CS314: FORMAL LANGUAGES AND AUTOMATA THEORY L. NADA ALZABEN. Lecture 1: Introduction

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1 CS314: FORMAL LANGUAGES AND AUTOMATA THEORY L. NADA ALZABEN Lecture 1: Introduction

2 Introduction to the course 2 Required Text Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, BY MICHAEL SIPSER. The slides are only used for presenting the lectures and it is NOT USED FOR STUDY.

3 Introduction to the course 3 In this course we will cover these topics within 15 weeks.

4 Introduction to the course 4 Assessment method Assessment Week Grade Percentage from overall grade Comments Work sheets 4 work sheets 10 10% Every sheet covers from 2-3 chapters Quiz #1 Quiz #2 TUE 25/4/1435 TUE 01/6/ % Midterm #1 Midterm #2 Project Class participation TUE 10/5/1435 TUE 15/6/1435 MON 28/6/1435 Every lecture % 15% 10% 5% Oral questions Final exam Total SUN 19/7/ % 100%

5 0.1 Introduction 5 The theory of computation is divided over three parts: Part1: Automata Part2: Computability Part3: Complexity. They are all combined in one question: Q: What are the fundamental capabilities and limitations of computers?

6 Introduction..(cont.) 6 Complexity Computability Automata Computer problems (hard easy) Hard problem options Determine (true-false) deals with definitions and properties of mathematical models of computation Finite automaton (text processing compilershardware design) Who benefits from hard computations Context-free grammar (PL AI)

7 7 Set: A set is a group of objects represented as a unit. Sets contain any object (numbers, symbols, other sets). Example: A={7, 21, 57} Properties (read page 3-5) Membership- subsets- proper subset- order and repetition (set or multiset) infinite set- empty set. Describing a set {n rule about n} Set operations (union intersection complement) Venn diagram

8 8 Sequences and tuples. Sequence is a list of objects in some order. Uses parentheses. Tuples are finite sequence Properties (read page 6) Power set pairs Cartesian product or cross product

9 9 Functions and Relations Functions are central to mathematics. A function is an object that sets up an input-output relationship. (ex. f a = b ) A function is also called mapping. Examples: abs(x), add(a, b),. Domain (possible inputs), range (possible outputs) The notation for that: f: D R Ex. abs: Z Z, what about the add:?? a function that uses all the element in the range is called onto the range Ways to describe a function: procedures use table

10 10 Ex. f: 0,1,2,3,4 {0,1,2,3,4} What is the function and what is the modulo (m)? Zm = 0,1,2,, m 1, a number modulo m is the reminder after division by m. g: Z4 X Z4 Z4 (two dimension table, function add) When f is A 1 x.x A k, for some sets A 1.. A k, the input to f is a k- tuple (a 1, a 2,., a k ) and we call the a i the arguments to f A function with k arguments is called a k-ary function, and k is called the arity of the function. Some binary functions use infix notation (use symbol) rather than prefix notation (ex. a+b, add(a,b) ) A predicate or property is a function whose range is TRUE or FALSE (ex. Even(4)=TRUE) A property whose domain is a set of k-tuples Ax xa is called a relation. (k-ary relation)(ex. 2-ary relation using less than (<) infix)

11 11 Example 0.10 Predicate function (range {true, false}) on set {SCISSORS, PAPERS, STONES} relation is beat. Express beat set Equivalence relation R (special type of binary relation) must meat three conditions: Reflexive if for every x, xrx Symettric if for every x and y, xry and yrx Transitive if for every x,y and z, xry and yrz then xrz Example: set S={1,2,3,4}, R1 is less than or equal R2 is has the same parity. Explain which is equivalence relation?

12 12 Graphs: Undirected graph (graph): set of points with lines connecting some of the points. (points= nodes or vertices) (lines=edges) Directed graph : points with arrows connected the points. Node degree is number of edges on a node. No more than one edge is allowed between two nodes.

13 13 Graphs (cont.): Order of edge pair (i,j) does not matter only in undirected graph How to write the formal description of a graph G with V nodes and E edges for (a) (b) graphs.

14 14 Graphs are mostly used to represent data. Labeling the nodes or the edges make it Labeled Graph. Subset graph A path is a sequences of nodes connected by edges. Simple path doesn t repeat any nodes. Connected graph if every two nodes have a path between them. A path is cycle if it starts and ends in the same node. A simple cycle contains at least 3 nodes and repeat only the first and last node. A graph is a tree if connected and have NO simple cycles. Tree properties: root, leaves,

15 15 In Directed graph: outdegree =number of arrows pointing from a node. indegree= number of arrows pointing to a node Write formal description of the graph: Directed path when all arrows point in the same direction as its steps. Directed graph is strongly connected if a directed path connects every two nodes.

16 16 Strings and languages: Alphabet is any nonempty finite set and its members are symbols. String over an alphabet is finite sequence of symbols of that alphabet. (ex: 1 = 0,1 ) The length of string is the number of symbols in the string w Length zero is called empty string ( ᵋ ) Reverse of w (w R ) Substring (z of w) if z appears consecutively within w Concatenation. Lexicographic ordering (same as dictionary ordering). Ex. {0,1}

17 17 Boolean logic: Boolean logic is a mathematical system built around True and False values. (Boolean values) (0,1) Boolean operations: (negation -NOT ( ), conjunction- AND( ), disjunction OR ( )) these operations are used with operands. Other operations:( XOR ( ), equality ( ), implication ( ))

18 18 Distributive law.

COMS 1003 Fall Introduction to Computer Programming in C. Bits, Boolean Logic & Discrete Math. September 13 th

COMS 1003 Fall Introduction to Computer Programming in C. Bits, Boolean Logic & Discrete Math. September 13 th COMS 1003 Fall 2005 Introduction to Computer Programming in C Bits, Boolean Logic & Discrete Math September 13 th Hello World! Logistics See the website: http://www.cs.columbia.edu/~locasto/ Course Web