# Introduction to Automata Theory. Reading: Chapter 1

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Introduction to Automata Theory Reading: Chapter 1 1

2 What is Automata Theory? Study of abstract computing devices, or machines Automaton = an abstract computing device Note: A device need not even be a physical hardware! A fundamental question in computer science: Find out what different models of machines can do and cannot do The theory of computation Computability vs. Complexity 2

3 (A pioneer of automata theory) Alan Turing ( ) Father of Modern Computer Science English mathematician Studied abstract machines called Turing machines even before computers existed Heard of the Turing test? 3

4 Theory of Computation: A Historical Perspective 1930s Alan Turing studies Turing machines Decidability Halting problem s Finite automata machines studied Noam Chomsky proposes the Chomsky Hierarchy for formal languages 1969 Cook introduces intractable problems or NP-Hard problems Modern computer science: compilers, computational & complexity theory evolve 4

5 Languages & Grammars Or words Languages: A language is a collection of sentences of finite length all constructed from a finite alphabet of symbols Grammars: A grammar can be regarded as a device that enumerates the sentences of a language - nothing more, nothing less N. Chomsky, Information and Control, Vol 2, 1959 Image source: Nowak et al. Nature, vol 417,

6 The Chomsky Hierachy A containment hierarchy of classes of formal languages Regular (DFA) Contextfree (PDA) Contextsensitive (LBA) Recursivelyenumerable (TM) 6

7 The Central Concepts of Automata Theory 7

8 Alphabet An alphabet is a finite, non-empty set of symbols We use the symbol (sigma) to denote an alphabet Examples: Binary: = {0,1} All lower case letters: = {a,b,c,..z} Alphanumeric: = {a-z, A-Z, 0-9} DNA molecule letters: = {a,c,g,t} 8

9 Strings A string or word is a finite sequence of symbols chosen from Empty string is ε (or epsilon ) Length of a string w, denoted by w, is equal to the number of (non- ε) characters in the string E.g., x = x = 6 x = 01 ε 0 ε 1 ε 00 ε x =? xy = concatentation of two strings x and y 9

10 Powers of an alphabet Let be an alphabet. k = the set of all strings of length k * = 0 U 1 U 2 U + = 1 U 2 U 3 U 10

11 Languages L is a said to be a language over alphabet, only if L * this is because * is the set of all strings (of all possible length including 0) over the given alphabet Examples: 1. Let L be the language of all strings consisting of n 0 s followed by n 1 s: L = {ε,01,0011,000111, } 2. Let L be the language of all strings of with equal number of 0 s and 1 s: L = {ε,01,10,0011,1100,0101,1010,1001, } Canonical ordering of strings in the language Definition: Ø denotes the Empty language Let L = {ε}; Is L=Ø? NO 11

12 The Membership Problem Given a string w *and a language L over, decide whether or not w L. Example: Let w = Q) Is w the language of strings with equal number of 0s and 1s? 12

13 Finite Automata Some Applications Software for designing and checking the behavior of digital circuits Lexical analyzer of a typical compiler Software for scanning large bodies of text (e.g., web pages) for pattern finding Software for verifying systems of all types that have a finite number of states (e.g., stock market transaction, communication/network protocol) 13

14 Finite Automata : Examples On/Off switch action state Modeling recognition of the word then Start state Transition Intermediate state Final state 14

15 Structural expressions Grammars Regular expressions E.g., unix style to capture city names such as Palo Alto CA : [A-Z][a-z]*([ ][A-Z][a-z]*)*[ ][A-Z][A-Z] Start with a letter A string of other letters (possibly empty) Should end w/ 2-letter state code Other space delimited words (part of city name) 15

16 Formal Proofs 16

17 Deductive Proofs From the given statement(s) to a conclusion statement (what we want to prove) Logical progression by direct implications Example for parsing a statement: If y 4, then 2 y y 2. given conclusion (there are other ways of writing this). 17

18 Example: Deductive proof Let Claim 1: If y 4, then 2 y y 2. Let x be any number which is obtained by adding the squares of 4 positive integers. Claim 2: Given x and assuming that Claim 1 is true, prove that 2 x x 2 Proof: 1) Given: x = a 2 + b 2 + c 2 + d 2 2) Given: a 1, b 1, c 1, d 1 3) a 2 1, b 2 1, c 2 1, d 2 1 (by 2) 4) x 4 (by 1 & 3) 5) 2 x x 2 (by 4 and Claim 1) implies or follows 18

19 On Theorems, Lemmas and Corollaries We typically refer to: A major result as a theorem An intermediate result that we show to prove a larger result as a lemma A result that follows from an already proven result as a corollary An example: Theorem: The height of an n-node binary tree is at least floor(lg n) Lemma: Level i of a perfect binary tree has 2 i nodes. Corollary: A perfect binary tree of height h has 2 h+1-1 nodes. 19

20 Quantifiers For all or For every Universal proofs Notation * =? There exists Used in existential proofs Notation * =? Implication is denoted by => E.g., IF A THEN B can also be written as A=>B * I wasn t able to locate the symbol for these notation in powerpoint. Sorry! Please follow the standard notation for these quantifiers. These will be presented in class. 20

21 Proving techniques By contradiction Start with the statement contradictory to the given statement E.g., To prove (A => B), we start with: (A and ~B) and then show that could never happen What if you want to prove that (A and B => C or D)? By induction (3 steps) Basis, inductive hypothesis, inductive step By contrapositive statement If A then B If ~B then ~A 21

22 Proving techniques By counter-example Show an example that disproves the claim Note: There is no such thing called a proof by example! So when asked to prove a claim, an example that satisfied that claim is not a proof 22

23 Different ways of saying the same thing If H then C : i. H implies C ii. iii. iv. H => C C if H H only if C v. Whenever H holds, C follows 23

24 If-and-Only-If statements A if and only if B (A <==> B) (if part) if B then A ( <= ) (only if part) A only if B ( => ) (same as if A then B ) If and only if is abbreviated as iff i.e., A iff B Example: Theorem: Let x be a real number. Then floor of x = ceiling of x if and only if x is an integer. Proofs for iff have two parts One for the if part & another for the only if part 24

25 Summary Automata theory & a historical perspective Chomsky hierarchy Finite automata Alphabets, strings/words/sentences, languages Membership problem Proofs: Deductive, induction, contrapositive, contradiction, counterexample If and only if Read chapter 1 for more examples and exercises 25

### Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi

Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the

### Automata Theory and Languages

Automata Theory and Languages SITE : http://www.info.univ-tours.fr/ mirian/ Automata Theory, Languages and Computation - Mírian Halfeld-Ferrari p. 1/1 Introduction to Automata Theory Automata theory :

### The Halting Problem is Undecidable

185 Corollary G = { M, w w L(M) } is not Turing-recognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine

### FORMAL LANGUAGES, AUTOMATA AND COMPUTATION

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION REDUCIBILITY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 1 / 20 THE LANDSCAPE OF THE CHOMSKY HIERARCHY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 2 / 20 REDUCIBILITY

### Finite Automata and Formal Languages

Finite Automata and Formal Languages TMV026/DIT321 LP4 2011 Lecture 14 May 19th 2011 Overview of today s lecture: Turing Machines Push-down Automata Overview of the Course Undecidable and Intractable Problems

### Theory of Computation

Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s

### Motivation. Automata = abstract computing devices. Turing studied Turing Machines (= computers) before there were any real computers

Motivation Automata = abstract computing devices Turing studied Turing Machines (= computers) before there were any real computers We will also look at simpler devices than Turing machines (Finite State

### Finite Automata. Reading: Chapter 2

Finite Automata Reading: Chapter 2 1 Finite Automaton (FA) Informally, a state diagram that comprehensively captures all possible states and transitions that a machine can take while responding to a stream

### Overview of E0222: Automata and Computability

Overview of E0222: Automata and Computability Deepak D Souza Department of Computer Science and Automation Indian Institute of Science, Bangalore. August 3, 2011 What this course is about What we study

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Compiler Construction

Compiler Construction Regular expressions Scanning Görel Hedin Reviderad 2013 01 23.a 2013 Compiler Construction 2013 F02-1 Compiler overview source code lexical analysis tokens intermediate code generation

### Turing Machines: An Introduction

CIT 596 Theory of Computation 1 We have seen several abstract models of computing devices: Deterministic Finite Automata, Nondeterministic Finite Automata, Nondeterministic Finite Automata with ɛ-transitions,

### Finite Automata and Formal Languages

Finite Automata and Formal Languages TMV026/DIT321 LP4 2011 Ana Bove Lecture 1 March 21st 2011 Course Organisation Overview of the Course Overview of today s lecture: Course Organisation Level: This course

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### Model 2.4 Faculty member + student

Model 2.4 Faculty member + student Course syllabus for Formal languages and Automata Theory. Faculty member information: Name of faculty member responsible for the course Office Hours Office Number Email

### Definition: String concatenation. Definition: String. Definition: Language (cont.) Definition: Language

CMSC 330: Organization of Programming Languages Regular Expressions and Finite Automata Introduction That s it for the basics of Ruby If you need other material for your project, come to office hours or

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

### Automata and Computability. Solutions to Exercises

Automata and Computability Solutions to Exercises Fall 25 Alexis Maciel Department of Computer Science Clarkson University Copyright c 25 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata

### Finite Automata. Reading: Chapter 2

Finite Automata Reading: Chapter 2 1 Finite Automata Informally, a state machine that comprehensively captures all possible states and transitions that a machine can take while responding to a stream (or

### Introduction to Turing Machines

Automata Theory, Languages and Computation - Mírian Halfeld-Ferrari p. 1/2 Introduction to Turing Machines SITE : http://www.sir.blois.univ-tours.fr/ mirian/ Automata Theory, Languages and Computation

CS5236 Advanced Automata Theory Frank Stephan Semester I, Academic Year 2012-2013 Advanced Automata Theory is a lecture which will first review the basics of formal languages and automata theory and then

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### 3515ICT Theory of Computation Turing Machines

Griffith University 3515ICT Theory of Computation Turing Machines (Based loosely on slides by Harald Søndergaard of The University of Melbourne) 9-0 Overview Turing machines: a general model of computation

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### Computing Functions with Turing Machines

CS 30 - Lecture 20 Combining Turing Machines and Turing s Thesis Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata

### Honors Class (Foundations of) Informatics. Tom Verhoeff. Department of Mathematics & Computer Science Software Engineering & Technology

Honors Class (Foundations of) Informatics Tom Verhoeff Department of Mathematics & Computer Science Software Engineering & Technology www.win.tue.nl/~wstomv/edu/hci c 2011, T. Verhoeff @ TUE.NL 1/20 Information

### Automata and Languages

Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful

### Computability Theory

CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 1-10.

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### 31 is a prime number is a mathematical statement (which happens to be true).

Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to

### Mathematical Induction. Lecture 10-11

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

### (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems.

3130CIT: Theory of Computation Turing machines and undecidability (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems. An undecidable problem

### SRM UNIVERSITY FACULTY OF ENGINEERING & TECHNOLOGY SCHOOL OF COMPUTING DEPARTMENT OF SOFTWARE ENGINEERING COURSE PLAN

Course Code : CS0355 SRM UNIVERSITY FACULTY OF ENGINEERING & TECHNOLOGY SCHOOL OF COMPUTING DEPARTMENT OF SOFTWARE ENGINEERING COURSE PLAN Course Title : THEORY OF COMPUTATION Semester : VI Course : June

### Regular Languages and Finite State Machines

Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries - some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection

### Introduction to Proofs

Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Introduction to Theory of Computation

Introduction to Theory of Computation Prof. (Dr.) K.R. Chowdhary Email: kr.chowdhary@iitj.ac.in Formerly at department of Computer Science and Engineering MBM Engineering College, Jodhpur Tuesday 28 th

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

### CSE 135: Introduction to Theory of Computation Decidability and Recognizability

CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28, 30-2014 High-Level Descriptions of Computation Instead of giving a Turing

### Regular Languages and Finite Automata

Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a

### Computational Models Lecture 8, Spring 2009

Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 1 Computational Models Lecture 8, Spring 2009 Encoding of TMs Universal Turing Machines The Halting/Acceptance

### A Formalization of the Turing Test Evgeny Chutchev

A Formalization of the Turing Test Evgeny Chutchev 1. Introduction The Turing test was described by A. Turing in his paper [4] as follows: An interrogator questions both Turing machine and second participant

### Models of Sequential Computation

Models of Sequential Computation 13 C H A P T E R Programming is often considered as an art, but it raises mostly practical questions like which language should I choose, which algorithm is adequate, and

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### Deterministic Finite Automata

1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function

### Pushdown Automata. place the input head on the leftmost input symbol. while symbol read = b and pile contains discs advance head remove disc from pile

Pushdown Automata In the last section we found that restricting the computational power of computing devices produced solvable decision problems for the class of sets accepted by finite automata. But along

### Formal Grammars and Languages

Formal Grammars and Languages Tao Jiang Department of Computer Science McMaster University Hamilton, Ontario L8S 4K1, Canada Bala Ravikumar Department of Computer Science University of Rhode Island Kingston,

### Mathematical Induction

MCS-236: Graph Theory Handout #A5 San Skulrattanakulchai Gustavus Adolphus College Sep 15, 2010 Mathematical Induction The following three principles governing N are equivalent. Ordinary Induction Principle.

### Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### Philadelphia University Faculty of Information Technology Department of Computer Science First Semester, 2007/2008.

Philadelphia University Faculty of Information Technology Department of Computer Science First Semester, 2007/2008 Course Syllabus Course Title: Theory of Computation Course Level: 3 Lecture Time: Course

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### Informatique Fondamentale IMA S8

Informatique Fondamentale IMA S8 Cours 1 - Intro + schedule + finite state machines Laure Gonnord http://laure.gonnord.org/pro/teaching/ Laure.Gonnord@polytech-lille.fr Université Lille 1 - Polytech Lille

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### Pushdown automata. Informatics 2A: Lecture 9. Alex Simpson. 3 October, 2014. School of Informatics University of Edinburgh als@inf.ed.ac.

Pushdown automata Informatics 2A: Lecture 9 Alex Simpson School of Informatics University of Edinburgh als@inf.ed.ac.uk 3 October, 2014 1 / 17 Recap of lecture 8 Context-free languages are defined by context-free

### Genetic programming with regular expressions

Genetic programming with regular expressions Børge Svingen Chief Technology Officer, Open AdExchange bsvingen@openadex.com 2009-03-23 Pattern discovery Pattern discovery: Recognizing patterns that characterize

### T-79.186 Reactive Systems: Introduction and Finite State Automata

T-79.186 Reactive Systems: Introduction and Finite State Automata Timo Latvala 14.1.2004 Reactive Systems: Introduction and Finite State Automata 1-1 Reactive Systems Reactive systems are a class of software

### LOGICAL INFERENCE & PROOFs. Debdeep Mukhopadhyay Dept of CSE, IIT Madras

LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a

### 24 Uses of Turing Machines

Formal Language and Automata Theory: CS2004 24 Uses of Turing Machines 24 Introduction We have previously covered the application of Turing Machine as a recognizer and decider In this lecture we will discuss

### Automata on Infinite Words and Trees

Automata on Infinite Words and Trees Course notes for the course Automata on Infinite Words and Trees given by Dr. Meghyn Bienvenu at Universität Bremen in the 2009-2010 winter semester Last modified:

### Context-Free Grammars

Context-Free Grammars Describing Languages We've seen two models for the regular languages: Automata accept precisely the strings in the language. Regular expressions describe precisely the strings in

### Quantum and Non-deterministic computers facing NP-completeness

Quantum and Non-deterministic computers facing NP-completeness Thibaut University of Vienna Dept. of Business Administration Austria Vienna January 29th, 2013 Some pictures come from Wikipedia Introduction

### 5HFDOO &RPSLOHU 6WUXFWXUH

6FDQQLQJ 2XWOLQH 2. Scanning The basics Ad-hoc scanning FSM based techniques A Lexical Analysis tool - Lex (a scanner generator) 5HFDOO &RPSLOHU 6WUXFWXUH 6RXUFH &RGH /H[LFDO \$QDO\VLV6FDQQLQJ 6\QWD[ \$QDO\VLV3DUVLQJ

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### The P versus NP Solution

The P versus NP Solution Frank Vega To cite this version: Frank Vega. The P versus NP Solution. 2015. HAL Id: hal-01143424 https://hal.archives-ouvertes.fr/hal-01143424 Submitted on 17 Apr

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy

Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy Kim S. Larsen Odense University Abstract For many years, regular expressions with back referencing have been used in a variety

### Context-Free Grammars (CFG)

Context-Free Grammars (CFG) SITE : http://www.sir.blois.univ-tours.fr/ mirian/ Automata Theory, Languages and Computation - Mírian Halfeld-Ferrari p. 1/2 An informal example Language of palindromes: L

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### CSC4510 AUTOMATA 2.1 Finite Automata: Examples and D efinitions Definitions

CSC45 AUTOMATA 2. Finite Automata: Examples and Definitions Finite Automata: Examples and Definitions A finite automaton is a simple type of computer. Itsoutputislimitedto yes to or no. It has very primitive

### Chapter 7 Uncomputability

Chapter 7 Uncomputability 190 7.1 Introduction Undecidability of concrete problems. First undecidable problem obtained by diagonalisation. Other undecidable problems obtained by means of the reduction

### Special course in Computer Science: Molecular Computing

Special course in Computer Science: Molecular Computing Lecture 7: DNA Computing by Splicing and by Insertion-Deletion, basics of formal languages and computability Vladimir Rogojin Department of IT, Åbo

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### conditional statement conclusion Vocabulary Flash Cards Chapter 2 (p. 66) Chapter 2 (p. 69) Chapter 2 (p. 66) Chapter 2 (p. 76)

biconditional statement conclusion Chapter 2 (p. 69) conditional statement conjecture Chapter 2 (p. 76) contrapositive converse Chapter 2 (p. 67) Chapter 2 (p. 67) counterexample deductive reasoning Chapter

### Fundamentals of Mathematics Lecture 6: Propositional Logic

Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### Practice Problems for Final Exam: Solutions CS 341: Foundations of Computer Science II Prof. Marvin K. Nakayama

Practice Problems for Final Exam: Solutions CS 341: Foundations of Computer Science II Prof. Marvin K. Nakayama 1. Short answers: (a) Define the following terms and concepts: i. Union, intersection, set

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER

THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER JASPER DEANTONIO Abstract. This paper is a study of the Turing Degrees, which are levels of incomputability naturally arising from sets of natural numbers.

### Computer Architecture Syllabus of Qualifying Examination

Computer Architecture Syllabus of Qualifying Examination PhD in Engineering with a focus in Computer Science Reference course: CS 5200 Computer Architecture, College of EAS, UCCS Created by Prof. Xiaobo

### Computing basics. Ruurd Kuiper

Computing basics Ruurd Kuiper October 29, 2009 Overview (cf Schaum Chapter 1) Basic computing science is about using computers to do things for us. These things amount to processing data. The way a computer

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### 6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010. Class 4 Nancy Lynch

6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 4 Nancy Lynch Today Two more models of computation: Nondeterministic Finite Automata (NFAs)

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### 6.080/6.089 GITCS Feb 12, 2008. Lecture 3

6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my

### A Fixed-Point Approach towards Efficient Models Conversion

ISSN: 1940-8978 A Fixed-Point Approach towards Efficient Models Conversion Stefan Andrei, Hikyoo Koh No. 2, June 2008 Computer Science Department Technical Reports - Lamar University Computer Science Department,

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### Chapter I Logic and Proofs

MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than

### Master s Thesis. One-Pass Strategies for Context-Free Games. Christian Cöster August 2015

Master s Thesis One-Pass Strategies for Context-Free Games Christian Cöster August 2015 Examiners: Prof. Dr. Thomas Schwentick Prof. Dr. Christoph Buchheim Technische Universität Dortmund Fakultät für

### CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### CS 341 Homework 9 Languages That Are and Are Not Regular

CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w

### 1.1 Logical Form and Logical Equivalence 1

Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic