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  Example:  {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 variables Formal 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=aaabbb v=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  = 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 Kleene star: L*=U n ≥ 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

   and    is an alphabet   is the set of all strings obtained by concatenating zero or more symbols from    always contains  then      Formal Languages, Lecture 2, slide 17

Finiteness  is always finite   and   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 x 0 =  x 1 = x = ab x 2 = 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  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} {a n b n | 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,...} {a n b n } = { , 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  a,b     a,b,aa,ab,ba,bb,aaa,aab,aba, abb, baa,bab,bba,bbb,aaaa,… ….} L = {a,aa,aab} is a language on  as L is a subset of   and is finite L = {a n b n :n>0} is also a subset of   but it is infinite Formal Languages, Lecture 2, slide 26

Concatenation of two Languages L 1 L 2 = {xy :x ε L 1 and y ε L 2 } Formal Languages, Lecture 2, slide 27