Deterministic Time and Space Complexity Measures
|
|
- Virginia Christine Sharp
- 5 years ago
- Views:
Transcription
1 Deterministic Complexity Measures: Time and Space Deterministic Time and Space Complexity Measures Definition Let M be any DTM with L(M) Σ, and let x Σ be any input. For the computation M(x), define the time function and the space function, denoted respectively by Time M and Space M, both of which map from Σ to N, as follows: Time M (x) = Space M (x) = m undefined if M(x) has exactly m + 1 configurations otherwise; number of tape cells in a largest configuration of M(x) if M(x) terminates undefined otherwise. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 1 / 22
2 Deterministic Complexity Measures: Time and Space Blum s Axioms Let ϕ 0,ϕ 1,ϕ 2,... be a fixed Gödelization (i.e., an effective enumeration) of all one-argument functions in IP, the class of all partial recursive (i.e., computable) functions. Let IR be the class of all total (i.e., everywhere defined) recursive functions. Let Φ IP be a function mapping from N Σ to N, and write Φ i (x) as a shorthand for Φ(i, x). We say that Φ is a Blum complexity measure if and only if the following two axioms are satisfied: Axiom 1: For each i N, D Φi = D ϕi. Axiom 2: The set {(i, x, m) Φi (x) = m} is decidable. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 2 / 22
3 Deterministic Complexity Measures: Time and Space Deterministic Time and Space Complexity Measures Definition (continued) Define the functions time M : N N and space M : N N by: max Time M (x) if Time M (x) is defined for x: x =n time M (n) = each x with x = n undefined otherwise; space M (n) = max Space M (x) if Space M (x) is defined for x: x =n each x with x = n undefined otherwise. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 3 / 22
4 Deterministic Complexity Measures: Time and Space Deterministic Time and Space Complexity Classes Definition Let t and s be functions in IR mapping from N to N. Define the following deterministic complexity classes with resource function t and s, respectively: DTIME(t) = A DSPACE(s) = A A = L(M) for some DTM M and, for each n N, time M (n) t(n) A = L(M) for some DTM M and, for each n N, space M (n) s(n) ;. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 4 / 22
5 Deterministic Complexity Measures: Time and Space Deterministic Time and Space Complexity Classes Remark: Note that a deterministic Turing machine M decides its language. If A = L(M) then both time M (n) and space M (n) are defined for each n N. In contrast, a nondeterministic Turing machine accepts its language. Thus, the nondeterministic case is treated slightly differently. The resource functions t and s in are called the names of DTIME(t) and DSPACE(s), respectively. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 5 / 22
6 Deterministic Complexity Measures: Time and Space Deterministic Time and Space Complexity Classes Remark: If M is a Turing machine with more than one tape, then Space M (x), the size of a largest configuration of M(x), is defined to be the maximum number of tape cells, where the maximum is taken both over all tapes and over all configurations in the computation. If there is a separate read-only input tape, then only the space used on the working tapes is to be taken into account (reasonable due to sublinear space functions such as logarithmic space). J. Rothe (HHU Düsseldorf) Kryptokomplexität I 6 / 22
7 Nondeterministic Complexity Measures: Time and Space Nondeterministic Time and Space Measures Definition Let M be any NTM with L(M) Σ, and let x Σ be any input. Let Time M (x,α) and Space M (x,α) denote the time and space functions for each path α in M(x). For the computation M(x), define the nondeterministic time and space function, denoted by NTime M and NSpace M, both of which map from Σ to N, as follows: min Time M(x,α) if x L(M) NTime M (x) = M(x) accepts on path α undefined otherwise; NSpace M (x) = min Space M (x,α) if x L(M) M(x) accepts on path α undefined otherwise. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 7 / 22
8 Nondeterministic Complexity Measures: Time and Space Nondeterministic Time and Space Measures Definition (continued) Let t and s be functions in IR mapping from N to N. We say that M accepts a set A in time t if for each x A, we have NTime M (x) t( x ), and for each x A, M does not accept x. We say that M accepts a set A in space s if for each x A, we have NSpace M (x) s( x ), and for each x A, M does not accept x. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 8 / 22
9 Nondeterministic Complexity Measures: Time and Space Nondeterministic Time and Space Complexity Classes Definition (continued) Define the following nondeterministic complexity classes with resource function t and s, respectively: NTIME(t) = A A = L(M) for some NTM M that accepts A in time t(n) ; NSPACE(s) = A A = L(M) for some NTM M that accepts A in space s(n). J. Rothe (HHU Düsseldorf) Kryptokomplexität I 9 / 22
10 Resource Functions Resource Functions Remark: It is reasonable to consider collections F of similar resource functions and to define the complexity class corresponding to F by DTIME(F) = DTIME(f) etc. f F Such a collection F contains all resource functions with a similar rate of growth. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 10 / 22
11 Resource Functions Resource Functions Remark: Consider the collections of functions mapping from N to N each: ILin contains all linear functions, IPol contains all polynomials, 2 ILin contains all exponential functions whose exponent is linear in n, and 2 IPol contains all exponential functions whose exponent is polynomial in n. More generally, for any function t : N N, define the collection of all functions linear in t (respectively, polynomial in t) by: ILin(t) = {f f = l t and l ILin}; IPol(t) = {f f = p t and p IPol}. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 11 / 22
12 Asymptotic Rate of Growth Asymptotic Rate-of-Growth Notation Definition For functions f and g mapping from N to N, define the following notation: f(n) ae g(n) to mean that f(n) g(n) is true for all but finitely many n N. Analogously, the notations < ae, ae, and > ae are defined. The subscript ae of ae, etc. stands for almost everywhere. Similarly, the notation f(n) io g(n) means that f(n) g(n) is true for infinitely many n N. Analogously, the notations < io, io, and > io are defined. The subscript io of io, etc. stands for infinitely often. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 12 / 22
13 Asymptotic Rate of Growth Asymptotic Rate-of-Growth Notation Definition For functions f and g mapping from N to N, define the following notation: f O(g) there is a real constant c > 0 such that f(n)+1 ae c (g(n)+1). f o(g) for all real constants c > 0, f(n)+1 < ae c (g(n)+1). J. Rothe (HHU Düsseldorf) Kryptokomplexität I 13 / 22
14 Asymptotic Rate of Growth Asymptotic Rate-of-Growth Notation Definition f g lim sup n f(n)+1 g(n)+1 <. Note that f O(g) f g. Intuitively, f g means that, by order of magnitude, f does not grow faster than g, with at most finitely many exceptions allowed. f g lim sup n f(n)+1 g(n)+1 = 0. Note that f o(g) f g. Intuitively, f g means that, by order of magnitude, g does grow strictly faster than f, with at most finitely many exceptions allowed. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 14 / 22
15 Asymptotic Rate of Growth Asymptotic Rate-of-Growth Notation Definition f io g lim inf n f(n)+1 g(n)+1 <. Intuitively, f io g means that, by order of magnitude, f does not grow faster than g, at least not for infinitely many arguments. f io g lim inf n f(n)+1 g(n)+1 = 0. Intuitively, f io g means that, by order of magnitude, g does grow strictly faster than f, at least for infinitely many arguments. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 15 / 22
16 Asymptotic Rate of Growth Asymptotic Rate-of-Growth Notation Definition Write f g for g f, f g for g f, f io g for g io f, and f io g for g io f. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 16 / 22
17 Some Central Worst-Case Complexity Classes Some Central Worst-Case Complexity Classes Space classes L = DSPACE(log) NL = NSPACE(log) LINSPACE = DSPACE(ILin) NLINSPACE = NSPACE(ILin) PSPACE = DSPACE(IPol) NPSPACE = NSPACE(IPol) EXPSPACE = DSPACE(2 IPol ) NEXPSPACE = NSPACE(2 IPol ) J. Rothe (HHU Düsseldorf) Kryptokomplexität I 17 / 22
18 Some Central Worst-Case Complexity Classes Some Central Worst-Case Complexity Classes Time classes REALTIME = DTIME(id) LINTIME = DTIME(ILin) P = DTIME(IPol) NP = NTIME(IPol) E = DTIME(2 ILin ) NE = NTIME(2 ILin ) EXP = DTIME(2 IPol ) NEXP = NTIME(2 IPol ) J. Rothe (HHU Düsseldorf) Kryptokomplexität I 18 / 22
19 Some Central Worst-Case Complexity Classes Polynomial versus Exponential Functions t(n) n = 10 n = 20 n = 30 n sec sec sec n sec.0004 sec.0009 sec n sec.008 sec.027 sec n 5.1 sec 3.2 sec 24.3 sec 2 n.001 sec 1.0 sec 17.9 min 3 n.059 sec 58 min 6.5 years Table: Comparing some functions (Garey & Johnson 1979) J. Rothe (HHU Düsseldorf) Kryptokomplexität I 19 / 22
20 Some Central Worst-Case Complexity Classes Polynomial versus Exponential Functions t(n) n = 40 n = 50 n = 60 n sec sec sec n sec.0025 sec.0036 sec n sec.125 sec.256 sec n min 5.2 min 13.0 min 2 n 12.7 days 35.7 years 366 centuries 3 n 3855 centuries centuries centuries Table: Comparing some functions (Garey & Johnson 1979) J. Rothe (HHU Düsseldorf) Kryptokomplexität I 20 / 22
21 Some Central Worst-Case Complexity Classes What if the Computers Get Faster? t i (n) Computer 100 times 1000 times today faster faster t 1 (n) = n N N N 1 t 2 (n) = n 2 N 2 10 N N 2 t 3 (n) = n 3 N N 3 10 N 3 t 4 (n) = n 5 N N N 4 t 5 (n) = 2 n N 5 N N t 6 (n) = 3 n N 6 N N Table: What if the computers get faster? (Garey & Johnson 1979) J. Rothe (HHU Düsseldorf) Kryptokomplexität I 21 / 22
22 Some Central Worst-Case Complexity Classes Time versus Space for Deterministic Classes Theorem 1 DTIME(t) DSPACE(t). 2 DSPACE(s) DTIME(2 ILin(s) ) if s log. Proof: See blackboard. J. Rothe (HHU Düsseldorf) Kryptokomplexität I 22 / 22
The space complexity of a standard Turing machine. The space complexity of a nondeterministic Turing machine
298 8. Space Complexity The space complexity of a standard Turing machine M = (Q,,,, q 0, accept, reject) on input w is space M (w) = max{ uav : q 0 w M u q av, q Q, u, a, v * } The space complexity of
More informationSpace Complexity. The space complexity of a program is how much memory it uses.
Space Complexity The space complexity of a program is how much memory it uses. Measuring Space When we compute the space used by a TM, we do not count the input (think of input as readonly). We say that
More informationLecture 5: The Landscape of Complexity Classes
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 5: The Landscape of Complexity Classes David Mix Barrington and Alexis Maciel July 21,
More informationLecture 3: Reductions and Completeness
CS 710: Complexity Theory 9/13/2011 Lecture 3: Reductions and Completeness Instructor: Dieter van Melkebeek Scribe: Brian Nixon Last lecture we introduced the notion of a universal Turing machine for deterministic
More informationIntroduction to Complexity Classes. Marcin Sydow
Denition TIME(f(n)) TIME(f(n)) denotes the set of languages decided by deterministic TM of TIME complexity f(n) Denition SPACE(f(n)) denotes the set of languages decided by deterministic TM of SPACE complexity
More informationIntroduction to Computational Complexity
Introduction to Computational Complexity A 10-lectures Graduate Course Martin Stigge, martin.stigge@it.uu.se Uppsala University, Sweden 13.7. - 17.7.2009 Martin Stigge (Uppsala University, SE) Computational
More informationCS Lecture 29 P, NP, and NP-Completeness. k ) for all k. Fall The class P. The class NP
CS 301 - Lecture 29 P, NP, and NP-Completeness Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata Equivalence of
More informationHierarchy theorems. Evgenij Thorstensen V18. Evgenij Thorstensen Hierarchy theorems V18 1 / 18
Hierarchy theorems Evgenij Thorstensen V18 Evgenij Thorstensen Hierarchy theorems V18 1 / 18 Comparing functions To prove results like TIME(f(n)) TIME(g(n)), we need a stronger notion of one function growing
More informationComplexity Theory 112. Space Complexity
Complexity Theory 112 Space Complexity We ve already seen the definition SPACE(f(n)): the languages accepted by a machine which uses O(f(n)) tape cells on inputs of length n. Counting only work space NSPACE(f(n))
More informationChapter 1 - Time and Space Complexity. deterministic and non-deterministic Turing machine time and space complexity classes P, NP, PSPACE, NPSPACE
Chapter 1 - Time and Space Complexity deterministic and non-deterministic Turing machine time and space complexity classes P, NP, PSPACE, NPSPACE 1 / 41 Deterministic Turing machines Definition 1.1 A (deterministic
More informationLecture 21: Space Complexity (The Final Exam Frontier?)
6.045 Lecture 21: Space Complexity (The Final Exam Frontier?) 1 conp NP MIN-FORMULA conp P NP FIRST-SAT TAUT P FACTORING SAT NP NP NP 2 VOTE VOTE VOTE For your favorite course on automata and complexity
More informationResource-Bounded Computation
Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space, other resources? Def: L is decidable within time
More informationTime Complexity. Definition. Let t : n n be a function. NTIME(t(n)) = {L L is a language decidable by a O(t(n)) deterministic TM}
Time Complexity Definition Let t : n n be a function. TIME(t(n)) = {L L is a language decidable by a O(t(n)) deterministic TM} NTIME(t(n)) = {L L is a language decidable by a O(t(n)) non-deterministic
More informationSpace Complexity. Master Informatique. Université Paris 5 René Descartes. Master Info. Complexity Space 1/26
Space Complexity Master Informatique Université Paris 5 René Descartes 2016 Master Info. Complexity Space 1/26 Outline Basics on Space Complexity Main Space Complexity Classes Deterministic and Non-Deterministic
More informationP vs. NP Classes. Prof. (Dr.) K.R. Chowdhary.
P vs. NP Classes Prof. (Dr.) K.R. Chowdhary Email: kr.chowdhary@iitj.ac.in Formerly at department of Computer Science and Engineering MBM Engineering College, Jodhpur Monday 10 th April, 2017 kr chowdhary
More informationIntroduction to Computational Complexity
Introduction to Computational Complexity Jiyou Li lijiyou@sjtu.edu.cn Department of Mathematics, Shanghai Jiao Tong University Sep. 24th, 2013 Computation is everywhere Mathematics: addition; multiplication;
More informationComputability and Complexity CISC462, Fall 2018, Space complexity 1
Computability and Complexity CISC462, Fall 2018, Space complexity 1 SPACE COMPLEXITY This material is covered in Chapter 8 of the textbook. For simplicity, we define the space used by a Turing machine
More informationCSCI 1590 Intro to Computational Complexity
CSCI 1590 Intro to Computational Complexity Space Complexity John E. Savage Brown University February 11, 2008 John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11,
More informationLecture 6: Oracle TMs, Diagonalization Limits, Space Complexity
CSE 531: Computational Complexity I Winter 2016 Lecture 6: Oracle TMs, Diagonalization Limits, Space Complexity January 22, 2016 Lecturer: Paul Beame Scribe: Paul Beame Diagonalization enabled us to separate
More informationLecture 2: Tape reduction and Time Hierarchy
Computational Complexity Theory, Fall 2010 August 27 Lecture 2: Tape reduction and Time Hierarchy Lecturer: Peter Bro Miltersen Scribe: Andreas Hummelshøj Jakobsen Tape reduction Theorem 1 (k-tapes 2-tape
More informationMTAT Complexity Theory October 13th-14th, Lecture 6
MTAT.07.004 Complexity Theory October 13th-14th, 2011 Lecturer: Peeter Laud Lecture 6 Scribe(s): Riivo Talviste 1 Logarithmic memory Turing machines working in logarithmic space become interesting when
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Ch 7 Distinguish between computability and complexity Articulate motivation questions
More informationCS5371 Theory of Computation. Lecture 23: Complexity VIII (Space Complexity)
CS5371 Theory of Computation Lecture 23: Complexity VIII (Space Complexity) Objectives Introduce Space Complexity Savitch s Theorem The class PSPACE Space Complexity Definition [for DTM]: Let M be a DTM
More information1 Computational Problems
Stanford University CS254: Computational Complexity Handout 2 Luca Trevisan March 31, 2010 Last revised 4/29/2010 In this lecture we define NP, we state the P versus NP problem, we prove that its formulation
More informationNotes for Lecture Notes 2
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
More informationPolynomial Time Computation. Topics in Logic and Complexity Handout 2. Nondeterministic Polynomial Time. Succinct Certificates.
1 2 Topics in Logic and Complexity Handout 2 Anuj Dawar MPhil Advanced Computer Science, Lent 2010 Polynomial Time Computation P = TIME(n k ) k=1 The class of languages decidable in polynomial time. The
More informationΘεωρητική Πληροφορική Ι (ΣΗΜΜΥ) Υπολογιστική Πολυπλοκότητα Εργαστήριο Λογικής και Επιστήμης Υπολογισμών Εθνικό Μετσόβιο Πολυτεχνείο
Θεωρητική Πληροφορική Ι (ΣΗΜΜΥ) Υπολογιστική Πολυπλοκότητα Εργαστήριο Λογικής και Επιστήμης Υπολογισμών Εθνικό Μετσόβιο Πολυτεχνείο 2016-2017 Πληροφορίες Μαθήματος Θεωρητική Πληροφορική Ι (ΣΗΜΜΥ) Αλγόριθμοι
More information: On the P vs. BPP problem. 30/12/2016 Lecture 12
03684155: On the P vs. BPP problem. 30/12/2016 Lecture 12 Time Hierarchy Theorems Amnon Ta-Shma and Dean Doron 1 Diagonalization arguments Throughout this lecture, for a TM M, we denote M t to be the machine
More informationComputational Complexity IV: PSPACE
Seminar on Theoretical Computer Science and Discrete Mathematics Aristotle University of Thessaloniki Context 1 Section 1: PSPACE 2 3 4 Time Complexity Time complexity of DTM M: - Increasing function t:
More informationCS5371 Theory of Computation. Lecture 23: Complexity VIII (Space Complexity)
CS5371 Theory of Computation Lecture 23: Complexity VIII (Space Complexity) Objectives Introduce Space Complexity Savitch s Theorem The class PSPACE Space Complexity Definition [for DTM]: Let M be a DTM
More informationAdvanced Topics in Theoretical Computer Science
Advanced Topics in Theoretical Computer Science Part 5: Complexity (Part II) 30.01.2014 Viorica Sofronie-Stokkermans Universität Koblenz-Landau e-mail: sofronie@uni-koblenz.de 1 Contents Recall: Turing
More informationTheory of Computation
Theory of Computation Unit 4-6: Turing Machines and Computability Decidability and Encoding Turing Machines Complexity and NP Completeness Syedur Rahman syedurrahman@gmail.com Turing Machines Q The set
More informationComplexity Theory. Knowledge Representation and Reasoning. November 2, 2005
Complexity Theory Knowledge Representation and Reasoning November 2, 2005 (Knowledge Representation and Reasoning) Complexity Theory November 2, 2005 1 / 22 Outline Motivation Reminder: Basic Notions Algorithms
More informationU.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan 9/6/2004. Notes for Lecture 3
U.C. Berkeley CS278: Computational Complexity Handout N3 Professor Luca Trevisan 9/6/2004 Notes for Lecture 3 Revised 10/6/04 1 Space-Bounded Complexity Classes A machine solves a problem using space s(
More informationCS601 DTIME and DSPACE Lecture 5. Time and Space functions: t,s : N N +
CS61 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 constantc, such that, 1. A = L(M)
More informationNotes on Space-Bounded Complexity
U.C. Berkeley CS172: Automata, Computability and Complexity Handout 6 Professor Luca Trevisan 4/13/2004 Notes on Space-Bounded Complexity These are notes for CS278, Computational Complexity, scribed by
More informationTime-Space Tradeoffs for SAT
Lecture 8 Time-Space Tradeoffs for SAT April 22, 2004 Lecturer: Paul Beame Notes: Definition 8.1. TIMESPACE(T (n), S(n)) = {L {0, 1} offline, multitape TM M such that L = L(M) and M uses time O(T (n))
More informationLecture 8: Alternatation. 1 Alternate Characterizations of the Polynomial Hierarchy
CS 710: Complexity Theory 10/4/2011 Lecture 8: Alternatation Instructor: Dieter van Melkebeek Scribe: Sachin Ravi In this lecture, we continue with our discussion of the polynomial hierarchy complexity
More informationMTAT Complexity Theory October 20th-21st, Lecture 7
MTAT.07.004 Complexity Theory October 20th-21st, 2011 Lecturer: Peeter Laud Lecture 7 Scribe(s): Riivo Talviste Polynomial hierarchy 1 Turing reducibility From the algorithmics course, we know the notion
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105-abc/ Logistics HW7 due tonight Thursday's class: REVIEW Final exam on Thursday Dec 8, 8am-11am, LEDDN AUD Note card allowed
More informationCS154, Lecture 17: conp, Oracles again, Space Complexity
CS154, Lecture 17: conp, Oracles again, Space Complexity Definition: conp = { L L NP } What does a conp computation look like? In NP algorithms, we can use a guess instruction in pseudocode: Guess string
More informationDIMACS Technical Report May Complexity Classes. Eric Allender 1;2. Dept. of Computer Science. Rutgers University
DIMACS Technical Report 98-23 May 1998 Complexity Classes by Eric Allender 1;2 Dept. of Computer Science Rutgers University New Brunswick, New Jersey 08903 Michael C. Loui 3 University of Illinois at Urbana-Champaign
More informationIntro to Theory of Computation
Intro to Theory of Computation LECTURE 22 Last time Review Today: Finish recursion theorem Complexity theory Exam 2 solutions out Homework 9 out Sofya Raskhodnikova L22.1 I-clicker question (frequency:
More informationComplexity of domain-independent planning. José Luis Ambite
Complexity of domain-independent planning José Luis Ambite 1 Decidability Decision problem: a problem with a yes/no answer e.g. is N prime? Decidable: if there is a program (i.e. a Turing Machine) that
More informationCould we potentially place A in a smaller complexity class if we consider other computational models?
Introduction to Complexity Theory Big O Notation Review Linear function: r(n) =O(n). Polynomial function: r(n) =2 O(1) Exponential function: r(n) =2 no(1) Logarithmic function: r(n) = O(log n) Poly-log
More informationNP, polynomial-time mapping reductions, and NP-completeness
NP, polynomial-time mapping reductions, and NP-completeness In the previous lecture we discussed deterministic time complexity, along with the time-hierarchy theorem, and introduced two complexity classes:
More informationOutline. Complexity Theory. Example. Sketch of a log-space TM for palindromes. Log-space computations. Example VU , SS 2018
Complexity Theory Complexity Theory Outline Complexity Theory VU 181.142, SS 2018 3. Logarithmic Space Reinhard Pichler Institute of Logic and Computation DBAI Group TU Wien 3. Logarithmic Space 3.1 Computational
More informationNotes on Space-Bounded Complexity
U.C. Berkeley CS172: Automata, Computability and Complexity Handout 7 Professor Luca Trevisan April 14, 2015 Notes on Space-Bounded Complexity These are notes for CS278, Computational Complexity, scribed
More informationReview of Basic Computational Complexity
Lecture 1 Review of Basic Computational Complexity March 30, 2004 Lecturer: Paul Beame Notes: Daniel Lowd 1.1 Preliminaries 1.1.1 Texts There is no one textbook that covers everything in this course. Some
More informationComputational Complexity
CS311 Computational Structures Computational Complexity Lecture 16 Andrew P. Black Andrew Tolmach 1 So, itʼs computable! But at what cost? Some things that are computable in principle are in practice intractable
More informationPrinciples of Knowledge Representation and Reasoning
Principles of Knowledge Representation and Reasoning Complexity Theory Bernhard Nebel, Malte Helmert and Stefan Wölfl Albert-Ludwigs-Universität Freiburg April 29, 2008 Nebel, Helmert, Wölfl (Uni Freiburg)
More informationTheory of Computation
Theory of Computation Dr. Sarmad Abbasi Dr. Sarmad Abbasi () Theory of Computation 1 / 38 Lecture 21: Overview Big-Oh notation. Little-o notation. Time Complexity Classes Non-deterministic TMs The Class
More informationDefinition: Alternating time and space Game Semantics: State of machine determines who
CMPSCI 601: Recall From Last Time Lecture Definition: Alternating time and space Game Semantics: State of machine determines who controls, White wants it to accept, Black wants it to reject. White wins
More informationLecture 3. 1 Terminology. 2 Non-Deterministic Space Complexity. Notes on Complexity Theory: Fall 2005 Last updated: September, 2005.
Notes on Complexity Theory: Fall 2005 Last updated: September, 2005 Jonathan Katz Lecture 3 1 Terminology For any complexity class C, we define the class coc as follows: coc def = { L L C }. One class
More informationLecture 12: Randomness Continued
CS 710: Complexity Theory 2/25/2010 Lecture 12: Randomness Continued Instructor: Dieter van Melkebeek Scribe: Beth Skubak & Nathan Collins In the last lecture we introduced randomized computation in terms
More informationCS294: Pseudorandomness and Combinatorial Constructions September 13, Notes for Lecture 5
UC Berkeley Handout N5 CS94: Pseudorandomness and Combinatorial Constructions September 3, 005 Professor Luca Trevisan Scribe: Gatis Midrijanis Notes for Lecture 5 In the few lectures we are going to look
More informationTuring Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q0, qaccept, qreject), where Q, Σ, Γ are all finite
The Church-Turing Thesis CS60001: Foundations of Computing Science Professor, Dept. of Computer Sc. & Engg., Turing Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q 0, q accept, q reject ), where
More informationTheory of Computation CS3102 Spring 2015 A tale of computers, math, problem solving, life, love and tragic death
Theory of Computation CS3102 Spring 2015 A tale of computers, math, problem solving, life, love and tragic death Robbie Hott www.cs.virginia.edu/~jh2jf Department of Computer Science University of Virginia
More informationTime Complexity. CS60001: Foundations of Computing Science
Time Complexity CS60001: Foundations of Computing Science Professor, Dept. of Computer Sc. & Engg., Measuring Complexity Definition Let M be a deterministic Turing machine that halts on all inputs. The
More informationIntractable Problems [HMU06,Chp.10a]
Intractable Problems [HMU06,Chp.10a] Time-Bounded Turing Machines Classes P and NP Polynomial-Time Reductions A 10 Minute Motivation https://www.youtube.com/watch?v=yx40hbahx3s 1 Time-Bounded TM s A Turing
More informationLecture 16: Time Complexity and P vs NP
6.045 Lecture 16: Time Complexity and P vs NP 1 Time-Bounded Complexity Classes Definition: TIME(t(n)) = { L there is a Turing machine M with time complexity O(t(n)) so that L = L(M) } = { L L is a language
More informationECE 695 Numerical Simulations Lecture 2: Computability and NPhardness. Prof. Peter Bermel January 11, 2017
ECE 695 Numerical Simulations Lecture 2: Computability and NPhardness Prof. Peter Bermel January 11, 2017 Outline Overview Definitions Computing Machines Church-Turing Thesis Polynomial Time (Class P)
More informationsatisfiability (sat) Satisfiability unsatisfiability (unsat or sat complement) and validity Any Expression φ Can Be Converted into CNFs and DNFs
Any Expression φ Can Be Converted into CNFs and DNFs φ = x j : This is trivially true. φ = φ 1 and a CNF is sought: Turn φ 1 into a DNF and apply de Morgan s laws to make a CNF for φ. φ = φ 1 and a DNF
More informationComputational Complexity Theory. Markus Bläser, Holger Dell Universität des Saarlandes Draft November 14, 2016 and forever
Computational Complexity Theory Markus Bläser, Holger Dell Universität des Saarlandes Draft November 14, 2016 and forever 2 1 Simple lower bounds and gaps Complexity theory is the science of classifying
More informationComputational Complexity III: Limits of Computation
: Limits of Computation School of Informatics Thessaloniki Seminar on Theoretical Computer Science and Discrete Mathematics Aristotle University of Thessaloniki Context 1 2 3 Computability vs Complexity
More informationTime-bounded computations
Lecture 18 Time-bounded computations We now begin the final part of the course, which is on complexity theory. We ll have time to only scratch the surface complexity theory is a rich subject, and many
More informationChapter 6: Turing Machines
Chapter 6: Turing Machines 6.1 The Turing Machine Definition A deterministic Turing machine (DTM) M is specified by a sextuple (Q, Σ, Γ, δ, s, f), where Q is a finite set of states; Σ is an alphabet of
More informationU.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan August 30, Notes for Lecture 1
U.C. Berkeley CS278: Computational Complexity Handout N1 Professor Luca Trevisan August 30, 2004 Notes for Lecture 1 This course assumes CS170, or equivalent, as a prerequisite. We will assume that the
More informationComputational Complexity CSCI-GA Subhash Khot Transcribed by Patrick Lin
Computational Complexity CSCI-GA 3350 Subhash Khot Transcribed by Patrick Lin Abstract. These notes are from a course in Computational Complexity, as offered in Spring 2014 at the Courant Institute of
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105-ab/ Today's learning goals Sipser Ch 7.2, 7.3 Distinguish between polynomial and exponential DTIME Define nondeterministic
More informationComputational Complexity Theory. Markus Bläser Universität des Saarlandes Draft January 23, 2014 and forever
Computational Complexity Theory Markus Bläser Universität des Saarlandes Draft January 23, 24 and forever 2 1 Simple lower bounds and gaps Complexity theory is the science of classifying problems with
More informationLogarithmic space. Evgenij Thorstensen V18. Evgenij Thorstensen Logarithmic space V18 1 / 18
Logarithmic space Evgenij Thorstensen V18 Evgenij Thorstensen Logarithmic space V18 1 / 18 Journey below Unlike for time, it makes sense to talk about sublinear space. This models computations on input
More informationArtificial Intelligence. 3 Problem Complexity. Prof. Dr. Jana Koehler Fall 2016 HSLU - JK
Artificial Intelligence 3 Problem Complexity Prof. Dr. Jana Koehler Fall 2016 Agenda Computability and Turing Machines Tractable and Intractable Problems P vs. NP Decision Problems Optimization problems
More informationCISC 4090 Theory of Computation
CISC 4090 Theory of Computation Complexity Professor Daniel Leeds dleeds@fordham.edu JMH 332 Computability Are we guaranteed to get an answer? Complexity How long do we have to wait for an answer? (Ch7)
More informationDefinition: Alternating time and space Game Semantics: State of machine determines who
CMPSCI 601: Recall From Last Time Lecture 3 Definition: Alternating time and space Game Semantics: State of machine determines who controls, White wants it to accept, Black wants it to reject. White wins
More informationBeyond NP [HMU06,Chp.11a] Tautology Problem NP-Hardness and co-np Historical Comments Optimization Problems More Complexity Classes
Beyond NP [HMU06,Chp.11a] Tautology Problem NP-Hardness and co-np Historical Comments Optimization Problems More Complexity Classes 1 Tautology Problem & NP-Hardness & co-np 2 NP-Hardness Another essential
More informationCS Lecture 28 P, NP, and NP-Completeness. Fall 2008
CS 301 - Lecture 28 P, NP, and NP-Completeness Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata Equivalence of
More informationMm7 Intro to distributed computing (jmp) Mm8 Backtracking, 2-player games, genetic algorithms (hps) Mm9 Complex Problems in Network Planning (JMP)
Algorithms and Architectures II H-P Schwefel, Jens M. Pedersen Mm6 Advanced Graph Algorithms (hps) Mm7 Intro to distributed computing (jmp) Mm8 Backtracking, 2-player games, genetic algorithms (hps) Mm9
More information1 Deterministic Turing Machines
Time and Space Classes Exposition by William Gasarch 1 Deterministic Turing Machines Turing machines are a model of computation. It is believed that anything that can be computed can be computed by a Turing
More informationLecture 22: PSPACE
6.045 Lecture 22: PSPACE 1 VOTE VOTE VOTE For your favorite course on automata and complexity Please complete the online subject evaluation for 6.045 2 Final Exam Information Who: You On What: Everything
More informationINAPPROX APPROX PTAS. FPTAS Knapsack P
CMPSCI 61: Recall From Last Time Lecture 22 Clique TSP INAPPROX exists P approx alg for no ε < 1 VertexCover MAX SAT APPROX TSP some but not all ε< 1 PTAS all ε < 1 ETSP FPTAS Knapsack P poly in n, 1/ε
More informationCS5371 Theory of Computation. Lecture 10: Computability Theory I (Turing Machine)
CS537 Theory of Computation Lecture : Computability Theory I (Turing Machine) Objectives Introduce the Turing Machine (TM)? Proposed by Alan Turing in 936 finite-state control + infinitely long tape A
More informationVariations of the Turing Machine
Variations of the Turing Machine 1 The Standard Model Infinite Tape a a b a b b c a c a Read-Write Head (Left or Right) Control Unit Deterministic 2 Variations of the Standard Model Turing machines with:
More informationThe Polynomial Hierarchy
The Polynomial Hierarchy Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Motivation..synthesizing circuits is exceedingly difficulty. It is even
More informationSpace Complexity. Huan Long. Shanghai Jiao Tong University
Space Complexity Huan Long Shanghai Jiao Tong University Acknowledgements Part of the slides comes from a similar course in Fudan University given by Prof. Yijia Chen. http://basics.sjtu.edu.cn/ chen/
More informationCOMPLEXITY THEORY. PSPACE = SPACE(n k ) k N. NPSPACE = NSPACE(n k ) 10/30/2012. Space Complexity: Savitch's Theorem and PSPACE- Completeness
15-455 COMPLEXITY THEORY Space Complexity: Savitch's Theorem and PSPACE- Completeness October 30,2012 MEASURING SPACE COMPLEXITY FINITE STATE CONTROL I N P U T 1 2 3 4 5 6 7 8 9 10 We measure space complexity
More informationReview of unsolvability
Review of unsolvability L L H To prove unsolvability: show a reduction. To prove solvability: show an algorithm. Unsolvable problems (main insight) Turing machine (algorithm) properties Pattern matching
More informationTime Complexity (1) CSCI Spring Original Slides were written by Dr. Frederick W Maier. CSCI 2670 Time Complexity (1)
Time Complexity (1) CSCI 2670 Original Slides were written by Dr. Frederick W Maier Spring 2014 Time Complexity So far we ve dealt with determining whether or not a problem is decidable. But even if it
More informationComputational Complexity Theory. Markus Bläser, Holger Dell, Karteek Sreenivasaiah Universität des Saarlandes Draft June 15, 2015 and forever
Computational Complexity Theory Markus Bläser, Holger Dell, Karteek Sreenivasaiah Universität des Saarlandes Draft June 15, 2015 and forever 2 1 Simple lower bounds and gaps Complexity theory is the science
More informationFinish K-Complexity, Start Time Complexity
6.045 Finish K-Complexity, Start Time Complexity 1 Kolmogorov Complexity Definition: The shortest description of x, denoted as d(x), is the lexicographically shortest string such that M(w) halts
More informationComputational complexity
COMS11700 Computational complexity Department of Computer Science, University of Bristol Bristol, UK 2 May 2014 COMS11700: Computational complexity Slide 1/23 Introduction If we can prove that a language
More informationCOSE215: Theory of Computation. Lecture 21 P, NP, and NP-Complete Problems
COSE215: Theory of Computation Lecture 21 P, NP, and NP-Complete Problems Hakjoo Oh 2017 Spring Hakjoo Oh COSE215 2017 Spring, Lecture 21 June 11, 2017 1 / 11 Contents 1 The classes P and N P Reductions
More informationCS151 Complexity Theory. Lecture 4 April 12, 2017
CS151 Complexity Theory Lecture 4 A puzzle A puzzle: two kinds of trees depth n...... cover up nodes with c colors promise: never color arrow same as blank determine which kind of tree in poly(n, c) steps?
More informationThe P versus NP Problem. Dean Casalena University of Cape Town CSLDEA001
The P versus NP Problem Dean Casalena University of Cape Town CSLDEA001 dean@casalena.co.za Contents 1. Introduction 2. Turing Machines and Syntax 2.1 Overview 2.2 Turing Machine Syntax. 2.3 Polynomial
More informationan efficient procedure for the decision problem. We illustrate this phenomenon for the Satisfiability problem.
1 More on NP In this set of lecture notes, we examine the class NP in more detail. We give a characterization of NP which justifies the guess and verify paradigm, and study the complexity of solving search
More informationChapter 7: Time Complexity
Chapter 7: Time Complexity 1 Time complexity Let M be a deterministic Turing machine that halts on all inputs. The running time or time complexity of M is the function f: N N, where f(n) is the maximum
More informationCOMP 382: Reasoning about algorithms
Fall 2014 Unit 4: Basics of complexity analysis Correctness and efficiency So far, we have talked about correctness and termination of algorithms What about efficiency? Running time of an algorithm For
More informationTheory of Computation. Ch.8 Space Complexity. wherein all branches of its computation halt on all
Definition 8.1 Let M be a deterministic Turing machine, DTM, that halts on all inputs. The space complexity of M is the function f : N N, where f(n) is the maximum number of tape cells that M scans on
More informationPSPACE, NPSPACE, L, NL, Savitch's Theorem. More new problems that are representa=ve of space bounded complexity classes
PSPACE, NPSPACE, L, NL, Savitch's Theorem More new problems that are representa=ve of space bounded complexity classes Outline for today How we'll count space usage Space bounded complexity classes New
More information