Theory of Computation
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Computational Complexity Theory Introduction to Computational
31/03/2012 Computational complexity theory Introduction to computational complexity theory Complexity (computability) theory theory deals with two aspects: Algorithm’s complexity. Problem’s complexity. References S. Cook, « The complexity of Theorem Proving Procedures », 1971. Garey and Johnson, « Computers and Intractability, A guide to the theory of NP-completeness », 1979. J. Carlier et Ph. Chrétienne « Problèmes d’ordonnancements : algorithmes et complexité », 1988. 1 31/03/2012 Basic Notions • Some problem is a “question” characterized by parameters and needs an answer. – Parameters description; – Properties that a solutions must satisfy; – An instance is obtained when the parameters are fixed to some values. • An algorithm: a set of instructions describing how some task can be achieved or a problem can be solved. • A program : the computational implementation of an algorithm. Algorithm’s complexity (I) • There may exists several algorithms for the same problem • Raised questions: – Which one to choose ? – How they are compared ? – How measuring the efficiency ? – What are the most appropriate measures, running time, memory space ? 2 31/03/2012 Algorithm’s complexity (II) • Running time depends on: – The data of the problem, – Quality of program..., – Computer type, – Algorithm’s efficiency, – etc. • Proceed by analyzing the algorithm: – Search for some n characterizing the data. – Compute the running time in terms of n. – Evaluating the number of elementary operations, (elementary operation = simple instruction of a programming language). Algorithm’s evaluation (I) • Any algorithm is composed of two main stages: initialization and computing one • The complexity parameter is the size data n (binary coding). Definition: Let be n>0 andT(n) the running time of an algorithm expressed in terms of the size data n, T(n) is of O(f(n)) iff n0 and some constant c such that: n n0, we have T(n) c f(n). -
Computability of Fraïssé Limits
COMPUTABILITY OF FRA¨ISSE´ LIMITS BARBARA F. CSIMA, VALENTINA S. HARIZANOV, RUSSELL MILLER, AND ANTONIO MONTALBAN´ Abstract. Fra¨ıss´estudied countable structures S through analysis of the age of S, i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fra¨ıss´elimit. We also show that degree spectra of relations on a sufficiently nice Fra¨ıss´elimit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fra¨ıss´elimit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal. Contents 1. Introduction1 1.1. Classical results about Fra¨ıss´elimits and background definitions4 2. Computable Ages5 3. Computable Fra¨ıss´elimits8 3.1. Computable properties of Fra¨ıss´elimits8 3.2. Existence of computable Fra¨ıss´elimits9 4. Examples 15 5. Upward closure of degree spectra of relations 18 6. Necessary conditions for spectral universality 20 6.1. Local finiteness 20 6.2. Finite realizability 21 7. A sufficient condition for spectral universality 22 7.1. The countable atomless Boolean algebra 23 References 24 1. Introduction Computable model theory studies the algorithmic complexity of countable structures, of their isomorphisms, and of relations on such structures. Since algorithmic properties often depend on data presentation, in computable model theory classically isomorphic structures can have different computability-theoretic properties. -
Submitted Thesis
Saarland University Faculty of Mathematics and Computer Science Bachelor’s Thesis Synthetic One-One, Many-One, and Truth-Table Reducibility in Coq Author Advisor Felix Jahn Yannick Forster Reviewers Prof. Dr. Gert Smolka Prof. Dr. Markus Bläser Submitted: September 15th, 2020 ii Eidesstattliche Erklärung Ich erkläre hiermit an Eides statt, dass ich die vorliegende Arbeit selbstständig ver- fasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverständniserklärung Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit veröffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrücken, September 15th, 2020 Abstract Reducibility is an essential concept for undecidability proofs in computability the- ory. The idea behind reductions was conceived by Turing, who introduced the later so-called Turing reduction based on oracle machines. In 1944, Post furthermore in- troduced with one-one, many-one, and truth-table reductions in comparison to Tur- ing reductions more specific reducibility notions. Post then also started to analyze the structure of the different reducibility notions and their computability degrees. Most undecidable problems were reducible from the halting problem, since this was exactly the method to show them undecidable. However, Post was able to con- struct also semidecidable but undecidable sets that do not one-one, many-one, or truth-table reduce from the halting problem. -
Module 34: Reductions and NP-Complete Proofs
Module 34: Reductions and NP-Complete Proofs This module 34 focuses on reductions and proof of NP-Complete problems. The module illustrates the ways of reducing one problem to another. Then, the module illustrates the outlines of proof of NP- Complete problems with some examples. The objectives of this module are To understand the concept of Reductions among problems To Understand proof of NP-Complete problems To overview proof of some Important NP-Complete problems. Computational Complexity There are two types of complexity theories. One is related to algorithms, known as algorithmic complexity theory and another related to complexity of problems called computational complexity theory [2,3]. Algorithmic complexity theory [3,4] aims to analyze the algorithms in terms of the size of the problem. In modules 3 and 4, we had discussed about these methods. The size is the length of the input. It can be recollected from module 3 that the size of a number n is defined to be the number of binary bits needed to write n. For example, Example: b(5) = b(1012) = 3. In other words, the complexity of the algorithm is stated in terms of the size of the algorithm. Asymptotic Analysis [1,2] refers to the study of an algorithm as the input size reaches a limit and analyzing the behaviour of the algorithms. The asymptotic analysis is also science of approximation where the behaviour of the algorithm is expressed in terms of notations such as big-oh, Big-Omega and Big- Theta. Computational complexity theory is different. Computational complexity aims to determine complexity of the problems itself. -
Computability and Complexity
Computability and Complexity Lecture Notes Winter Semester 2016/2017 Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria [email protected] July 18, 2016 Contents List of Definitions5 List of Theorems7 List of Theses9 List of Figures9 Notions, Notations, Translations 12 1. Introduction 18 I. Computability 23 2. Finite State Machines and Regular Languages 24 2.1. Deterministic Finite State Machines...................... 25 2.2. Nondeterministic Finite State Machines.................... 30 2.3. Minimization of Finite State Machines..................... 38 2.4. Regular Languages and Finite State Machines................. 43 2.5. The Expressiveness of Regular Languages................... 58 3. Turing Complete Computational Models 63 3.1. Turing Machines................................ 63 3.1.1. Basics.................................. 63 3.1.2. Recognizing Languages........................ 68 3.1.3. Generating Languages......................... 72 3.1.4. Computing Functions.......................... 74 3.1.5. The Church-Turing Thesis....................... 79 Contents 3 3.2. Turing Complete Computational Models.................... 80 3.2.1. Random Access Machines....................... 80 3.2.2. Loop and While Programs....................... 84 3.2.3. Primitive Recursive and µ-recursive Functions............ 95 3.2.4. Further Models............................. 106 3.3. The Chomsky Hierarchy............................ 111 3.4. Real Computers................................ -
31 Summary of Computability Theory
CS:4330 Theory of Computation Spring 2018 Computability Theory Summary Haniel Barbosa Readings for this lecture Chapters 3-5 and Section 6.2 of [Sipser 1996], 3rd edition. A hierachy of languages n m B Regular: a b n n B Deterministic Context-free: a b n n n 2n B Context-free: a b [ a b n n n B Turing decidable: a b c B Turing recognizable: ATM 1 / 12 Why TMs? B In 1900: Hilbert posed 23 “challenge problems” in Mathematics The 10th problem: Devise a process according to which it can be decided by a finite number of operations if a given polynomial has an integral root. It became necessary to have a formal definition of “algorithms” to define their expressivity. 2 / 12 Church-Turing Thesis B In 1936 Church and Turing independently defined “algorithm”: I λ-calculus I Turing machines B Intuitive notion of algorithms = Turing machine algorithms B “Any process which could be naturally called an effective procedure can be realized by a Turing machine” th B We now know: Hilbert’s 10 problem is undecidable! 3 / 12 Algorithm as Turing Machine Definition (Algorithm) An algorithm is a decider TM in the standard representation. B The input to a TM is always a string. B If we want an object other than a string as input, we must first represent that object as a string. B Strings can easily represent polynomials, graphs, grammars, automata, and any combination of these objects. 4 / 12 How to determine decidability / Turing-recognizability? B Decidable / Turing-recognizable: I Present a TM that decides (recognizes) the language I If A is mapping reducible to -
CS 2210: Theory of Computation
CS 2210: Theory of Computation Spring, 2019 Administrative Information • Background survey • Textbook: E. Rich, Automata, Computability, and Complexity: Theory and Applications, Prentice-Hall, 2008. • Book website: http://www.cs.utexas.edu/~ear/cs341/automatabook/ • My Office Hours: – Monday, 6:00-8:00pm, Searles 224 – Tuesday, 1:00-2:30pm, Searles 222 • TAs – Anjulee Bhalla: Hours TBA, Searles 224 – Ryan St. Pierre, Hours TBA, Searles 224 What you can expect from the course • How to do proofs • Models of computation • What’s the difference between computability and complexity? • What’s the Halting Problem? • What are P and NP? • Why do we care whether P = NP? • What are NP-complete problems? • Where does this make a difference outside of this class? • How to work the answers to these questions into the conversation at a cocktail party… What I will expect from you • Problem Sets (25%): – Goal: Problems given on Mondays and Wednesdays – Due the next Monday – Graded by following Monday – A learning tool, not a testing tool – Collaboration encouraged; more on this in next slide • Quizzes (15%) • Exams (2 non-cumulative, 30% each): – Closed book, closed notes, but… – Can bring in 8.5 x 11 page with notes on both sides • Class participation: Tiebreaker Other Important Things • Go to the TA hours • Study and work on problem sets in groups • Collaboration Issues: – Level 0 (In-Class Problems) • No restrictions – Level 1 (Homework Problems) • Verbal collaboration • But, individual write-ups – Level 2 (not used in this course) • Discussion with TAs only – Level 3 (Exams) • Professor clarifications only Right now… • What does it mean to study the “theory” of something? • Experience with theory in other disciplines? • Relationship to practice? – “In theory, theory and practice are the same. -
The Problem with Threads
The Problem with Threads Edward A. Lee Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2006-1 http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-1.html January 10, 2006 Copyright © 2006, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Acknowledgement This work was supported in part by the Center for Hybrid and Embedded Software Systems (CHESS) at UC Berkeley, which receives support from the National Science Foundation (NSF award No. CCR-0225610), the State of California Micro Program, and the following companies: Agilent, DGIST, General Motors, Hewlett Packard, Infineon, Microsoft, and Toyota. The Problem with Threads Edward A. Lee Professor, Chair of EE, Associate Chair of EECS EECS Department University of California at Berkeley Berkeley, CA 94720, U.S.A. [email protected] January 10, 2006 Abstract Threads are a seemingly straightforward adaptation of the dominant sequential model of computation to concurrent systems. Languages require little or no syntactic changes to sup- port threads, and operating systems and architectures have evolved to efficiently support them. Many technologists are pushing for increased use of multithreading in software in order to take advantage of the predicted increases in parallelism in computer architectures. -
Computer Architecture: Dataflow (Part I)
Computer Architecture: Dataflow (Part I) Prof. Onur Mutlu Carnegie Mellon University A Note on This Lecture n These slides are from 18-742 Fall 2012, Parallel Computer Architecture, Lecture 22: Dataflow I n Video: n http://www.youtube.com/watch? v=D2uue7izU2c&list=PL5PHm2jkkXmh4cDkC3s1VBB7- njlgiG5d&index=19 2 Some Required Dataflow Readings n Dataflow at the ISA level q Dennis and Misunas, “A Preliminary Architecture for a Basic Data Flow Processor,” ISCA 1974. q Arvind and Nikhil, “Executing a Program on the MIT Tagged- Token Dataflow Architecture,” IEEE TC 1990. n Restricted Dataflow q Patt et al., “HPS, a new microarchitecture: rationale and introduction,” MICRO 1985. q Patt et al., “Critical issues regarding HPS, a high performance microarchitecture,” MICRO 1985. 3 Other Related Recommended Readings n Dataflow n Gurd et al., “The Manchester prototype dataflow computer,” CACM 1985. n Lee and Hurson, “Dataflow Architectures and Multithreading,” IEEE Computer 1994. n Restricted Dataflow q Sankaralingam et al., “Exploiting ILP, TLP and DLP with the Polymorphous TRIPS Architecture,” ISCA 2003. q Burger et al., “Scaling to the End of Silicon with EDGE Architectures,” IEEE Computer 2004. 4 Today n Start Dataflow 5 Data Flow Readings: Data Flow (I) n Dennis and Misunas, “A Preliminary Architecture for a Basic Data Flow Processor,” ISCA 1974. n Treleaven et al., “Data-Driven and Demand-Driven Computer Architecture,” ACM Computing Surveys 1982. n Veen, “Dataflow Machine Architecture,” ACM Computing Surveys 1986. n Gurd et al., “The Manchester prototype dataflow computer,” CACM 1985. n Arvind and Nikhil, “Executing a Program on the MIT Tagged-Token Dataflow Architecture,” IEEE TC 1990. -
Decidable. (Recall It’S Not “Regular”.)
15-251: Great Theoretical Ideas in Computer Science Fall 2016 Lecture 6 September 15, 2016 Turing & the Uncomputable Comparing the cardinality of sets 퐴 ≤ 퐵 if there is an injection (one-to-one map) from 퐴 to 퐵 퐴 ≥ 퐵 if there is a surjection (onto map) from 퐴 to 퐵 퐴 = 퐵 if there is a bijection from 퐴 to 퐵 퐴 > |퐵| if there is no surjection from 퐵 to 퐴 (or equivalently, there is no injection from 퐴 to 퐵) Countable and uncountable sets countable countably infinite uncountable One slide guide to countability questions You are given a set 퐴 : is it countable or uncountable 퐴 ≤ |ℕ| or 퐴 > |ℕ| 퐴 ≤ |ℕ| : • Show directly surjection from ℕ to 퐴 • Show that 퐴 ≤ |퐵| where 퐵 ∈ {ℤ, ℤ x ℤ, ℚ, Σ∗, ℚ[x], …} 퐴 > |ℕ| : • Show directly using a diagonalization argument • Show that 퐴 ≥ | 0,1 ∞| Proving sets countable using computation For example, f(n) = ‘the nth prime’. You could write a program (Turing machine) to compute f. So this is a well-defined rule. Or: f(n) = the nth rational in our listing of ℚ. (List ℤ2 via the spiral, omit the terms p/0, omit rationals seen before…) You could write a program to compute this f. Poll Let 퐴 be the set of all languages over Σ = 1 ∗ Select the correct ones: - A is finite - A is infinite - A is countable - A is uncountable Another thing to remember from last week Encoding different objects with strings Fix some alphabet Σ . We use the ⋅ notation to denote the encoding of an object as a string in Σ∗ Examples: is the encoding a TM 푀 is the encoding a DFA 퐷 is the encoding of a pair of TMs 푀1, 푀2 is the encoding a pair 푀, 푥, where 푀 is a TM, and 푥 ∈ Σ∗ is an input to 푀 Uncountable to uncomputable The real number 1/7 is “computable”. -
Ece585 Lec2.Pdf
ECE 485/585 Microprocessor System Design Lecture 2: Memory Addressing 8086 Basics and Bus Timing Asynchronous I/O Signaling Zeshan Chishti Electrical and Computer Engineering Dept Maseeh College of Engineering and Computer Science Source: Lecture based on materials provided by Mark F. Basic I/O – Part I ECE 485/585 Outline for next few lectures Simple model of computation Memory Addressing (Alignment, Byte Order) 8088/8086 Bus Asynchronous I/O Signaling Review of Basic I/O How is I/O performed Dedicated/Isolated /Direct I/O Ports Memory Mapped I/O How do we tell when I/O device is ready or command complete? Polling Interrupts How do we transfer data? Programmed I/O DMA ECE 485/585 Simplified Model of a Computer Control Control Data, Address, Memory Data Path Microprocessor Keyboard Mouse [Fetch] Video display [Decode] Printer [Execute] I/O Device Hard disk drive Audio card Ethernet WiFi CD R/W DVD ECE 485/585 Memory Addressing Size of operands Bytes, words, long/double words, quadwords 16-bit half word (Intel: word) 32-bit word (Intel: doubleword, dword) 0x107 64-bit double word (Intel: quadword, qword) 0x106 Note: names are non-standard 0x105 SUN Sparc word is 32-bits, double is 64-bits 0x104 0x103 Alignment 0x102 Can multi-byte operands begin at any byte address? 0x101 Yes: non-aligned 0x100 No: aligned. Low order address bit(s) will be zero ECE 485/585 Memory Operand Alignment …Intel IA speak (i.e. word = 16-bits = 2 bytes) 0x107 0x106 0x105 0x104 0x103 0x102 0x101 0x100 Aligned Unaligned Aligned Unaligned Aligned Unaligned word word Double Double Quad Quad address address word word word word -----0 address address address address -----00 ----000 ECE 485/585 Memory Operand Alignment Why do we care? Unaligned memory references Can cause multiple memory bus cycles for a single operand May also span cache lines Requiring multiple evictions, multiple cache line fills Complicates memory system and cache controller design Some architectures restrict addresses to be aligned Even in architectures without alignment restrictions (e.g. -
Computability on the Integers
Computability on the integers Laurent Bienvenu ( LIAFA, CNRS & Université de Paris 7 ) EJCIM 2011 Amiens, France 30 mars 2011 1. Basic objects of computability The formalization and study of the notion of computable function is what computability theory is about. As opposed to complexity theory, we do not care about efficiency, just about feasibility. Computable. functions What does it means for a function f : N ! N to be computable? 1. Basic objects of computability 3/79 As opposed to complexity theory, we do not care about efficiency, just about feasibility. Computable. functions What does it means for a function f : N ! N to be computable? The formalization and study of the notion of computable function is what computability theory is about. 1. Basic objects of computability 3/79 Computable. functions What does it means for a function f : N ! N to be computable? The formalization and study of the notion of computable function is what computability theory is about. As opposed to complexity theory, we do not care about efficiency, just about feasibility. 1. Basic objects of computability 3/79 But for us, it is now obvious that computable = realizable by a program / algorithm. Surprisingly, the first acceptable formalization (Turing machines) is still one of the best (if not the best) we know today. The. intuition At the time these questions were first considered (1930’s), computers did not exist (at least in the modern sense). 1. Basic objects of computability 4/79 Surprisingly, the first acceptable formalization (Turing machines) is still one of the best (if not the best) we know today.