CSC312Automata Theory
Lecture # 2
Languages
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Alphabets: An alphabet is a finite set of symbols, usually letters, digits, and punctuations.
Valid/In-valid alphabets: An alphabet may contain letters consisting of group of symbols for example Σ= {a, ba, bab, d}.
Remarks: While defining an alphabet of letters consisting of more than one symbols, no letter should be started with the letter of the same alphabet i.e. one letter should not be the prefix of another. However, a letter may be ended in a letter of same alphabet.
Valid alphabet :Invalid alphabet :
, ,a ba c
Alphabets and Strings
, ,a ab c
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String or word: A finite sequence of letters/alphabets
Examples: “cat”, “dog”, “house”, “read” …
Defined over an alphabet: zcba ,,,,
Alphabets and Strings
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Alphabets and Strings
We will use small alphabets:
Strings
abbaw
bbbaaav
abu
ba,
baaabbbaaba
baba
abba
ab
a
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String Operations
m
n
bbbv
aaaw
21
21
bbbaaa
abba
mn bbbaaawv 2121
Concatenation
abbabbbaaa
Let we have following strings
Reverse
12aaaw nR abba
aaabbb
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String Length
Length:
Examples:
naaaw 21
nw
1
2
4
a
aa
abba
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Length of Concatenation
Example:
vuuv
853
8
5,
3,
vuuv
aababaabuv
vabaabv
uaabu
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Empty String
A string with no letters: Observations:
Note-1: A language that does not contain any word at all is denoted by or { }. This language doesn’t contain any word not even the NULL string. i.e. { } ≠ {}
or or 0
w w w
abba abba abba
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Empty String
Note-2: Suppose a language L doesn’t contain NULL then
L = L + but L ≠ L + {}.
Important : NULL is identity element with respect to concatenation.
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SubstringSubstring of string:
a subsequence of consecutive characters
String Substring
bbab
b
abba
ab
abbab
abbab
abbab
abbab
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Prefix and Suffix Let the string is Prefixes Suffixes
abbab
abbab
abba
abb
ab
a
b
ab
bab
bbab
abbab uvw
prefix
suffix
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Another Operation
- w repeated n time; that is,
Example:
Definition:
n
n wwww
abbaabbaabba 2
0w
0abba
nw
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The * Operation
: the set of all possible strings from alphabet , called closure of alphabets also
known as Kleene star operator or Kleene star closure.
i.e. infinitely many words each of finite length.
*
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
14
The + Operation
: the set of all possible strings from alphabet except , also known as
Kleene plus operator.
Note : are infinite
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
* ,,,,,,,, aabaaabbbaabaaba
* and
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LanguagesA language is a set of strings ORA language is any subset of , usually
denoted by L. It may be finite or infinite. Example:
Languages:
If a string w is in L, we say that w is a sentence of L.
*
,,,,,,,,*
,
aaabbbaabaaba
ba
},,,,,{
,,
aaaaaaabaababaabba
aabaaa
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Note that:
}{}{
0}{
1}{
0
Sets
Set size
Set size
String length
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Another Example
An infinite language }0:{ nbaL nn
aaaaabbbbb
aabb
ab
L Labb
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Operations on LanguagesThe usual set operations
Complement:
aaaaaabbbaaaaaba
ababbbaaaaaba
aaaabbabaabbbaaaaaba
,,,,
}{,,,
},,,{,,,
LL *
,,,,,,, aaabbabaabbaa
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ReverseDefinition:
Examples:
ConcatenationDefinition:
Examples:
}:{ LwwL RR ababbaabababaaabab R ,,,,
{ : 0}
{ : 0}
n n
R n n
L a b n
L b a n
2121 ,: LyLxxyLL
, , ,
, , , , ,
a ab ba b aa
ab aaa abb abaa bab baaa
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Another OperationDefinition:
L concatenated with itself n times.
Special case:
n
n LLLL
bbbbbababbaaabbabaaabaaa
babababa
,,,,,,,
,,,, 3
0
0, ,
L
a bba aaa
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More Examples
}0:{ nbaL nn
}0,:{2 mnbabaL mmnn
2Laabbaaabbb
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Star-Closure (Kleene *)
Definition:
Example:
210* LLLL
,,,,
,,,,
,,
,
*,
abbbbabbaaabbaaa
bbbbbbaabbaa
bbabba
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Positive Closure
Definition:
Note: L+ includes if and only if L includes
*
21
L
LLL
,,,,
,,,,
,,
,
abbbbabbaaabbaaa
bbbbbbaabbaa
bba
bba
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Lexicographical OrderAssume that the symbols in are themselves ordered. Definition: A set of strings is in lexicographical order if -The strings are grouped first according to their length. -Then, within each group, the strings are ordered “alphabetically” according to the ordering of the symbols.
Ex: Let the alphabet beThe set of all strings in Lexicographical order is, a, b, aa, ab, ba, bb, aaa, …., bbb, aaaa, …, bbbb, ….
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Lexicographical Order
{ , }a b
