3. Alphabets and Strings
A common way to talk about words, number, pairs of
words, etc. is by representing them as strings
To define strings, we start with an alphabet
Examples
An alphabet is a finite set of symbols.
S1 = {a, b, c, d, …, z}: the set of letters in English
S2 = {0, 1, …, 9}: the set of (base 10) digits
S3 = {a, b, …, z, #}: the set of letters plus the
special symbol #
S4 = {(, )}: the set of open and closed brackets
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4. Strings
The empty string will be denoted by E
Examples
A string over alphabet S is a finite sequence of symbols in S.
abfbz is a string over S1 = {a, b, c, d, …, z}
9021 is a string over S2 = {0, 1, …, 9}
ab#bc is a string over S3 = {a, b, …, z, #}
))()(() is a string over S4 = {(, )}
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5. Cont.…
We will use small alphabets:
Strings
5
abbaw
bbbaaav
abu
ba,
baaabbbaaba
baba
abba
ab
a
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9. Recursive Definition of Length
For any letter:
For any string :
Example:
9
1a
1 wwawa
4
1111
111
11
1
a
ab
abbabba
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10. 4. Length of Concatenation
Example:
10
vuuv
853
8
5,
3,
vuuv
aababaabuv
vabaabv
uaabu
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11. Proof of Concatenation Length
• Claim:
• Proof: By induction on the length
• Induction basis:
• From definition of length:
11
vuuv
v
1v
vuuuv 1
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12. Empty String
A string with no letters:
Observations:
12
abbaabbaabba
www
0
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13. Substring
Substring of string:
a subsequence of consecutive characters
String Substring
13
bbab
b
abba
ab
abbab
abbab
abbab
abbab
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14. Prefix and Suffix
Given string
Prefixes Suffixes
14
abbab
abbab
abba
abb
ab
a
b
ab
bab
bbab
abbab uvw
prefix
suffix
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15. Another Operation
• Example:
• Definition:
•
15
n
n
wwww
abbaabbaabba 2
0
w
0
abba
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16. The * Operation
• : the set of all possible strings from alphabet
16
*
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
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17. 17
The + Operation
: the set of all possible strings from
alphabet except
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
*
,,,,,,,, aabaaabbbaabaaba
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18. 1.2 Languages
Languages can be used to describe problems with
“yes/no” answers, for example:
A language is a set of strings over an alphabet.
L1 = The set of all strings over S1 that contain
the substring “fool”
L2 = The set of all strings over S2 that are divisible by 7
= {7, 14, 21, …}
L3 = The set of all strings of the form s#s where s is any
string over {a, b, …, z}
L4 = The set of all strings over S4 where every ( can be
matched with a subsequent )
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19. Cont.…
A language is any subset of
Example 1:
Languages:
19
*
,,,,,,,,*
,
aaabbbaabaaba
ba
},,,,,{
,,
aaaaaaabaababaabba
aabaaa
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20. Example 2
An infinite language
is aabbaa ?
20
}0:{ nbaL nn
aaaaabbbbb
aabb
ab
L Labb
L
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29. Grammar
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A grammar is a mechanism used for describing
languages. This is one of the most simple but yet
powerful mechanism.
The grammar for English tells us what are the
words in it and the rules to construct sentences.
30. Formal definitions of a Grammar
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A grammar G is defined as a quadruple
31. Example
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Consider the grammar , where N = {S}, ={a, b} and P is
the set of the following production rules .
Some terminal strings are generated by this grammar together
with their derivation is given below
33. Cont.…
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Example: Find a grammar for the language
It is possible to find a grammar for L by modifying the previous
grammar since we need to generate an extra b at the
end of the string .
We can do this by adding a production S Bb
where the non-terminal B generates
