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COMPLEXITY THEORY Review Lecture 13: Space Hierarchy and Gaps
COMPLEXITY THEORY Review Lecture 13: Space Hierarchy and Gaps Markus Krotzsch¨ Knowledge-Based Systems TU Dresden, 26th Dec 2018 Markus Krötzsch, 26th Dec 2018 Complexity Theory slide 2 of 19 Review: Time Hierarchy Theorems Time Hierarchy Theorem 12.12 If f , g : N N are such that f is time- → constructible, and g log g o(f ), then · ∈ DTime (g) ( DTime (f ) ∗ ∗ Nondeterministic Time Hierarchy Theorem 12.14 If f , g : N N are such that f → is time-constructible, and g(n + 1) o(f (n)), then A Hierarchy for Space ∈ NTime (g) ( NTime (f ) ∗ ∗ In particular, we find that P , ExpTime and NP , NExpTime: , L NL P NP PSpace ExpTime NExpTime ExpSpace ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ , Markus Krötzsch, 26th Dec 2018 Complexity Theory slide 3 of 19 Markus Krötzsch, 26th Dec 2018 Complexity Theory slide 4 of 19 Space Hierarchy Proving the Space Hierarchy Theorem (1) Space Hierarchy Theorem 13.1: If f , g : N N are such that f is space- → constructible, and g o(f ), then For space, we can always assume a single working tape: ∈ Tape reduction leads to a constant-factor increase in space • DSpace(g) ( DSpace(f ) Constant factors can be eliminated by space compression • Proof: Again, we construct a diagonalisation machine . We define a multi-tape TM Therefore, DSpacek(f ) = DSpace1(f ). D D for inputs of the form , w (other cases do not matter), assuming that , w = n hM i |hM i| Compute f (n) in unary to mark the available space on the working tape Space turns out to be easier to separate – we get: • Space Hierarchy Theorem 13.1: If f , g : N N are such that f is space- Initialise a separate countdown tape with the largest binary number that can be → • constructible, and g o(f ), then written in f (n) space ∈ Simulate on , w , making sure that only previously marked tape cells are • M hM i DSpace(g) ( DSpace(f ) used Time-bound the simulation using the content of the countdown tape by • Challenge: TMs can run forever even within bounded space. -
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”. -
Complexity Theory Lecture 9 Co-NP Co-NP-Complete
Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP. -
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. -
Computational Complexity and Intractability: an Introduction to the Theory of NP Chapter 9 2 Objectives
1 Computational Complexity and Intractability: An Introduction to the Theory of NP Chapter 9 2 Objectives . Classify problems as tractable or intractable . Define decision problems . Define the class P . Define nondeterministic algorithms . Define the class NP . Define polynomial transformations . Define the class of NP-Complete 3 Input Size and Time Complexity . Time complexity of algorithms: . Polynomial time (efficient) vs. Exponential time (inefficient) f(n) n = 10 30 50 n 0.00001 sec 0.00003 sec 0.00005 sec n5 0.1 sec 24.3 sec 5.2 mins 2n 0.001 sec 17.9 mins 35.7 yrs 4 “Hard” and “Easy” Problems . “Easy” problems can be solved by polynomial time algorithms . Searching problem, sorting, Dijkstra’s algorithm, matrix multiplication, all pairs shortest path . “Hard” problems cannot be solved by polynomial time algorithms . 0/1 knapsack, traveling salesman . Sometimes the dividing line between “easy” and “hard” problems is a fine one. For example, . Find the shortest path in a graph from X to Y (easy) . Find the longest path (with no cycles) in a graph from X to Y (hard) 5 “Hard” and “Easy” Problems . Motivation: is it possible to efficiently solve “hard” problems? Efficiently solve means polynomial time solutions. Some problems have been proved that no efficient algorithms for them. For example, print all permutation of a number n. However, many problems we cannot prove there exists no efficient algorithms, and at the same time, we cannot find one either. 6 Traveling Salesperson Problem . No algorithm has ever been developed with a Worst-case time complexity better than exponential . -
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. -
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. -
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. -
On the Complexity of Numerical Analysis
On the Complexity of Numerical Analysis Eric Allender Peter B¨urgisser Rutgers, the State University of NJ Paderborn University Department of Computer Science Department of Mathematics Piscataway, NJ 08854-8019, USA DE-33095 Paderborn, Germany [email protected] [email protected] Johan Kjeldgaard-Pedersen Peter Bro Miltersen PA Consulting Group University of Aarhus Decision Sciences Practice Department of Computer Science Tuborg Blvd. 5, DK 2900 Hellerup, Denmark IT-parken, DK 8200 Aarhus N, Denmark [email protected] [email protected] Abstract In Section 1.3 we discuss our main technical contribu- tions: proving upper and lower bounds on the complexity We study two quite different approaches to understand- of PosSLP. In Section 1.4 we present applications of our ing the complexity of fundamental problems in numerical main result with respect to the Euclidean Traveling Sales- analysis. We show that both hinge on the question of under- man Problem and the Sum-of-Square-Roots problem. standing the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program 1.1 Polynomial Time Over the Reals producing an integer N, decide whether N>0. We show that PosSLP lies in the counting hierarchy, and combining The Blum-Shub-Smale model of computation over the our results with work of Tiwari, we show that the Euclidean reals provides a very well-studied complexity-theoretic set- Traveling Salesman Problem lies in the counting hierarchy ting in which to study the computational problems of nu- – the previous best upper bound for this important problem merical analysis. -
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/. -
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]. -
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].