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2
A language is a set of strings
String: A sequence of letters
Examples: “cat”, “dog”, “house”, …
Defined over an alphabet: zcba ,,,,
3
Alphabets and StringsWe will use small alphabets:
Strings
abbaw
bbbaaav
abu
ba,
baaabbbaaba
baba
abba
ab
a
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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
Prefixes Suffixesabbab
abbab
abba
abb
ab
a
b
ab
bab
bbab
abbab uvw
prefix
suffix
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The * Operation : the set of all possible strings from alphabet
*
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
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The + Operation : the set of all possible strings from alphabet except
,,,,,,,,,*
,
aabaaabbbaabaaba
ba
*
,,,,,,,, aabaaabbbaabaaba
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LanguagesA language is any subset of
Example:
Languages:
*
,,,,,,,,*
,
aaabbbaabaaba
ba
},,,,,{
,,
aaaaaaabaababaabba
aabaaa
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Operations on LanguagesThe usual set operations
Complement:
aaaaaabbbaaaaaba
ababbbaaaaaba
aaaabbabaabbbaaaaaba
,,,,
}{,,,
},,,{,,,
LL *
,,,,,,, aaabbabaabbaa
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Another OperationDefinition:
Special case:
n
n LLLL
bbbbbababbaaabbabaaabaaa
babababa
,,,,,,,
,,,, 3
0
0
,, aaabbaa
L
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Star-Closure (Kleene *)
Definition:
Example:
210* LLLL
,,,,
,,,,
,,
,
*,
abbbbabbaaabbaaa
bbbbbbaabbaa
bbabba