Automata and Formal Language

Course Outline Chapter 1: Introduction  Chapter 2: Finite Automata  Chapter 3: Regular Language and Regular Grammar  Chapter 4: Properties of Regular Language  Chapter 5: Context-Fr

Automata and Formal Language

Quan Thanh Tho qttho@dit.hcmut.edu.vn

Course Overview

 An introduction to the fundamental theories and algorithms for computing on digital computer.

acceptable output based on self-made

decision

programming language syntax.

Course Outline

 Chapter 1: Introduction

 Chapter 2: Finite Automata

 Chapter 3: Regular Language and Regular Grammar

 Chapter 4: Properties of Regular Language

 Chapter 5: Context-Free Grammar

 Chapter 6: Simplification of Context-Free Grammar

 Chapter 7: Pushdown Automata

 Assignment: 30%

 Final Exam: 70%

Related Issues

 Digital Circuit Design

 Compiler

 Programming Languages

Required Background

 Set and Graph Theory

 Induction and Contradiction-based Methods

Three Basic Concepts

 Languages

 Grammars

 Automata

Language Concatenation

L1L2 = {xy | x∈L1, y∈L2}

Ln = L L L (n times)

L0 = {λ}

Language Concatenation (cont’d)

• Example 1.4:

L = {anbn | n ≥ 0}

L2 = {anbnambm | n ≥ 0, m ≥ 0}

Closure Operators

 Star-Closure: L* = L0 ∪ L1 ∪ L2

 Positive-Closure: L+ = L1 ∪ L2

<noun> → boy | dog

<verb> → runs | walks

Hands-on Exercise 1.1:

 Find all of possible sentences of the grammar given in Example 1.5

Formal Grammar

• Formal grammar:

G = (V, T, S, P) V: finite set of variables

T: finite set of terminal symbols

S∈V: start variable

P: finite set of productions

Formal Grammar (cont’d)

• Productions: x → y, x∈(V ∪T )+, y∈(V ∪T )*

• Example 1.6:

G = ({S}, {a, b}, S, P) P: S → aSb

S → λ

Directly Derive

• w = uxv derives z = uyv if xy is a production

w ⇒ z w1 ⇒* wn (w1 ⇒ w2 ⇒ ⇒ wn | w1 = wn) w1 ⇒+ wn

Directly Derive (cont’d)

Generated Language

G = (V, T, S, P) L(G) = {w∈T* | S ⇒* w}

• Example 1.8:

Take into consideration the grammar given in Example 1.6:

L(G) = {anbn | n ≥ 0}

Sequential Forms

• S , w1 , w2 , , wn (containing variables) with S ⇒ w1 ⇒ w2 ⇒ ⇒ wn

⇒ w is a derivation

• Example 1.10: S ⇒ aSb ⇒ aaSbb ⇒ aabb

aabb: sentence aaSbb: sentential form

 “An automaton (plural: automata) is a self-operating

machine The word is sometimes used to describe a robot, more specifically an autonomous robot Used colloquially, it refers to a mindless follower.” (Wiki)

 An abstract model of digital computer:

Automaton Structure

Control unit Input file

Output

Storage

Input File

 Input file: is

 Divided into squares or cells, each of which holds a symbol of the alphabet

 The symbols are to be read from left to right.

 The end of input file is detectable

 The automaton cannot change the contents of the input file.

 A device consisting of unlimited cells.

 Each cell hold a symbol from an alphabet (it is not necessary

to be the same alphabet as the input one)

 The automaton can read and change the storage cells

Control Unit

 Having a finite set of infernal states

 Can be any one of its infernal states

 Can change from one infernal state to other.

Automation Operations

• Transition function:

current state × input symbol × storage info → next state

Output may be produced Info in the storage may be changed

• Configuration: current state × input symbol × storage info

• Move: current configuration → next configuration

Automaton Types

• Accepter: yes/no output

Transducer: string of symbols as output

• Deterministic: single possible move at one point Non-deterministic: multiple possible moves

• Exercise 4, 5, 6, 8, 9, 12, 15, 17 of Section 1.2 - Linz’s book.