toc review
TRANSCRIPT
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Sets
•
Alphabet: Set of symbols• A ={a, b, c, d}
• Σ = {0, 1, 2}
• Language• Set of strings
•
L = {ab, cd, da, acc, dddd, ε}• = {1, 011, 102}
• ε:• !he empty string"
• #as 0 symbols
• $ε$ = 0
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Strings
•A substring
• %art of a string
• & = abcd substrings of &: {a, b, c, d, ab, bc, cd, ab
abcd}
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'leene star
•Applies to sets
• Symbols
• Strings
• A={a, b, c} A( = {ε, a, b, c, aa, ab, ac, ba, bb,
abc)}
• L= {01, 00, 11} L(= {ε, 0100, 01, 1101)}
•
A language L o*er alphabet A is a subset of A*
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+achines Automata
•
-sed to decide &hether an input string belongs to a lan
• .initestate machine• egular Languages
•%ushdo&n automaton
• /ontetfree Languages
• !uring machine• ecursi*elyenumerable languages
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.initestate machine
•
, A, , 30, .4• : .inite set of states
• ={30, 31, a2}
• A: 5nput alphabet 6set of symbols7• A = {0, 1}
• transition function• 6, A7
• %artial *s" !otal 6complete7
• 30 : 5nitial state 630∈ 7
• 30
• .: set of 8nal states 6.⊆7•
. = {a2}
•
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.initestate !ransducer
•, A, , 30, ., , 4
• : utput alphabet 6set of symbols7
• 6, A7 (
•
Is FST more powerful than FSA?
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egular Languages 67
•
Languages that can ha*e .SA• .inite and in8nite
• L1 ∈ and L2 ∈ L1 ∪ L2 ∈
• L1 ∈ and L2 ∈ L1 ∩ L2 ∈
• L1 ∈ and L2 ∈ L1 L2 ∈
• L1 ∈ ¬L1 ∈
•
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%umping Lemma 6for regularlanguages7•
;escribes conditions that must be met by all reglanguages
• -sed to pro*e that a language is not regular"• Assume that a language is regular
• Sho& that it *iolates the pumping lemma
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%umping Lemma 6for regularlanguages7•
All su
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%;A: .SA B StacC• Languages that re3uire counting canDt be decided by .SA
• anbn
• , A, , , 30, E0 , .4
• : .inite set of states• {30, 31, 32, 3F}
• A: 5nput alphabet 6set of symbols7• {a, b}
• : StacC alphabet 6set of symbols7• {A, G}
• transition function• 6, A∪ {}7 and (
• 30 : 5nitial state 630∈ 7
• 30
• E0: 5nitial stacC symbol 6E0∈ 7
•
.: set of 8nal states 6.⊆7• {3F}
•
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;eterministic /ontetfreeLanguages 6•
Languages that be recogni@ed by a ;%;A, but no
• :• L1 ∈ ¬L1 ∈
•
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/onstructing .SA /omplement
1" /omplete• +aCe the .SA total 6complete7
2" S&ap 8nal and non8nal states
•
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/onstructing %;A /omplement
1" Hliminating loops
2" /omplete+aCe the %;A total 6complete7
F" S&apping 8nal and non8nal states
•
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!uring +achine• , A, , E0, 30, .4
• : .inite set of states•
{30, 31, 32, 3F, 3I, 3J, 3K}• A: 5nput alphabet 6set of symbols7
• {a, b}
• : +emory alphabet 6set of symbols7• {A, }
• E0: 5nitial memory symbol 6E0∈ 7
•
• transition function• 6., 7 , , {,L,S}2
• 30 : 5nitial state 630∈ 7
• 30
• .: set of 8nal states 6.⊆7• {3K}
•
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+ultitape !uring +achine• , A, , E0, 30, .4
• : .inite set of states•
{30, 31, 32, 3F, 3I, 3J, 3K}• A: 5nput alphabet 6set of symbols7
• {a, b}
• : +emory alphabet 6set of symbols7• {A, }
• E0: 5nitial memory symbol 6E0∈ 7
•
• transition function• 6., 7 , , {,L,S} B1
• 30 : 5nitial state 630∈ 7
• 30
• .: set of 8nal states 6.⊆7• {3K}
•
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L={anb2nan $ n> 1}