Theory of Computation

Lecture #2 Mahmoud Ali Ahmed

Languages & GrammarBefore discussing languages & grammar let us deal

with some related issues.

Alphabet: is defined as a nonempty finite set

of symbols or letters such as {a1, a2, …, ak}.

Particularly the alphabet for theory of

computation is mostly the binary alphabet {0, 1}.

Set of alphabet represented by

Languages & Grammar

A string (word or sentence): is defied as a

finite sequence of symbols over the set of

alphabet (∑), e.g. aaabbbabbb is string over

∑ = {a, b}. A string might be represent by w.

A string may have no symbol at all, in this

case it is called the empty string / null string

and denoted by or

Note:Some times string (word/sentence) can be

represented as function. e.g.

A string x = x0 x1 x2 x3 … xn, which can be

viewed as function as:

x : [n] | x(k) = xk.

where n is the length of the string x, which

denoted by |x|, e.g. |aabbaab| = 7

Operations on stringsFor an alphabet Σ, given any two strings u: [m] → Σ

and v: [n] → Σ, the concatenation u・ v (also written

as uv) of u and v is the string uv: [m + n] → Σ,

defined such that:

nmimifmiv

miifiuiuv

1)(,1)(

)(

In particular uuu

Formal Languages

Formal vs. Natural Languages

Natural Languages: used by human to communicate.

e.g. Arabic, English, …

syntax is very complicated not completely specified

Formal LanguagesFormal vs. Natural Languages

Formal Language: specified by a well-defined sets

of grammatical rules

e.g. programming languages

syntax completely defined by given grammar.

Why study Formal Languages?

Programming languages are formal.

very useful for pattern matching.

Important for research in natural language

processing (NLP) with computers.

Closely tied to study of abstract machines.

Formal Languages

Let be any finite set of symbols (possibly

none), called an alphabet.

A word (string) over is any fine

sequence of letters from .

A formal language is simply a set of strings

Formal Languages* defined as the set of all possible words

over , including the empty word (denoted

by ).

ExampleLet , then},{ ba

},,,,,,,,,,{* baaabaaaabbbaabaaba

A language over is defined as strings over i.e. a subset of

*

Examples: ∑ = {a, b} aabaaa ,,,

8, * xbax

oddisxbax *,

)()(, * xNxNbax ba

Example

,,,,,,,,,* aabaaabbbaabaaba

Let ., ba Then

aabaaaL ,,

The language

is a finite language on .

Note: Language can be infinite, and the most

interesting languages are infinite.

New language can be constructed using any of the

set operations.

As an example the union of tow languages over ,

is also a language over .

Having {as, so} and {if, soon, possible} as tow

languages over the concatenation of the tow

languages give another language over as follows.

{as, so} {if, soon, possible} =

{asif, assoon, aspossible, soif, sosoon, sopossible}.

Regular Expression & Language

Definition: A regular expression over , that

corresponding to a language, can be recursively

defined as follows:

• ε is a Regular Expression indicates the language containing an

empty string. (L (ε) = {ε})

• φ is a Regular Expression denoting an empty language. (L (φ) = { })

• a is a Regular Expression where L = {a}.

Regular Expression & Language

• If x is a Regular Expression denoting the

language L(x) and y is a Regular Expression denoting the

language L(y), then

o x + y is a Regular Expression corresponding to the

language L(x) L(y)∪  where L(x + y) = L(x) L(y)∪ .

o x . x is a Regular Expression corresponding to the

language L(x) . L(y) where L(x. y) = L(x) . L(y).o R* is a Regular Expression corresponding to the language

L(R*)where L(R*) = (L(R))*.• if we apply any of the above rules several times, they are

Regular Expressions.

A language over is a regular language if there is

some regular expression over corresponding to it.

Examples of regular expression over

**011

1*

** 110

MeaningExpressionAll words over contains exactly a single 0.

All words over containing at least one 1.

All words over that containing a substring 110.

}1,0{

(00 + 01 + 10 + 11)*All string of 0’s and 1’s of even length can be obtained by concatenating any combination of the strings 00, 01, 10 and 11 including ε

MeaningExpression*)( All words over ∑ with an even length

*)( All words over ∑ of length a multiple of three

101100 All words over ∑ starts and ends with the same symbol

011 All strings containing over ∑ that begin with 1 and ending with 01

(0|(1(01*0)*1))* All strings containing over ∑ contains binary numbers that are multiples of 3

Formal Grammar

Formal definition

,,,, PSTVG where,

A generative grammar, which firstly proposed

by Noam Chomsky in 1950s, considered as a

grammar G that defined as a quadruple.

is a finite set of objects called variables.V

is a finite set of objects called terminal symbols.T

is a special symbol called start variable.VS

is a finite set of productions.P

V TIt is assumed that and are nonempty and disjoint

Definition (L(G))

wSTwGL :)(

Let be a grammar. Then the set ,,,, PSTVG

is the language generated by . G

Example

,,,, PSTVG Consider a grammar G that defined as

where,V = {S, B} T = {a, b, c}P consists of the following production rule

S aBSc S abc

Ba aB Bb bb

L(G) may be derived to be consisted of the

following strings:

L(G) = {abc, aabbcc, aaabbbccc, …}

Therefore L(G) can be represented as

L(G) = {an bn cn | n>0}

L(G) = {w ϵ {a, b, c}* | na(w) = nb(w) = nc(w)}

Representation of Language using Graphs

Transition Graph over is a finite directed graph in

which every arrow (edge) is labeled by some word

(possibly the empty word ( )).

There is a least one vertex , labeled by a ( ) sign

(such vertices are called initial vertices), and a

(possibly empty set) of vertices, labeled by a ( )

sign (called the final vertices). A vertex can be both

initial and final.

T *w

''

''

ba , *

a b bab,,

ExampleConsider the following transition graphs over

ba,

ba

All words over beginning with a followed by a sb'

ba ,aaba ,

ba,

bb

All words over containing two consecutive ( ) or two consecutive ( )

sa'sb'

bbaa,baab,bbaa ,baab,

All words over containing an even number of ( ) and even number of ( )

sa'sb'

b

aa

ba,

ba,

All words over either starting with ( ) or containing ( ) .

baa