsbased on combinatorial principles which guarantee the 3-colorability, thk is often by a natural self-reduction which existence of solutions to the problems. Many interesting produces a polynomially equivalent decision problem. search problems not known to be solvable in polynomial However, it may also happen that the related decision time are contained in these classes, and a number of them problem is not computationally equivalent to the original are complete problems. We consider the question of the rel- search problem. Thk is particularly important in the case ative complexity of these search problem classes. We prove when a solution is guaranteed to exist for the search prob- several separations which show that in a generic relativized lem. For example, consider the following prc)blems: world, the search classes are distinct and there is a standard search problem in each of them that is not computation- 1.Given a list al ,. -
Counting Below #P: Classes, Problems and Descriptive Complexity
Msc Thesis Counting below #P : Classes, problems and descriptive complexity Angeliki Chalki µΠλ8 Graduate Program in Logic, Algorithms and Computation National and Kapodistrian University of Athens Supervised by Aris Pagourtzis Advisory Committee Dimitris Fotakis Aris Pagourtzis Stathis Zachos September 2016 Abstract In this thesis, we study counting classes that lie below #P . One approach, the most regular in Computational Complexity Theory, is the machine-based approach. Classes like #L, span-L [1], and T otP ,#PE [38] are defined establishing space and time restrictions on Turing machine's computational resources. A second approach is Descriptive Complexity's approach. It characterizes complexity classes by the type of logic needed to express the languages in them. Classes deriving from this viewpoint, like #FO [44], #RHΠ1 [16], #RΣ2 [44], are equivalent to #P , the class of AP - interriducible problems to #BIS, and some subclass of the problems owning an FPRAS respectively. A great objective of such an investigation is to gain an understanding of how “efficient counting" relates to these already defined classes. By “efficient counting" we mean counting solutions of a problem using a polynomial time algorithm or an FPRAS. Many other interesting properties of the classes considered and their problems have been examined. For example alternative definitions of counting classes using relation-based op- erators, and the computational difficulty of complete problems, since complete problems capture the difficulty of the corresponding class. Moreover, in Section 3.5 we define the log- space analog of the class T otP and explore how and to what extent results can be transferred from polynomial time to logarithmic space computation. -
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. -
Parity Detection of Propagating Microwave Fields
Parity Detection of Propagating Microwave Fields Jean-Claude Besse,1, ∗ Simone Gasparinetti,1, y Michele C. Collodo,1 Theo Walter,1 Ants Remm,1 Jonas Krause,1 Christopher Eichler,1 and Andreas Wallraff1 1Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland (Dated: December 23, 2019) The parity of the number of elementary excitations present in a quantum system provides impor- tant insights into its physical properties. Parity measurements are used, for example, to tomographi- cally reconstruct quantum states or to determine if a decay of an excitation has occurred, information which can be used for quantum error correction in computation or communication protocols. Here we demonstrate a versatile parity detector for propagating microwaves, which distinguishes between radiation fields containing an even or odd number n of photons, both in a single-shot measurement and without perturbing the parity of the detected field. We showcase applications of the detector for direct Wigner tomography of propagating microwaves and heralded generation of Schr¨odinger cat states. This parity detection scheme is applicable over a broad frequency range and may prove useful, for example, for heralded or fault-tolerant quantum communication protocols. I. INTRODUCTION qubits for measurement based entanglement generation [13], for elements of error correction [14{16], and entan- In quantum physics, the parity P of a wavefunction glement stabilization [17], an experiment which was also governs whether a system has an even or odd number of performed with ions [18, 19]. excitations n. The parity affects, for example, the sys- tem's statistical properties such as the likelihood of tran- sitions occurring between distinct quantum states [1,2].