Example of Regular Expression

Ex. 6:
Find a regular expression corresponding to the language of all strings over the alphabet { a, b } that contain no more than one occurence of the string aa.

Solution:
If there is one substring aa in a string of the language, then that aa can be followed by any number of b. If an a comes after that aa, then that a must be preceded by b because otherwise there are two occurences of aa. Hence any string that follows aa is represented by ( b + ba )*. On the other hand if an a precedes the aa, then it must be followed by b. Hence a string preceding the aa can be represented by ( b + ab )*. Hence if a string of the language contains aa then it corresponds to the regular expression ( b + ab )*aa( b + ba )* .

If there is no aa but at least one a exists in a string of the language, then applying the same argument as for aa to a, ( b + ab )*a( b + ba )* is obtained as a regular expression corresponding to such strings.

If there may not be any a in a string of the language, then applying the same argument as for aa to , ( b + ab )*( b + ba )* is obtained as a regular expression corresponding to such strings.
Altogether ( b + ab )*( + a + aa )( b + ba )* is a regular expression for the language.

More examples

Ex. 4:
Describe as simply as possible in English the language corresponding to the regular expression a*b(a*ba*b)*a* .

Solution:
A string in the language can start and end with a or b, it has at least one b, and after the first b all the b's in the string appear in pairs. Any numbe of a's can appear any place in the string. Thus simply put, it is the set of strings over the alphabet { a, b } that contain an odd number of b's.
We will continue examples in next post..

Examples of Regular Grammars and Regular Expressions

Ex. 2:
Find a regular expression corresponding to the language of all strings over the alphabet { a, b } that contain exactly two a's.

Solution:
A string in this language must have at least two a's. Since any string of b's can be placed in front of the first a, behind the second a and between the two a's, and since an arbitrasry string of b's can be represented by the regular expression b*, b*a b*a b* is a regular expression for this language.


Ex. 3:
Find a regular expression corresponding to the language of all strings over the alphabet { a, b } that do not end with ab.

Solution:
Any string in a language over { a , b } must end in a or b. Hence if a string does not end with ab then it ends with a or if it ends with b the last b must be preceded by a symbol b. Since it can have any string in front of the last a or bb, ( a + b )*( a + bb ) is a regular expression for the language

Examples of Regular Expressions and Regular Languages.

examples of Regular expressions and Regular Languages.
Ex.:
Let r1 and r2 be arbitrary regular expressions over some alphabet. Find a simple (the shortest and with the smallest nesting of * and +) regular expression which is equal to each of the following regular expressions.

(a) (r1 + r2 + r1r2 + r2r1)*
(b) (r1(r1 + r2)*)+

Solution:
One general strategy to approach this type of question is to try to see whether or not they are equal to simple regular expressions that are familiar to us such as a, a*, a+, (a + b)*, (a + b)+ etc.

(a) Since (r1 + r2)* represents all strings consisting of strings of r1 and/or r2 , r1r2 + r2r1 in the given regular expression is redundant, that is, they do not produce any strings that are not represented by (r1 + r2)*. Thus (r1 + r2 + r1r2 + r2r1)* is reduced to (r1 + r2)*.

(b) (r1(r1 + r2)*)+ means that all the strings represented by it must consist of one or more strings of (r1(r1 + r2)*). However, the strings of (r1(r1 + r2)*) start with a string of r1 followed by any number of strings taken arbitrarily from r1 and/or r2.
Thus anything that comes after the first r1 in (r1(r1 + r2)*)+ is represented by (r1 + r2)*. Hence (r1(r1 + r2)*) also represents the strings of (r1(r1 + r2)*)+, and conversely (r1(r1 + r2)*)+ represents the strings represented by (r1(r1 + r2)*). Hence (r1(r1 + r2)*)+ is reduced to (r1(r1 + r2)*).

In our next Posts, we will see some more good examples of Regular expressions and Regular Languages.

Examples of regular expression and regular languages

So far, we have discussed several important formal definitions. Now we will see some good examples of them. We will discuss other important definitions time to time.

Examples of regular expression and regular languages corresponding to them

•( a + b )2 corresponds to the language {aa, ab, ba, bb}, that is the set of strings of length 2 over the alphabet {a, b}.

In general ( a + b )k corresponds to the set of strings of length k over the alphabet {a, b}. ( a + b )* corresponds to the set of all strings over the alphabet {a, b}.

•a*b* corresponds to the set of strings consisting of zero or more a's followed by zero or more b's.

•a*b+a* corresponds to the set of strings consisting of zero or more a's followed by one or more b's followed by zero or more a's.

•( ab )+ corresponds to the language {ab, abab, ababab, ... }, that is, the set of strings of repeated ab's.


Note:
A regular expression is not unique for a language. That is, a regular language, in general, corresponds to more than one regular expression. For example (a + b)* and ( a*b* )* correspond to the set of all strings over the alphabet {a, b}.


Definition of Equality of Regular Expressions

Regular expressions are equal if and only if they correspond to the same language.

Thus for example (a + b)* = (a*b*)*, because they both represent the language of all strings over the alphabet {a, b}.

In general, it is not easy to see by inspection whether or not two regular expressions are equal.

Deterministic Finite Automata (DFA)

DFAs are equivalent in computing power to NFAs (nondeterministic finite automata).On the other hand, DFAs are of strictly limited power in the languages they can recognize — many simple languages, including any problem that requires more than constant space to solve, cannot be recognized by a DFA.
The classical example of a simply described language that no DFA can recognize is the language consisting of strings of the form anbn — some finite number of a's, followed by an equal number of b's. It can be shown that no DFA can have enough states to recognize such a language.
See you later with our next post….

Automata Step by Step

Today, we will discuss “Regular expressions”.

Regular expressions:

A regular expression (also "RegEx" or "regex”) is a string that is used to describe or match a set of strings, according to certain syntax rules. The specific syntax rules vary depending on the specific implementation, programming language, or library in use. Additionally, the functionality of regex implementations can vary between versions

In another words, a regular expression, often called a pattern, is an expression that describes a set of strings. They are usually used to give a concise description of a set, without having to list all elements. For example, the set containing the three strings Handel, Händel, and Haendel can be described by the pattern "H(ä|ae?)ndel" (or alternatively, it is said that the pattern matches each of the three strings). In most formalism, if there is any regex that matches a particular set then there are an infinite number of such expressions.

Automata Step by Step

Now we will define “Sequence”

Sequence:
A sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence.

Now we will define “Language”.

Language(or Formal Language):

A Language is a set of strings over a given input alphabet.
As an example of formal language, an alphabet might be {a, b}and a string over that alphabet might be “ababba”

Automata Step by Step

Let’s see the definitions of “String”, “Alphabet”,” Sequence” and “Language”. Remember all above terms are connected. You can’t understand any of them without knowing the definitions of all of them.

String:

In computer programming and formal language theory, (and other branches of mathematics), a string is an ordered sequence of symbols. These symbols are chosen from a predetermined set.

Example: Empty String (contains no symbols)

Length of Empty String is zero.

Now we will define “Alphabet”.

Alphabet:
An alphabet is a finite set of symbols.
Examples:
Binary Alphabet = {0, 1}
Unary alphabet= {1}
ASCII Alphabet= {…..A, B, C………, a, b, c…………}
Decimal Alphabet= {0, 1, 2…….8, 9}