Glossary of Complexity Classes
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Notes for Lecture 2
Notes on Complexity Theory Last updated: December, 2011 Lecture 2 Jonathan Katz 1 Review The running time of a Turing machine M on input x is the number of \steps" M takes before it halts. Machine M is said to run in time T (¢) if for every input x the running time of M(x) is at most T (jxj). (In particular, this means it halts on all inputs.) The space used by M on input x is the number of cells written to by M on all its work tapes1 (a cell that is written to multiple times is only counted once); M is said to use space T (¢) if for every input x the space used during the computation of M(x) is at most T (jxj). We remark that these time and space measures are worst-case notions; i.e., even if M runs in time T (n) for only a fraction of the inputs of length n (and uses less time for all other inputs of length n), the running time of M is still T . (Average-case notions of complexity have also been considered, but are somewhat more di±cult to reason about. We may cover this later in the semester; or see [1, Chap. 18].) Recall that a Turing machine M computes a function f : f0; 1g¤ ! f0; 1g¤ if M(x) = f(x) for all x. We will focus most of our attention on boolean functions, a context in which it is more convenient to phrase computation in terms of languages. A language is simply a subset of f0; 1g¤. -
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. -
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. -
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. -
Valuative Characterization of Central Extensions of Algebraic Tori on Krull
Valuative characterization of central extensions of algebraic tori on Krull domains ∗ Haruhisa Nakajima † Department of Mathematics, J. F. Oberlin University Tokiwa-machi, Machida, Tokyo 194-0294, JAPAN Abstract Let G be an affine algebraic group with an algebraic torus G0 over an alge- braically closed field K of an arbitrary characteristic p. We show a criterion for G to be a finite central extension of G0 in terms of invariant theory of all regular 0 actions of any closed subgroup H containing ZG(G ) on affine Krull K-schemes such that invariant rational functions are locally fractions of invariant regular func- tions. Consider an affine Krull H-scheme X = Spec(R) and a prime ideal P of R with ht(P) = ht(P ∩ RH ) = 1. Let I(P) denote the inertia group of P un- der the action of H. The group G is central over G0 if and only if the fraction e(P, P ∩ RH )/e(P, P ∩ RI(P)) of ramification indices is equal to 1 (p = 0) or to the p-part of the order of the group of weights of G0 on RI(P)) vanishing on RI(P))/P ∩ RI(P) (p> 0) for an arbitrary X and P. MSC: primary 13A50, 14R20, 20G05; secondary 14L30, 14M25 Keywords: Krull domain; ramification index; algebraic group; algebraic torus; char- acter group; invariant theory 1 Introduction 1.A. We consider affine algebraic groups and affine schemes over a fixed algebraically closed field K of an arbitrary characteristic p. For an affine group G, denote by G0 its identity component. -
EXPSPACE-Hardness of Behavioural Equivalences of Succinct One
EXPSPACE-hardness of behavioural equivalences of succinct one-counter nets Petr Janˇcar1 Petr Osiˇcka1 Zdenˇek Sawa2 1Dept of Comp. Sci., Faculty of Science, Palack´yUniv. Olomouc, Czech Rep. [email protected], [email protected] 2Dept of Comp. Sci., FEI, Techn. Univ. Ostrava, Czech Rep. [email protected] Abstract We note that the remarkable EXPSPACE-hardness result in [G¨oller, Haase, Ouaknine, Worrell, ICALP 2010] ([GHOW10] for short) allows us to answer an open complexity ques- tion for simulation preorder of succinct one counter nets (i.e., one counter automata with no zero tests where counter increments and decrements are integers written in binary). This problem, as well as bisimulation equivalence, turn out to be EXPSPACE-complete. The technique of [GHOW10] was referred to by Hunter [RP 2015] for deriving EXPSPACE-hardness of reachability games on succinct one-counter nets. We first give a direct self-contained EXPSPACE-hardness proof for such reachability games (by adjust- ing a known PSPACE-hardness proof for emptiness of alternating finite automata with one-letter alphabet); then we reduce reachability games to (bi)simulation games by using a standard “defender-choice” technique. 1 Introduction arXiv:1801.01073v1 [cs.LO] 3 Jan 2018 We concentrate on our contribution, without giving a broader overview of the area here. A remarkable result by G¨oller, Haase, Ouaknine, Worrell [2] shows that model checking a fixed CTL formula on succinct one-counter automata (where counter increments and decre- ments are integers written in binary) is EXPSPACE-hard. Their proof is interesting and nontrivial, and uses two involved results from complexity theory. -
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. -
On the NP-Completeness of the Minimum Circuit Size Problem
On the NP-Completeness of the Minimum Circuit Size Problem John M. Hitchcock∗ A. Pavany Department of Computer Science Department of Computer Science University of Wyoming Iowa State University Abstract We study the Minimum Circuit Size Problem (MCSP): given the truth-table of a Boolean function f and a number k, does there exist a Boolean circuit of size at most k computing f? This is a fundamental NP problem that is not known to be NP-complete. Previous work has studied consequences of the NP-completeness of MCSP. We extend this work and consider whether MCSP may be complete for NP under more powerful reductions. We also show that NP-completeness of MCSP allows for amplification of circuit complexity. We show the following results. • If MCSP is NP-complete via many-one reductions, the following circuit complexity amplifi- Ω(1) cation result holds: If NP\co-NP requires 2n -size circuits, then ENP requires 2Ω(n)-size circuits. • If MCSP is NP-complete under truth-table reductions, then EXP 6= NP \ SIZE(2n ) for some > 0 and EXP 6= ZPP. This result extends to polylog Turing reductions. 1 Introduction Many natural NP problems are known to be NP-complete. Ladner's theorem [14] tells us that if P is different from NP, then there are NP-intermediate problems: problems that are in NP, not in P, but also not NP-complete. The examples arising out of Ladner's theorem come from diagonalization and are not natural. A canonical candidate example of a natural NP-intermediate problem is the Graph Isomorphism (GI) problem. -
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. -
NP-Completeness Part I
NP-Completeness Part I Outline for Today ● Recap from Last Time ● Welcome back from break! Let's make sure we're all on the same page here. ● Polynomial-Time Reducibility ● Connecting problems together. ● NP-Completeness ● What are the hardest problems in NP? ● The Cook-Levin Theorem ● A concrete NP-complete problem. Recap from Last Time The Limits of Computability EQTM EQTM co-RE R RE LD LD ADD HALT ATM HALT ATM 0*1* The Limits of Efficient Computation P NP R P and NP Refresher ● The class P consists of all problems solvable in deterministic polynomial time. ● The class NP consists of all problems solvable in nondeterministic polynomial time. ● Equivalently, NP consists of all problems for which there is a deterministic, polynomial-time verifier for the problem. Reducibility Maximum Matching ● Given an undirected graph G, a matching in G is a set of edges such that no two edges share an endpoint. ● A maximum matching is a matching with the largest number of edges. AA maximummaximum matching.matching. Maximum Matching ● Jack Edmonds' paper “Paths, Trees, and Flowers” gives a polynomial-time algorithm for finding maximum matchings. ● (This is the same Edmonds as in “Cobham- Edmonds Thesis.) ● Using this fact, what other problems can we solve? Domino Tiling Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling The Setup ● To determine whether you can place at least k dominoes on a crossword grid, do the following: ● Convert the grid into a graph: each empty cell is a node, and any two adjacent empty cells have an edge between them. -
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/.