Automata
Alphabets, Strings and Languages
Example : Consider the string 011 over the binary alphabet. All the prefixes, suffixes and substrings of this string are listed below.
Prefixes: e, 0, 01, 011.
Suffixes: e, 1, 11, 011.
Substrings: e, 0, 1, 01, 11, 011.
Note that x is a prefix (suffix or substring) to x, for any string x and e is a prefix (suffix or substring) to any string.
A string x is a proper prefix (suffix) of string y if x is a prefix (suffix) of y and x y.
In the above example, all prefixes except 011 are proper prefixes.
Powers of Strings : For any string x and integer , we use to denote the string formed by sequentially concatenating n copies of x. We can also give an inductive definition of as follows: = e, if n = 0 ; otherwise
Example : If x = 011, then = 011011011, = 011 and
Powers of Alphabets :
We write (for some integer k) to denote the set of strings of length k with symbols from . In other words, = { w | w is a string over and | w | = k}. Hence, for any alphabet, denotes the set of all strings of length zero. That is, = { e }. For the binary alphabet { 0, 1 } we have the following.
The set of all strings over an alphabet is denoted by . That is,
The set contains all the strings that can be generated by iteratively concatenating symbols from any number of times.
Example : If = { a, b }, then = { e, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, …}.
Please note that if , then that is . It may look odd that one can proceed from the empty set to a non-empty set by iterated concatenation. But there is a reason for this and we accept this convention.
The set of all nonempty strings over an alphabet is denoted by . That is,
Note that is infinite. It contains no infinite strings but strings of arbitrary lengths.
Reversal :
For any string the reversal of the string is .
An inductive definition of reversal can be given as follows:
Languages :
A language over an alphabet is a set of strings over
Example : Consider the string 011 over the binary alphabet. All the prefixes, suffixes and substrings of this string are listed below.
Prefixes: e, 0, 01, 011.
Suffixes: e, 1, 11, 011.
Substrings: e, 0, 1, 01, 11, 011.
Note that x is a prefix (suffix or substring) to x, for any string x and e is a prefix (suffix or substring) to any string.
A string x is a proper prefix (suffix) of string y if x is a prefix (suffix) of y and x y.
In the above example, all prefixes except 011 are proper prefixes.
Powers of Strings : For any string x and integer , we use to denote the string formed by sequentially concatenating n copies of x. We can also give an inductive definition of as follows: = e, if n = 0 ; otherwise
Example : If x = 011, then = 011011011, = 011 and
Powers of Alphabets :
We write (for some integer k) to denote the set of strings of length k with symbols from . In other words, = { w | w is a string over and | w | = k}. Hence, for any alphabet, denotes the set of all strings of length zero. That is, = { e }. For the binary alphabet { 0, 1 } we have the following.
The set of all strings over an alphabet is denoted by . That is,
The set contains all the strings that can be generated by iteratively concatenating symbols from any number of times.
Example : If = { a, b }, then = { e, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, …}.
Please note that if , then that is . It may look odd that one can proceed from the empty set to a non-empty set by iterated concatenation. But there is a reason for this and we accept this convention.
The set of all nonempty strings over an alphabet is denoted by . That is,
Note that is infinite. It contains no infinite strings but strings of arbitrary lengths.
Reversal :
For any string the reversal of the string is .
An inductive definition of reversal can be given as follows:
Languages :
A language over an alphabet is a set of strings over