Lecture Two: Formal Languages Formal Languages, Lecture 2, slide 1 Amjad Ali

Lecture Two:

Formal Languages

Formal Languages, Lecture 2, slide 1

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 Languages, Lecture 2, slide 2

Formal Language

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

Formal Languages, Lecture 2, slide 3

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

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

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

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 as the name of the

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

Formal Languages, Lecture 2, slide 7

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

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

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

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

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

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

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

Kleene Star The Kleene closure of an alphabet , written as

*, is the language of all strings over {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 * means x is a string over Unless = {}, * is infinite

Formal Languages, Lecture 2, slide 15

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

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

Finiteness

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

Formal Languages, Lecture 2, slide 18

Numbers

We use N to denote the set of natural numbers: N = {0, 1, …}

Formal Languages, Lecture 2, slide 19

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 does not contain the parentheses)

(ab)7 = ababababababab

Formal Languages, Lecture 2, slide 20

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

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

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

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

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

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 Sas 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

Concatenation of two Languages

L1L2 = {xy :x ε L1 and y ε L2 }

Formal Languages, Lecture 2, slide 27