Logic and Complexity
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LSPACE VS NP IS AS HARD AS P VS NP 1. Introduction in Complexity
LSPACE VS NP IS AS HARD AS P VS NP FRANK VEGA Abstract. The P versus NP problem is a major unsolved problem in com- puter science. This consists in knowing the answer of the following question: Is P equal to NP? Another major complexity classes are LSPACE, PSPACE, ESPACE, E and EXP. Whether LSPACE = P is a fundamental question that it is as important as it is unresolved. We show if P = NP, then LSPACE = NP. Consequently, if LSPACE is not equal to NP, then P is not equal to NP. According to Lance Fortnow, it seems that LSPACE versus NP is easier to be proven. However, with this proof we show this problem is as hard as P versus NP. Moreover, we prove the complexity class P is not equal to PSPACE as a direct consequence of this result. Furthermore, we demonstrate if PSPACE is not equal to EXP, then P is not equal to NP. In addition, if E = ESPACE, then P is not equal to NP. 1. Introduction In complexity theory, a function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem [6]. A functional problem F is defined as a binary relation (x; y) 2 R over strings of an arbitrary alphabet Σ: R ⊂ Σ∗ × Σ∗: A Turing machine M solves F if for every input x such that there exists a y satisfying (x; y) 2 R, M produces one such y, that is M(x) = y [6]. -
The Polynomial Hierarchy
ij 'I '""T', :J[_ ';(" THE POLYNOMIAL HIERARCHY Although the complexity classes we shall study now are in one sense byproducts of our definition of NP, they have a remarkable life of their own. 17.1 OPTIMIZATION PROBLEMS Optimization problems have not been classified in a satisfactory way within the theory of P and NP; it is these problems that motivate the immediate extensions of this theory beyond NP. Let us take the traveling salesman problem as our working example. In the problem TSP we are given the distance matrix of a set of cities; we want to find the shortest tour of the cities. We have studied the complexity of the TSP within the framework of P and NP only indirectly: We defined the decision version TSP (D), and proved it NP-complete (corollary to Theorem 9.7). For the purpose of understanding better the complexity of the traveling salesman problem, we now introduce two more variants. EXACT TSP: Given a distance matrix and an integer B, is the length of the shortest tour equal to B? Also, TSP COST: Given a distance matrix, compute the length of the shortest tour. The four variants can be ordered in "increasing complexity" as follows: TSP (D); EXACTTSP; TSP COST; TSP. Each problem in this progression can be reduced to the next. For the last three problems this is trivial; for the first two one has to notice that the reduction in 411 j ;1 17.1 Optimization Problems 413 I 412 Chapter 17: THE POLYNOMIALHIERARCHY the corollary to Theorem 9.7 proving that TSP (D) is NP-complete can be used with DP. -
Complexity Theory
Complexity Theory Course Notes Sebastiaan A. Terwijn Radboud University Nijmegen Department of Mathematics P.O. Box 9010 6500 GL Nijmegen the Netherlands [email protected] Copyright c 2010 by Sebastiaan A. Terwijn Version: December 2017 ii Contents 1 Introduction 1 1.1 Complexity theory . .1 1.2 Preliminaries . .1 1.3 Turing machines . .2 1.4 Big O and small o .........................3 1.5 Logic . .3 1.6 Number theory . .4 1.7 Exercises . .5 2 Basics 6 2.1 Time and space bounds . .6 2.2 Inclusions between classes . .7 2.3 Hierarchy theorems . .8 2.4 Central complexity classes . 10 2.5 Problems from logic, algebra, and graph theory . 11 2.6 The Immerman-Szelepcs´enyi Theorem . 12 2.7 Exercises . 14 3 Reductions and completeness 16 3.1 Many-one reductions . 16 3.2 NP-complete problems . 18 3.3 More decision problems from logic . 19 3.4 Completeness of Hamilton path and TSP . 22 3.5 Exercises . 24 4 Relativized computation and the polynomial hierarchy 27 4.1 Relativized computation . 27 4.2 The Polynomial Hierarchy . 28 4.3 Relativization . 31 4.4 Exercises . 32 iii 5 Diagonalization 34 5.1 The Halting Problem . 34 5.2 Intermediate sets . 34 5.3 Oracle separations . 36 5.4 Many-one versus Turing reductions . 38 5.5 Sparse sets . 38 5.6 The Gap Theorem . 40 5.7 The Speed-Up Theorem . 41 5.8 Exercises . 43 6 Randomized computation 45 6.1 Probabilistic classes . 45 6.2 More about BPP . 48 6.3 The classes RP and ZPP . -
The Classes FNP and TFNP
Outline The classes FNP and TFNP C. Wilson1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function Problems What are Function Problems? Function Problems FSAT Defined Total Functions TSP Defined Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function -
Syllabus Computability Theory
Syllabus Computability Theory Sebastiaan A. Terwijn Institute for Discrete Mathematics and Geometry Technical University of Vienna Wiedner Hauptstrasse 8–10/E104 A-1040 Vienna, Austria [email protected] Copyright c 2004 by Sebastiaan A. Terwijn version: 2020 Cover picture and above close-up: Sunflower drawing by Alan Turing, c copyright by University of Southampton and King’s College Cambridge 2002, 2003. iii Emil Post (1897–1954) Alonzo Church (1903–1995) Kurt G¨odel (1906–1978) Stephen Cole Kleene Alan Turing (1912–1954) (1909–1994) Contents 1 Introduction 1 1.1 Preliminaries ................................ 2 2 Basic concepts 3 2.1 Algorithms ................................. 3 2.2 Recursion .................................. 4 2.2.1 Theprimitiverecursivefunctions . 4 2.2.2 Therecursivefunctions . 5 2.3 Turingmachines .............................. 6 2.4 Arithmetization............................... 10 2.4.1 Codingfunctions .......................... 10 2.4.2 Thenormalformtheorem . 11 2.4.3 The basic equivalence and Church’s thesis . 13 2.4.4 Canonicalcodingoffinitesets. 15 2.5 Exercises .................................. 15 3 Computable and computably enumerable sets 19 3.1 Diagonalization............................... 19 3.2 Computablyenumerablesets . 19 3.3 Undecidablesets .............................. 22 3.4 Uniformity ................................. 24 3.5 Many-onereducibility ........................... 25 3.6 Simplesets ................................. 26 3.7 Therecursiontheorem ........................... 28 3.8 Exercises -
On the Weakness of Sharply Bounded Polynomial Induction
On the Weakness of Sharply Bounded Polynomial Induction Jan Johannsen Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg May 4, 1994 Abstract 0 We shall show that if the theory S2 of sharply bounded polynomial induction is extended 0 by symbols for certain functions and their defining axioms, it is still far weaker than T2 , which has ordinary sharply bounded induction. Furthermore, we show that this extended 0 b 0 system S2+ cannot Σ1-define every function in AC , the class of functions computable by polynomial size constant depth circuits. 1 Introduction i i The theory S2 of bounded arithmetic and its fragments S2 and T2 (i ≥ 0) were defined in [Bu]. The language of these theories comprises the usual language of arithmetic plus additional 1 |x|·|y| symbols for the functions ⌊ 2 x⌋, |x| := ⌈log2(x + 1)⌉ and x#y := 2 . Quantifiers of the form ∀x≤t , ∃x≤t are called bounded quantifiers. If the bounding term t is furthermore of the form b b |s|, the quantifier is called sharply bounded. The classes Σi and Πi of the bounded arithmetic hierarchy are defined in analogy to the classes of the usual arithmetical hierarchy, where the i counts alternations of bounded quantifiers, ignoring the sharply bounded quantifiers. i S2 is axiomatized by a finite set of open axioms (called BASIC) plus the schema of poly- b nomial induction (PIND) for Σi -formulae ϕ: 1 ϕ(0) ∧ ∀x ( ϕ(⌊ x⌋) → ϕ(x) ) → ∀x ϕ(x) 2 i b T2 is the same with the ordinary induction scheme for Σi -formulae replacing PIND. -
An Introduction to Computability Theory
AN INTRODUCTION TO COMPUTABILITY THEORY CINDY CHUNG Abstract. This paper will give an introduction to the fundamentals of com- putability theory. Based on Robert Soare's textbook, The Art of Turing Computability: Theory and Applications, we examine concepts including the Halting problem, properties of Turing jumps and degrees, and Post's Theo- rem.1 In the paper, we will assume the reader has a conceptual understanding of computer programs. Contents 1. Basic Computability 1 2. The Halting Problem 2 3. Post's Theorem: Jumps and Quantifiers 3 3.1. Arithmetical Hierarchy 3 3.2. Degrees and Jumps 4 3.3. The Zero-Jump, 00 5 3.4. Post's Theorem 5 Acknowledgments 8 References 8 1. Basic Computability Definition 1.1. An algorithm is a procedure capable of being carried out by a computer program. It accepts various forms of numerical inputs including numbers and finite strings of numbers. Computability theory is a branch of mathematical logic that focuses on algo- rithms, formally known in this area as computable functions, and studies the degree, or level of computability that can be attributed to various sets (these concepts will be formally defined and elaborated upon below). This field was founded in the 1930s by many mathematicians including the logicians: Alan Turing, Stephen Kleene, Alonzo Church, and Emil Post. Turing first pioneered computability theory when he introduced the fundamental concept of the a-machine which is now known as the Turing machine (this concept will be intuitively defined below) in his 1936 pa- per, On computable numbers, with an application to the Entscheidungsproblem[7]. -
LANDSCAPE of COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M
LANDSCAPE OF COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M. Faramawi, MBA Dr. Kenneth W. Regan A complete language for EXPSPACE: PIM, “Polynomial Ideal Arithmetical Hierarchy (AH) Membership”—the simplest natural completeness level that is known PIM not to have polynomial‐size circuits. TOT = {M : M is total, K(2) TOT e e ∑n Πn i.e. halts for all inputs} EXPSPACE Succinct 3SAT Unknown ∑2 Π2 but Commonly Believed: RECRE L ≠ NL …………………….. L ≠ PH NEXP co‐NEXP = ∃∀.REC = ∀∃.REC P ≠ NP ∩ co‐NP ……… P ≠ PSPACE nxn Chess p p K D NP ≠ ∑ 2 ∩Π 2 ……… NP ≠ EXP D = {Turing machines M : EXP e Best Known Separations: Me does not accept e} = RE co‐RE the complement of K. AC0 ⊂ ACC0 ⊂ PP, also TC0 ⊂ PP QBF REC (“Diagonal Language”) For any fixed k, NC1 ⊂ PSPACE, …, NL ⊂ PSPACE = ∃.REC = ∀.REC there is a PSPACE problem in this P ⊂ EXP, NP ⊂ NEXP BQP: Bounded‐Error Quantum intersection that PSPACE ⊂ EXPSPACE Polynomial Time. Believed larger can NOT be Polynomial Hierarchy (PH) than P since it has FACTORING, solved by circuits p p k but not believed to contain NP. of size O(n ) ∑ 2 Π 2 L WS5 p p ∑ 2 Π 2 The levels of AH and poly. poly. BPP: Bounded‐Error Probabilistic TAUT = ∃∀ P = ∀∃ P SAT Polynomial Time. Many believe BPP = P. NC1 PH are analogous, NLIN except that we believe NP co‐NP 0 NP ∩ co‐NP ≠ P and NTIME [n2] TC QBF PARITY ∑p ∩Πp NP PP Probabilistic NP FACT 2 2 ≠ P , which NP ACC0 stand in contrast to P PSPACE Polynomial Time MAJ‐SAT CVP RE ∩ co‐RE = REC and 0 AC RE P P ∑2 ∩Π2 = REC GAP NP co‐NP REG NL poly 0 0 ∃ . -
The Relative Complexity of NP Search Problems
The Relative Complexity of NP Search Problems PAUL BEAME* STEPHEN COOKT JEFF EDMONDS$ Computer Science and Engineering Computer Science Dept. I. C.S.I. University of Washington University of Toronto Berkeley, CA 94704- ”1198 beame@cs. Washington. edu sacook@cs. toronto. edu edmonds@icsi .berke,ley.edu RUSSELL IMPAGLIAZZO TONIANN PITASSI$ Computer Science Dept. Mathematics and Computer Science UCSD University of Pittsburgh russelltlcs .ucsd.edu toniflcs .pitt, edu Abstract might be that of finding a 3-coloring of the graph if it ex- ists. One can always reduce a search problem to a related Papadimitriou introduced several classes of NP search prob- decision problem and, as in the reduction of 3-coloring to lemsbased on combinatorial principles which guarantee the 3-colorability, thk is often by a natural self-reduction which existence of solutions to the problems. Many interesting produces a polynomially equivalent decision problem. search problems not known to be solvable in polynomial However, it may also happen that the related decision time are contained in these classes, and a number of them problem is not computationally equivalent to the original are complete problems. We consider the question of the rel- search problem. Thk is particularly important in the case ative complexity of these search problem classes. We prove when a solution is guaranteed to exist for the search prob- several separations which show that in a generic relativized lem. For example, consider the following prc)blems: world, the search classes are distinct and there is a standard search problem in each of them that is not computation- 1.Given a list al ,. -
Counting Below #P: Classes, Problems and Descriptive Complexity
Msc Thesis Counting below #P : Classes, problems and descriptive complexity Angeliki Chalki µΠλ8 Graduate Program in Logic, Algorithms and Computation National and Kapodistrian University of Athens Supervised by Aris Pagourtzis Advisory Committee Dimitris Fotakis Aris Pagourtzis Stathis Zachos September 2016 Abstract In this thesis, we study counting classes that lie below #P . One approach, the most regular in Computational Complexity Theory, is the machine-based approach. Classes like #L, span-L [1], and T otP ,#PE [38] are defined establishing space and time restrictions on Turing machine's computational resources. A second approach is Descriptive Complexity's approach. It characterizes complexity classes by the type of logic needed to express the languages in them. Classes deriving from this viewpoint, like #FO [44], #RHΠ1 [16], #RΣ2 [44], are equivalent to #P , the class of AP - interriducible problems to #BIS, and some subclass of the problems owning an FPRAS respectively. A great objective of such an investigation is to gain an understanding of how “efficient counting" relates to these already defined classes. By “efficient counting" we mean counting solutions of a problem using a polynomial time algorithm or an FPRAS. Many other interesting properties of the classes considered and their problems have been examined. For example alternative definitions of counting classes using relation-based op- erators, and the computational difficulty of complete problems, since complete problems capture the difficulty of the corresponding class. Moreover, in Section 3.5 we define the log- space analog of the class T otP and explore how and to what extent results can be transferred from polynomial time to logarithmic space computation. -
Contributions to the Study of Resource-Bounded Measure
Contributions to the Study of ResourceBounded Measure Elvira Mayordomo Camara Barcelona abril de Contributions to the Study of ResourceBounded Measure Tesis do ctoral presentada en el Departament de Llenguatges i Sistemes Informatics de la Universitat Politecnica de Catalunya para optar al grado de Do ctora en Ciencias Matematicas por Elvira Mayordomo Camara Dirigida p or el do ctor Jose Luis Balcazar Navarro Barcelona abril de This dissertation was presented on June st in the Departament de Llenguatges i Sistemes Informatics Universitat Politecnica de Catalunya The committee was formed by the following members Prof Ronald V Bo ok University of California at Santa Barbara Prof Ricard Gavalda Universitat Politecnica de Catalunya Prof Mario Ro drguez Universidad Complutense de Madrid Prof Marta Sanz Universitat Central de Barcelona Prof Jacob o Toran Universitat Politecnica de Catalunya Acknowledgements Iwant to thank sp ecially two p eople Jose Luis Balcazar and Jack Lutz Without their help and encouragement this dissertation would never have b een written Jose Luis Balcazar was my advisor and taughtmemostofwhatIknow ab out Structural Complexity showing me with his example how to do research I visited Jack Lutz in Iowa State University in several o ccasions from His patience and enthusiasm in explaining to me the intuition behind his theory and his numerous ideas were the starting p ointofmostoftheresults in this dissertation Several coauthors have collab orated in the publications included in the dierent chapters Jack Lutz Steven -
On Infinite Variants of NP-Complete Problems
Journal of Computer and System Sciences SS1432 journal of computer and system sciences 53, 180193 (1996) article no. 0060 View metadata, citation and similarTaking papers at core.ac.uk It to the Limit: On Infinite Variants brought to you by CORE of NP-Complete Problems provided by Elsevier - Publisher Connector Tirza Hirst and David Harel* Department of Applied Mathematics H Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel Received November 1, 1993 We first set up a general way of obtaining infinite versions We define infinite, recursive versions of NP optimization problems. of NP maximization and minimization problems. If P is the For example, MAX CLIQUE becomes the question of whether a recursive problem that asks for a maximum by graph contains an infinite clique. The present paper was motivated by trying to understand what makes some NP problems highly max |[wÄ : A<,(wÄ , S)]|, undecidable in the infinite case, while others remain on low levels of S the arithmetical hierarchy. We prove two results; one enables using knowledge about the infinite case to yield implications to the finite where ,(wÄ , S) is first-order and A is a finite structure (say, case, and the other enables implications in the other direction. a graph), we define P as the problem that asks whether Moreover, taken together, the two results provide a method for proving there is an S, such that the set [wÄ : A<,(w, S)] is infinite, (finitary) problems to be outside the syntactic class MAX NP, and, where A is an infinite recursive structure. Thus, for example, hence, outside MAX SNP too, by showing that their infinite versions are 1 Max Clique becomes the question of whether a recursive 71-complete.