Top Banner
Lecture Two: Formal Languages Formal Languages, Lecture 2, slide 1 Amjad Ali
27

Lecture Two: Formal Languages

Feb 16, 2016

Download

Documents

huslu

Lecture Two: Formal Languages. Amjad Ali. Formal Language. It is an abstraction of the general characteristics of programming languages It consists of a set of symbols and some rules of formation of sentences Sentences are formed by grouping the symbols. Formal Language. - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Lecture  Two: Formal Languages

Lecture Two:Formal Languages

Formal Languages, Lecture 2, slide 1

Amjad Ali

Page 2: Lecture  Two: Formal Languages

Formal Language It is an abstraction of the general

characteristics of programming languages

It consists of a set of symbols and some rules of formation of sentences

Sentences are formed by grouping the symbols

Formal Languages, Lecture 2, slide 2

Page 3: Lecture  Two: Formal Languages

Formal Language

A formal language is the set of all strings permitted by the rules of formation

Formal Languages, Lecture 2, slide 3

Page 4: Lecture  Two: Formal Languages

What is a language? A system for the expression of

certain ideas, facts, or concepts, including a set of symbols and rules for their manipulation

Formal Languages, Lecture 2, slide 4

Page 5: Lecture  Two: Formal Languages

Mathematical definition of a language This shall require us to understand

the following concepts first Alphabets Strings Concatenation of strings etc.

Formal Languages, Lecture 2, slide 5

Page 6: Lecture  Two: Formal Languages

Alphabet An ALPHABET is a nonempty set of

symbols It is denoted by S Example:

S = {a,b}where a and b are symbols

Formal Languages, Lecture 2, slide 6

Page 7: Lecture  Two: Formal Languages

Alphabets An alphabet is any finite set of

symbols {0,1}: binary alphabet {0,1,2,3,4,5,6,7,8,9}: decimal

alphabet ASCII, Unicode: machine-text

alphabets Or just {a,b}: enough for many

examples {}: a legal but not usually interesting

alphabet We will usually use S as the name of the

alphabet we’re considering, as in S = {a,b}

Formal Languages, Lecture 2, slide 7

Page 8: Lecture  Two: Formal Languages

Strings Strings are constructed from the

individual symbols Strings are finite sequences of

symbols from the alphabet Example : aabba, ababaaa,

abbbaaa, etc are the strings formed by t he symbols of the alphabet

Formal Languages, Lecture 2, slide 8

Page 9: Lecture  Two: Formal Languages

Symbols And Variables Sometimes we will use variables that stand for

strings: x = abbb In programming languages, syntax helps

distinguish symbols from variables String x = "abbb";

In formal language, we rely on context and naming conventions to tell them apart

We'll use the first letters, like a, b, and c, as symbols

The last few, like x, y, and z, will be string variablesFormal Languages, Lecture 2,

slide 9

Page 10: Lecture  Two: Formal Languages

Assumptions Lower case letters a,b,c,… are

used for elements of the alphabet Lower case letters u,v,w,… for

string names eg w=aabbaba This indicates that w is a string

having specific value aabbaba

Formal Languages, Lecture 2, slide 10

Page 11: Lecture  Two: Formal Languages

Empty String The empty string is written as Like "" in some programming

languages || = 0 Don't confuse empty set and

empty string: {} {} {}

Formal Languages, Lecture 2, slide 11

Page 12: Lecture  Two: Formal Languages

Concatenation The concatenation of two strings x and

y is the string containing all the symbols of x in order, followed by all the symbols of y in order

We show concatenation just by writing the strings next to each other

If x = abc and y = def, then xy = abcdef For any x, x = x = x

Formal Languages, Lecture 2, slide 12

Page 13: Lecture  Two: Formal Languages

Concatenation of the strings Two strings are concatenated by

appending the symbols of one string to the end of the other string

Example u=aaabbbv=abbabba

Concatenated string uv=aaabbbabbabba

Formal Languages, Lecture 2, slide 13

Page 14: Lecture  Two: Formal Languages

Length of the string The length of the string is the

number of symbols in the string |w| = 5 if w = aabaa Empty String has no symbols and

is denoted by l |l| = 0

Automata Theory, Lecture 2, slide 14

Page 15: Lecture  Two: Formal Languages

Kleene Star The Kleene closure of an alphabet S, written as

S*, is the language of all strings over S {a}* is the set of all strings of zero or more

as: {, a, aa, aaa, …}

{a,b}* is the set of all strings of zero or more symbols, each of which is either a or b= {, a, b, aa, bb, ab, ba, aaa, …}

x S* means x is a string over S Unless S = {}, S* is infinite

Formal Languages, Lecture 2, slide 15

Page 16: Lecture  Two: Formal Languages

Kleene Star Iterating a language L L ={ε} L =L L =L·L L =L ·L Kleene star: L*=Un≥ 0 L Example: {a,b}* = {ε,a,b,aa,bb,ab,ba, aab,

…} all finite sequences over {a,b}.

Formal Languages, Lecture 2, slide 16

Page 17: Lecture  Two: Formal Languages

S+ and S*

S is an alphabet S* is the set of all strings obtained

by concatenating zero or more symbols from S

S* always contains l then S+ = S* - {l}

Formal Languages, Lecture 2, slide 17

Page 18: Lecture  Two: Formal Languages

Finiteness S is always finite S* and S+ are always infinite

Formal Languages, Lecture 2, slide 18

Page 19: Lecture  Two: Formal Languages

Numbers We use N to denote the set of

natural numbers: N = {0, 1, …}

Formal Languages, Lecture 2, slide 19

Page 20: Lecture  Two: Formal Languages

Exponents We use N to denote the set of natural numbers:

N = {0, 1, …} Exponent n concatenates a string with itself n

times If x = ab, then

x0 = x1 = x = ab x2 = xx = abab, etc.

We use parentheses for grouping exponentiations (assuming that S does not contain the parentheses)

(ab)7 = abababababababFormal Languages, Lecture 2, slide 20

Page 21: Lecture  Two: Formal Languages

Languages A language is a set of strings over

some fixed alphabet Not restricted to finite sets: in fact,

finite sets are not usually interesting languages

All our alphabets are finite, and all our strings are finite, but most of the languages we're interested in are infinite

Formal Languages, Lecture 2, slide 21

Page 22: Lecture  Two: Formal Languages

Language A language L is defined very

generally as a subset of S A string in a language L will be

called a sentence of L

Formal Languages, Lecture 2, slide 22

Page 23: Lecture  Two: Formal Languages

Set Formers A set written with extra constraints or

conditions limiting the elements of the set

Not the rigorous definitions we're looking for, but a useful notation anyway:{x {a, b}* | |x| ≤ 2} = {, a, b, aa, bb, ab, ba}

{xy | x {a, aa} and y {b, bb}} = {ab, abb, aab, aabb}

{x {a, b}* | x contains one a and two bs} = {abb, bab, bba}

{anbn | n ≥ 1} = {ab, aabb, aaabbb, aaaabbbb, ...}Formal Languages, Lecture 2, slide 23

Page 24: Lecture  Two: Formal Languages

Free Variables in Set Formers Unless otherwise constrained,

exponents in a set former are assumed to range over all N

Examples{(ab)n} = {, ab, abab, ababab, abababab, ...}

{anbn} = {, ab, aabb, aaabbb, aaaabbbb, ...}

Formal Languages, Lecture 2, slide 24

Page 25: Lecture  Two: Formal Languages

The Quest Set formers are relatively informal They can be vague, ambiguous, or

self-contradictory A big part of our quest in the study

of formal language is to develop better tools for defining languages

Formal Languages, Lecture 2, slide 25

Page 26: Lecture  Two: Formal Languages

Problem S = {a,b} S* = {l,a,b,aa,ab,ba,bb,aaa,aab,aba, abb, baa,bab,bba,bbb,aaaa,… ….}

L = {a,aa,aab} is a language on S as L is a subset of S* and is finite

L = {anbn:n>0} is also a subset of S* but it is infinite

Formal Languages, Lecture 2, slide 26

Page 27: Lecture  Two: Formal Languages

Concatenation of two Languages L1L2 = {xy :x ε L1 and y ε L2 }

Formal Languages, Lecture 2, slide 27