toc review

8/18/2019 TOC Review

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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

• 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

• .: 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

• .: set of 8nal states 6.⊆7• {3K}

•


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L={anb2nan $ n> 1}

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