Lecture 5: Logspace reductions and completeness
|
|
- Mabel Stokes
- 6 years ago
- Views:
Transcription
1 Coputational Coplexity Theory, Fall 2010 Septeber 8 Lecture 5: Logspace reductions and copleteness Lecturer: Kristoffer Arnsfelt Hansen Scribe: Claes Højer Jensen Recall the given relation between classes: L NL P NP PSPACE EXP NEXP EXPSPACE... These classes are defined for two reasons: First to study coputation under resources such as tie and space, but also just as iportantly to help us understand the coputational coplexity of concrete coputational probles that we are interested in. It turns out that all the classes above have natural coplete probles. Recalling the notion of NP-copleteness, this is based on polynoial tie reduction. This is however not useable for lesser classes such as NL and P, so a weaker notion of reductions is needed. Definition 1. The language L f associated with a function f : Σ is L f = {< x, bin(j), y > x Σ, j N, y, f(x) j = y} L f will here work as a substitute for the funciton f. Definition 2. We say f is P-coputable if: f(x) = n O(1) and L f P. f is L-coputable if: f(x) = n O(1) and L f L. Definition 3. Let A Σ and B be languages. We say A p B (p stands for polynoial tie, stands for any-one, i.e. any instances ay be apped to one instance) if there is a P-coputable f so that x Σ, x A f(x) B We say A log B if there is an L-coputable f so that x Σ, x A f(x) B Let C be a class of languages. Definition 4. L is log -hard for C if for all L C : L log L. Definition 5. L is log -coplete for C if L is log hard for C, L C. Lea 6. (Transitivity) If A log B and B log C then A log Proof. Let the reductions be given by L-coputable functions f and g, i.e. we have L f L and L g L. Define h(x) = g(f(x)), and let M f, M g be logspace achines for L f, L g. We ust show that h is L-coputable. Note that since we have liited space available, we can t just copute and put f(x) on a tape, as an interediate result. We will now construct a logspace achine M h for L h. C 1
2 On input < x, bin(j), y >, M h will siulate M g with a fictitious input tape containing < f(x), bin(j), y >. We will aintain the position of the tapehead on the fictitious tape by a counter, stored on the work tape. Whenever M g requires the current sybol on the fictitious input tape, M h will run the achine M f to copute the sybol. M f will be able to read x fro the input tape of M h. Space analysis on input x: for M g : O(log( f(x) )) = O(log( x O(1) )) = O(log( x )). for counter: O(log( f(x) )) = O(log( x O(1) )) = O(log( x )). for M f : O(log( x )). Thus the space used by M h is O(log( x )). We know that NP is closed under reductions; so is all the other classes we have considered. Lea 7. Closure under reductions If B C and A log B then A C, for C = L, NL, P,..., Exaple of log-coplete proble in NL: Definition 8. STCON Instance: Directed graph G = (V, E), s, t V Question: Is there a path fro s to t? Theore 9. STCON is coplete for NL Proof. STCON NL: For this we have the following algorith: 1. v := s 2. for i = 0 to n 1 3. guess u V 4. if (v, u) / E, reject 5. if u = t, accept 6. v = u Space usage: O(log n), since we only need to store node index v, counter i, etc. NL-hardness: Let L NL, and let M be a nondeterinistic O(log n) space bounded achibe for L. We will assue that M has a unique accepting configuration. This we can easily do by odifying M, to clearing the work tape and place tape heads at cell 1, before accepting. Reduction: On input x, output: G: configuration graph of M on input x s: initial confiduration 2
3 t: accept configuration Since M is O(log n) space bounded, each configuration is encoded by a string of length O(log n). We can thus output the configuration graph, by looping over check pairs of string encoding configurations, and check if M can go fro one configuration to the next in 1 step. This can be done in O(log n) space. Theore 10. (Ieran-Szelepcsenyi) Let S(n) log(n). Then (recall that co - C = {L L C}) co - NSPACE(S(n)) = NSPACE(S(n)) Proof. We will prove that STCON NL. Then the stateent of the theore follows by running the NL algorith on the configuration graph of a O(S(n)) space bounded nondeterinistic achine. We first note that siply switching accept and reject in the algorith fro the proof of STCON is in NL does not work: this would not copute the copleent of STCON. Crucial idea: Let V i = set of vertices reachable fro s in i steps, and let C i = V i. If we are given C i, then we can nondeterinistically enuerate the vertices of V i and afterwards we can know for sure if we enuerated exactly the vertices in V i as follows: 1. count := 0 2. for u V 3. nondeterinistically guess a path fro s to u of length i, vertex by vertex. (done with earlier shown algorith) 4. if found 5. count := count if count = C i, correct enueration! Algorith for STCON : 1. C cur := 1 //C 0 = 1 2. for i = 1 to n 1 //copute C i, given C i 1 3. C prev := C cur, C cur = 0 4. for all v V //test if v V i 5. count := 0, include := false 6. for all u V //enuerate V i 1 7. nondeterinistically guess a patch fro s to u of length i 1 vertex by vertex 8. if (found) 9. count := count if ((u, v) E or u = v) 11. include := true 12. if count! = C prev reject // Enueration of V i 1 failed 13. if (include) C cur := C cur + 1 // At this point we have coputed C n 1 correctly. 3
4 14. count := for all u V // Enuerate V n 1 to see if t V n nondeterinistically guess a path fro s to u of length n 1 vertex by vertex 17. if(found) 18. count := count if(u = t) reject 20. if (count = C cur accept) // If enueration was succesful we can tell for sure if t V n 1 Boolean Circuits Inputs: x 1,..., x n 0, 1 Circuit: Directed acyclic graph. Nodes correspond to gates, edges to wires. indegree 0 nodes: labeled by variables indegree 1 nodes: labeled by 1 indegree 2 nodes: labeled by, specified output node C coputes a function f : {0, 1} {0, 1}. Circuit value proble, CVP: Instance: Circuit C on n inputs and concrete input x {0, 1} n. Question: Is C(x) = 1? Circuit-SAT: Instance: Circuit C. Question: Does there exist x 0, 1 n such that C(x) = 1? Theore 11. CVP is log coplete for P Proof. To show that CVP P, given a circuit together with an input, we siple evaluate the circuit botto up. To show log -hardness, let L P, and let M be p(n) tie bounded single-tape TM accepting L. Build circuit by tableu ethod. 4
5 In each coordinate (i, t) we encode the content of the tape in cell i, whether the tape head is scanning the cell, and if so, the state of the achine. Now given this inforation we can copute the contents of the cell just by looking at the 3 previous cells below, in the neighbor positions (see figure). Theore 12. Circuit-SAT is log coplete for NP Proof. This can again by proven by the tableau ethod. We have additional inputs to describe the nondeterinistic choices. These are fed to each coordinate. The reduction will on input x construct this circuit, and then hardwire x, leaving the inputs for the nondeterinistic choices be the only inputs left. 5
Handout 7. and Pr [M(x) = χ L (x) M(x) =? ] = 1.
Notes on Coplexity Theory Last updated: October, 2005 Jonathan Katz Handout 7 1 More on Randoized Coplexity Classes Reinder: so far we have seen RP,coRP, and BPP. We introduce two ore tie-bounded randoized
More informationDSPACE(n)? = NSPACE(n): A Degree Theoretic Characterization
DSPACE(n)? = NSPACE(n): A Degree Theoretic Characterization Manindra Agrawal Departent of Coputer Science and Engineering Indian Institute of Technology, Kanpur 208016, INDIA eail: anindra@iitk.ernet.in
More informationLecture 3: Nondeterminism, NP, and NP-completeness
CSE 531: Computational Complexity I Winter 2016 Lecture 3: Nondeterminism, NP, and NP-completeness January 13, 2016 Lecturer: Paul Beame Scribe: Paul Beame 1 Nondeterminism and NP Recall the definition
More informationReducibility and Completeness
Reducibility and Copleteness Chapter 28 of the forthcoing CRC Handbook on Algoriths and Theory of Coputation Eric Allender 1 Rutgers University Michael C. Loui 2 University of Illinois at Urbana-Chapaign
More informationLecture 7: Polynomial time hierarchy
Computational Complexity Theory, Fall 2010 September 15 Lecture 7: Polynomial time hierarchy Lecturer: Kristoffer Arnsfelt Hansen Scribe: Mads Chr. Olesen Recall that adding the power of alternation gives
More informationCSE200: Computability and complexity Space Complexity
CSE200: Computability and complexity Space Complexity Shachar Lovett January 29, 2018 1 Space complexity We would like to discuss languages that may be determined in sub-linear space. Lets first recall
More informationUmans Complexity Theory Lectures
Umans Complexity Theory Lectures Lecture 8: Introduction to Randomized Complexity: - Randomized complexity classes, - Error reduction, - in P/poly - Reingold s Undirected Graph Reachability in RL Randomized
More informationLecture 8: Complete Problems for Other Complexity Classes
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 8: Complete Problems for Other Complexity Classes David Mix Barrington and Alexis Maciel
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 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 informationLecture 7: The Polynomial-Time Hierarchy. 1 Nondeterministic Space is Closed under Complement
CS 710: Complexity Theory 9/29/2011 Lecture 7: The Polynomial-Time Hierarchy Instructor: Dieter van Melkebeek Scribe: Xi Wu In this lecture we first finish the discussion of space-bounded nondeterminism
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 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 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 informationL S is not p m -hard for NP. Moreover, we prove for every L NP P, that there exists a sparse S EXP such that L S is not p m -hard for NP.
Properties of NP-Coplete Sets Christian Glaßer, A. Pavan, Alan L. Selan, Saik Sengupta January 15, 2004 Abstract We study several properties of sets that are coplete for NP. We prove that if L is an NP-coplete
More informationSpace is a computation resource. Unlike time it can be reused. Computational Complexity, by Fu Yuxi Space Complexity 1 / 44
Space Complexity Space is a computation resource. Unlike time it can be reused. Computational Complexity, by Fu Yuxi Space Complexity 1 / 44 Synopsis 1. Space Bounded Computation 2. Logspace Reduction
More informationBoolean circuits. Lecture Definitions
Lecture 20 Boolean circuits In this lecture we will discuss the Boolean circuit model of computation and its connection to the Turing machine model. Although the Boolean circuit model is fundamentally
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 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 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 informationNotes for Lecture 3... x 4
Stanford University CS254: Computational Complexity Notes 3 Luca Trevisan January 14, 2014 Notes for Lecture 3 In this lecture we introduce the computational model of boolean circuits and prove that polynomial
More informationAdvanced topic: Space complexity
Advanced topic: Space complexity CSCI 3130 Formal Languages and Automata Theory Siu On CHAN Chinese University of Hong Kong Fall 2016 1/28 Review: time complexity We have looked at how long it takes to
More informationThe 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 informationLecture Notes Each circuit agrees with M on inputs of length equal to its index, i.e. n, x {0, 1} n, C n (x) = M(x).
CS 221: Computational Complexity Prof. Salil Vadhan Lecture Notes 4 February 3, 2010 Scribe: Jonathan Pines 1 Agenda P-/NP- Completeness NP-intermediate problems NP vs. co-np L, NL 2 Recap Last time, we
More informationCircuits. Lecture 11 Uniform Circuit Complexity
Circuits Lecture 11 Uniform Circuit Complexity 1 Recall 2 Recall Non-uniform complexity 2 Recall Non-uniform complexity P/1 Decidable 2 Recall Non-uniform complexity P/1 Decidable NP P/log NP = P 2 Recall
More informationComplexity. Complexity Theory Lecture 3. Decidability and Complexity. Complexity Classes
Complexity Theory 1 Complexity Theory 2 Complexity Theory Lecture 3 Complexity For any function f : IN IN, we say that a language L is in TIME(f(n)) if there is a machine M = (Q, Σ, s, δ), such that: L
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 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 informationComplete problems for classes in PH, The Polynomial-Time Hierarchy (PH) oracle is like a subroutine, or function in
Oracle Turing Machines Nondeterministic OTM defined in the same way (transition relation, rather than function) oracle is like a subroutine, or function in your favorite PL but each call counts as single
More informationCMPT 710/407 - Complexity Theory Lecture 4: Complexity Classes, Completeness, Linear Speedup, and Hierarchy Theorems
CMPT 710/407 - Complexity Theory Lecture 4: Complexity Classes, Completeness, Linear Speedup, and Hierarchy Theorems Valentine Kabanets September 13, 2007 1 Complexity Classes Unless explicitly stated,
More informationLecture 3 (Notes) 1. The book Computational Complexity: A Modern Approach by Sanjeev Arora and Boaz Barak;
Topics in Theoretical Computer Science March 7, 2016 Lecturer: Ola Svensson Lecture 3 (Notes) Scribes: Ola Svensson Disclaimer: These notes were written for the lecturer only and may contain inconsistent
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 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 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 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 information1 Bounding the Margin
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost
More information8.1 Theore. For every s log, DSPACE(s 2 Pol ) = R p;bit (DSPACE(s Pol)): 8.2 Corollary. PSPACE = R p;bit (POLYLOGSPACE): The class POLYLOGSPACE sees t
(Kiel) for soe useful inforation concerning string logic; and to Martin Bottcher (Mainz), Gerhard Buntrock (Wurzburg), and Diana Ee (Wurzburg) for fruitful discussions on the subject of this paper. References
More informationList Scheduling and LPT Oliver Braun (09/05/2017)
List Scheduling and LPT Oliver Braun (09/05/207) We investigate the classical scheduling proble P ax where a set of n independent jobs has to be processed on 2 parallel and identical processors (achines)
More informationLecture 4 : Quest for Structure in Counting Problems
CS6840: Advanced Complexity Theory Jan 10, 2012 Lecture 4 : Quest for Structure in Counting Problems Lecturer: Jayalal Sarma M.N. Scribe: Dinesh K. Theme: Between P and PSPACE. Lecture Plan:Counting problems
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 information6.841/18.405J: Advanced Complexity Wednesday, February 12, Lecture Lecture 3
6.841/18.405J: Advanced Complexity Wednesday, February 12, 2003 Lecture Lecture 3 Instructor: Madhu Sudan Scribe: Bobby Kleinberg 1 The language MinDNF At the end of the last lecture, we introduced the
More informationCSC 5170: Theory of Computational Complexity Lecture 4 The Chinese University of Hong Kong 1 February 2010
CSC 5170: Theory of Computational Complexity Lecture 4 The Chinese University of Hong Kong 1 February 2010 Computational complexity studies the amount of resources necessary to perform given computations.
More information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
More informationLecture 20: PSPACE. November 15, 2016 CS 1010 Theory of Computation
Lecture 20: PSPACE November 15, 2016 CS 1010 Theory of Computation Recall that PSPACE = k=1 SPACE(nk ). We will see that a relationship between time and space complexity is given by: P NP PSPACE = NPSPACE
More informationSOLUTION: SOLUTION: SOLUTION:
Convert R and S into nondeterministic finite automata N1 and N2. Given a string s, if we know the states N1 and N2 may reach when s[1...i] has been read, we are able to derive the states N1 and N2 may
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 informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
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 informationITCS:CCT09 : Computational Complexity Theory Apr 8, Lecture 7
ITCS:CCT09 : Computational Complexity Theory Apr 8, 2009 Lecturer: Jayalal Sarma M.N. Lecture 7 Scribe: Shiteng Chen In this lecture, we will discuss one of the basic concepts in complexity theory; namely
More information198:538 Complexity of Computation Lecture 16 Rutgers University, Spring March 2007
198:538 Complexity of Computation Lecture 16 Rutgers University, Spring 2007 8 March 2007 In this lecture we discuss Shamir s theorem that PSPACE is the set of languages that have interactive proofs with
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 information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More informationCS151 Complexity Theory
Introduction CS151 Complexity Theory Lecture 5 April 13, 2004 Power from an unexpected source? we know PEXP, which implies no polytime algorithm for Succinct CVAL poly-size Boolean circuits for Succinct
More information1 PSPACE-Completeness
CS 6743 Lecture 14 1 Fall 2007 1 PSPACE-Completeness Recall the NP-complete problem SAT: Is a given Boolean formula φ(x 1,..., x n ) satisfiable? The same question can be stated equivalently as: Is the
More informationCOL 352 Introduction to Automata and Theory of Computation Major Exam, Sem II , Max 80, Time 2 hr. Name Entry No. Group
COL 352 Introduction to Automata and Theory of Computation Major Exam, Sem II 2015-16, Max 80, Time 2 hr Name Entry No. Group Note (i) Write your answers neatly and precisely in the space provided with
More informationThe Frequent Paucity of Trivial Strings
The Frequent Paucity of Trivial Strings Jack H. Lutz Departent of Coputer Science Iowa State University Aes, IA 50011, USA lutz@cs.iastate.edu Abstract A 1976 theore of Chaitin can be used to show that
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 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 informationLecture 1: Course Overview and Turing machine complexity
CSE 531: Computational Complexity I Winter 2016 Lecture 1: Course Overview and Turing machine complexity January 6, 2016 Lecturer: Paul Beame Scribe: Paul Beame 1 Course Outline 1. Basic properties of
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 informationA Self-Organizing Model for Logical Regression Jerry Farlow 1 University of Maine. (1900 words)
1 A Self-Organizing Model for Logical Regression Jerry Farlow 1 University of Maine (1900 words) Contact: Jerry Farlow Dept of Matheatics Univeristy of Maine Orono, ME 04469 Tel (07) 866-3540 Eail: farlow@ath.uaine.edu
More informationThe Immerman-Szelepcesnyi Theorem and a hard problem for EXPSPACE
The Immerman-Szelepcesnyi Theorem and a hard problem for EXPSPACE Outline for today A new complexity class: co-nl Immerman-Szelepcesnyi: NoPATH is complete for NL Introduction to Vector Addition System
More informationCSE 200 Lecture Notes Turing machine vs. RAM machine vs. circuits
CSE 200 Lecture Notes Turing machine vs. RAM machine vs. circuits Chris Calabro January 13, 2016 1 RAM model There are many possible, roughly equivalent RAM models. Below we will define one in the fashion
More informationIntroduction to Complexity Theory. Bernhard Häupler. May 2, 2006
Introduction to Complexity Theory Bernhard Häupler May 2, 2006 Abstract This paper is a short repetition of the basic topics in complexity theory. It is not intended to be a complete step by step introduction
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 information1 Circuit Complexity. CS 6743 Lecture 15 1 Fall Definitions
CS 6743 Lecture 15 1 Fall 2007 1 Circuit Complexity 1.1 Definitions A Boolean circuit C on n inputs x 1,..., x n is a directed acyclic graph (DAG) with n nodes of in-degree 0 (the inputs x 1,..., x n ),
More informationComputational and Statistical Learning Theory
Coputational and Statistical Learning Theory Proble sets 5 and 6 Due: Noveber th Please send your solutions to learning-subissions@ttic.edu Notations/Definitions Recall the definition of saple based Radeacher
More informationJASS 06 Report Summary. Circuit Complexity. Konstantin S. Ushakov. May 14, 2006
JASS 06 Report Summary Circuit Complexity Konstantin S. Ushakov May 14, 2006 Abstract Computer science deals with many computational models. In real life we have normal computers that are constructed using,
More informationComplexity: Some examples
Algorithms and Architectures III: Distributed Systems H-P Schwefel, Jens M. Pedersen Mm6 Distributed storage and access (jmp) Mm7 Introduction to security aspects (hps) Mm8 Parallel complexity (hps) Mm9
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 informationE0 370 Statistical Learning Theory Lecture 5 (Aug 25, 2011)
E0 370 Statistical Learning Theory Lecture 5 Aug 5, 0 Covering Nubers, Pseudo-Diension, and Fat-Shattering Diension Lecturer: Shivani Agarwal Scribe: Shivani Agarwal Introduction So far we have seen how
More informationUniversality for Nondeterministic Logspace
Universality for Nondeterministic Logspace Vinay Chaudhary, Anand Kumar Sinha, and Somenath Biswas Department of Computer Science and Engineering IIT Kanpur September 2004 1 Introduction The notion of
More informationCSC 5170: Theory of Computational Complexity Lecture 9 The Chinese University of Hong Kong 15 March 2010
CSC 5170: Theory of Computational Complexity Lecture 9 The Chinese University of Hong Kong 15 March 2010 We now embark on a study of computational classes that are more general than NP. As these classes
More informationCS256 Applied Theory of Computation
CS256 Applied Theory of Computation Compleity Classes III John E Savage Overview Last lecture on time-bounded compleity classes Today we eamine space-bounded compleity classes We prove Savitch s Theorem,
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 informationCS5371 Theory of Computation. Lecture 24: Complexity IX (PSPACE-complete, L, NL, NL-complete)
CS5371 Theory of Computation Lecture 24: Complexity IX (PSPACE-complete, L, NL, NL-complete) Objectives PSPACE-complete languages + Examples The classes L and NL NL-complete languages + Examples PSPACE-complete
More informationLecture 9: Connecting PH, P/poly and BPP
Comutational Comlexity Theory, Fall 010 Setember Lecture 9: Connecting PH, P/oly and BPP Lecturer: Kristoffer Arnsfelt Hansen Scribe: Martin Sergio Hedevang Faester Although we do not know how to searate
More informationIntroduction to Computational Complexity
Introduction to Computational Complexity George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 400 George Voutsadakis (LSSU) Computational Complexity September
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 informationTheory of Computation Space Complexity. (NTU EE) Space Complexity Fall / 1
Theory of Computation Space Complexity (NTU EE) Space Complexity Fall 2016 1 / 1 Space Complexity Definition 1 Let M be a TM that halts on all inputs. The space complexity of M is f : N N where f (n) is
More information1 Rademacher Complexity Bounds
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #10 Scribe: Max Goer March 07, 2013 1 Radeacher Coplexity Bounds Recall the following theore fro last lecture: Theore 1. With probability
More informationIntelligent Systems: Reasoning and Recognition. Perceptrons and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley osig 1 Winter Seester 2018 Lesson 6 27 February 2018 Outline Perceptrons and Support Vector achines Notation...2 Linear odels...3 Lines, Planes
More information1 Identical Parallel Machines
FB3: Matheatik/Inforatik Dr. Syaantak Das Winter 2017/18 Optiizing under Uncertainty Lecture Notes 3: Scheduling to Miniize Makespan In any standard scheduling proble, we are given a set of jobs J = {j
More informationAlgorithms for parallel processor scheduling with distinct due windows and unit-time jobs
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 57, No. 3, 2009 Algoriths for parallel processor scheduling with distinct due windows and unit-tie obs A. JANIAK 1, W.A. JANIAK 2, and
More informationLecture 4. 1 Circuit Complexity. Notes on Complexity Theory: Fall 2005 Last updated: September, Jonathan Katz
Notes on Complexity Theory: Fall 2005 Last updated: September, 2005 Jonathan Katz Lecture 4 1 Circuit Complexity Circuits are directed, acyclic graphs where nodes are called gates and edges are called
More informationCSCE 551 Final Exam, April 28, 2016 Answer Key
CSCE 551 Final Exam, April 28, 2016 Answer Key 1. (15 points) Fix any alphabet Σ containing the symbol a. For any language L Σ, define the language a\l := {w Σ wa L}. Show that if L is regular, then a\l
More informationMath 262A Lecture Notes - Nechiporuk s Theorem
Math 6A Lecture Notes - Nechiporuk s Theore Lecturer: Sa Buss Scribe: Stefan Schneider October, 013 Nechiporuk [1] gives a ethod to derive lower bounds on forula size over the full binary basis B The lower
More informationPolygonal Designs: Existence and Construction
Polygonal Designs: Existence and Construction John Hegean Departent of Matheatics, Stanford University, Stanford, CA 9405 Jeff Langford Departent of Matheatics, Drake University, Des Moines, IA 5011 G
More informationLecture 10: Boolean Circuits (cont.)
CSE 200 Computability and Complexity Wednesday, May, 203 Lecture 0: Boolean Circuits (cont.) Instructor: Professor Shachar Lovett Scribe: Dongcai Shen Recap Definition (Boolean circuit) A Boolean circuit
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 informationDe Morgan s a Laws. De Morgan s laws say that. (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2.
De Morgan s a Laws De Morgan s laws say that (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2. Here is a proof for the first law: φ 1 φ 2 (φ 1 φ 2 ) φ 1 φ 2 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 a Augustus DeMorgan
More informationUmans Complexity Theory Lectures
Umans Complexity Theory Lectures Lecture 12: The Polynomial-Time Hierarchy Oracle Turing Machines Oracle Turing Machine (OTM): Deterministic multitape TM M with special query tape special states q?, q
More informationP = k T IME(n k ) Now, do all decidable languages belong to P? Let s consider a couple of languages:
CS 6505: Computability & Algorithms Lecture Notes for Week 5, Feb 8-12 P, NP, PSPACE, and PH A deterministic TM is said to be in SP ACE (s (n)) if it uses space O (s (n)) on inputs of length n. Additionally,
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 informationGeneralized Queries on Probabilistic Context-Free Grammars
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 1, JANUARY 1998 1 Generalized Queries on Probabilistic Context-Free Graars David V. Pynadath and Michael P. Wellan Abstract
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 informationMa/CS 117c Handout # 5 P vs. NP
Ma/CS 117c Handout # 5 P vs. NP We consider the possible relationships among the classes P, NP, and co-np. First we consider properties of the class of NP-complete problems, as opposed to those which are
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 informationIntroduction to Discrete Optimization
Prof. Friedrich Eisenbrand Martin Nieeier Due Date: March 9 9 Discussions: March 9 Introduction to Discrete Optiization Spring 9 s Exercise Consider a school district with I neighborhoods J schools and
More informationA Simplified Analytical Approach for Efficiency Evaluation of the Weaving Machines with Automatic Filling Repair
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 0 A Siplified Analytical Approach for Efficiency Evaluation of the eaving
More information