Unit-4: Regular properties B. Srivathsan Chennai Mathematical - - PowerPoint PPT Presentation

Unit-4: Regular properties

• B. Srivathsan

Chennai Mathematical Institute

NPTEL-course July - November 2015

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Module 2: A gentle introduction to automata

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{ p1 } { p1,p2 } { p2 } {}

request=1 ready request=1 busy request=0 ready request=0 busy

AP = set of atomic propositions AP-INF = set of infinite words over PowerSet(AP) A property over AP is a subset of AP-INF

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Goal: Need finite descriptions of properties

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Goal: Need finite descriptions of properties Here: Finite state automata to describe sets of words

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Goal: Need finite descriptions of properties Here: Finite state automata to describe sets of finite words

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Alphabet: { a,b }

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...}

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L1

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L1 q0 q1 q2 q3 q4

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L1 q0 q1 q2 q3 q4 a b

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L1 q0 q1 q2 q3 q4 a b b a

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L1 q0 q1 q2 q3 q4 a b b a a b

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L1 q0 q1 q2 q3 q4 a b b a a b b a

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L1 q0 q1 q2 q3 q4 a b b a a b b a a,b

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Alphabet: { a,b } L2 = { a, aa, ab, aaa, aab, aba, abb , ...}

L2 is the set of all words starting with a

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Alphabet: { a,b } L2 = { a, aa, ab, aaa, aab, aba, abb , ...}

L2 is the set of all words starting with a

Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L2

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Alphabet: { a,b } L2 = { a, aa, ab, aaa, aab, aba, abb , ...}

L2 is the set of all words starting with a

Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L2 q0 q1 q2 a b a,b a,b

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Alphabet: { a,b } L2 = { a, aa, ab, aaa, aab, aba, abb , ...}

L2 is the set of all words starting with a

Design a TS with actions { a,b } and mark some states as accepting so that the set of all paths from an initial state to an accepting state equals L2 q0 q1 q2 a b a,b a,b

Finite Automaton

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Coming next: Some terminology

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Alphabet

Σ = { a, b }

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Alphabet

Σ = { a, b } Σ · Σ = { a, b } · { a, b }

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Alphabet

Σ = { a, b } Σ · Σ = { a, b } · { a, b } = { aa, ab, ba, bb }

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Alphabet

Σ = { a, b } Σ · Σ = { a, b } · { a, b } = { aa, ab, ba, bb } Σ1 = words of length 1 Σ2 = words of length 2

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Alphabet

Σ = { a, b } Σ · Σ = { a, b } · { a, b } = { aa, ab, ba, bb } Σ1 = words of length 1 Σ2 = words of length 2 Σ3 = words of length 3

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Alphabet

Σ = { a, b } Σ · Σ = { a, b } · { a, b } = { aa, ab, ba, bb } Σ1 = words of length 1 Σ2 = words of length 2 Σ3 = words of length 3 Σk = words of length k

. . . . . .

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Alphabet

Σ = { a, b } Σ · Σ = { a, b } · { a, b } = { aa, ab, ba, bb } Σ0 = { ε } (empty word, with length 0) Σ1 = words of length 1 Σ2 = words of length 2 Σ3 = words of length 3 Σk = words of length k

. . . . . .

aba · ε

= aba

ε · bbb

= bbb

w · ε

= w

ε · w

= w

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Alphabet

Σ = { a, b } Σ · Σ = { a, b } · { a, b } = { aa, ab, ba, bb } Σ0 = { ε } (empty word, with length 0) Σ1 = words of length 1 Σ2 = words of length 2 Σ3 = words of length 3 Σk = words of length k

. . . . . .

Σ∗ =

• i ≥ 0 Σi

= set of all finite length words

aba · ε

= aba

ε · bbb

= bbb

w · ε

= w

ε · w

= w

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Σ∗ = set of all words over Σ

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Σ∗ = set of all words over Σ

Any set of words is called a language

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Σ∗ = set of all words over Σ

Any set of words is called a language { ab, abab, ababab, ...} words starting with an a words starting with a b { ε, b, bb, bbb, ...} { ε, ab, abab, ababab, ...} { ε, bbb, bbbbbb, (bbb)3, ...} words starting and ending with an a { ε, ab, aabb, aaabbb,a4b4 ...}

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Σ∗ = set of all words over Σ

Any set of words is called a language { ab, abab, ababab, ...} words starting with an a aΣ∗ words starting with a b { ε, b, bb, bbb, ...} { ε, ab, abab, ababab, ...} { ε, bbb, bbbbbb, (bbb)3, ...} words starting and ending with an a { ε, ab, aabb, aaabbb,a4b4 ...}

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Σ∗ = set of all words over Σ

Any set of words is called a language { ab, abab, ababab, ...} words starting with an a aΣ∗ words starting with a b bΣ∗ { ε, b, bb, bbb, ...} { ε, ab, abab, ababab, ...} { ε, bbb, bbbbbb, (bbb)3, ...} words starting and ending with an a { ε, ab, aabb, aaabbb,a4b4 ...}

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Σ∗ = set of all words over Σ

Any set of words is called a language { ab, abab, ababab, ...} words starting with an a aΣ∗ words starting with a b bΣ∗ { ε, b, bb, bbb, ...} b∗ { ε, ab, abab, ababab, ...} { ε, bbb, bbbbbb, (bbb)3, ...} words starting and ending with an a { ε, ab, aabb, aaabbb,a4b4 ...}

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Σ∗ = set of all words over Σ

Any set of words is called a language { ab, abab, ababab, ...} words starting with an a aΣ∗ words starting with a b bΣ∗ { ε, b, bb, bbb, ...} b∗ { ε, ab, abab, ababab, ...}

(ab)∗

{ ε, bbb, bbbbbb, (bbb)3, ...} words starting and ending with an a { ε, ab, aabb, aaabbb,a4b4 ...}

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Σ∗ = set of all words over Σ

Any set of words is called a language { ab, abab, ababab, ...} words starting with an a aΣ∗ words starting with a b bΣ∗ { ε, b, bb, bbb, ...} b∗ { ε, ab, abab, ababab, ...}

(ab)∗

{ ε, bbb, bbbbbb, (bbb)3, ...}

(bbb)∗

words starting and ending with an a { ε, ab, aabb, aaabbb,a4b4 ...}

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Σ∗ = set of all words over Σ

Any set of words is called a language { ab, abab, ababab, ...} words starting with an a aΣ∗ words starting with a b bΣ∗ { ε, b, bb, bbb, ...} b∗ { ε, ab, abab, ababab, ...}

(ab)∗

{ ε, bbb, bbbbbb, (bbb)3, ...}

(bbb)∗

words starting and ending with an a aΣ∗a { ε, ab, aabb, aaabbb,a4b4 ...}

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In this module...

Task: Design Finite Automata for some languages

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Words Languages Finite Automata

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Alphabet: { a,b } L1 = { ab, abab, ababab, ...} Design a Finite automaton for L1 q0 q1 q2 q3 q4 a b b a a b b a a,b

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Alphabet: { a,b } L3 = { ε, ab, abab, ababab, ...} Design a Finite automaton for L3 q0 q1 q2 a b b a a,b

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Alphabet: { a,b }

Σ∗ = { ε, a, b, aa, ab, ba, bb ...}

Design a Finite automaton for Σ∗ q0 a,b

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Alphabet: { a,b } a∗ = { ε, a, aa, aaa, aaaa, a5, ...}

a∗ is the set of all words having only a

Design a Finite automaton for a∗

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Alphabet: { a,b } a∗ = { ε, a, aa, aaa, aaaa, a5, ...}

a∗ is the set of all words having only a

Design a Finite automaton for a∗ q0 a

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Alphabet: { a,b } a∗ = { ε, a, aa, aaa, aaaa, a5, ...}

a∗ is the set of all words having only a

Design a Finite automaton for a∗ q0 a Non-deterministic automaton

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Transition Systems

Deterministic

Single initial state and

Non-deterministic

Multiple initial states

• r

s r s r1 r2 rm

a a a a

. . .

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Transition Systems

Deterministic

Single initial state and

Non-deterministic

Multiple initial states

• r

s r s r1 r2 rm

a a a a

. . .

Same applies in the case of Finite Automata

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} Design a Finite automaton for ab∗

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} Design a Finite automaton for ab∗ q0 q1 a b

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} Design a Finite automaton for ab∗ q0 q1 a b Non-deterministic automaton

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} ba∗ = { b, ba, ba2, ba3, ba4, ...} Design a Finite automaton for ab∗ ∪ ba∗

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} ba∗ = { b, ba, ba2, ba3, ba4, ...} Design a Finite automaton for ab∗ ∪ ba∗ q0 q1 q2 a b a b

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} ba∗ = { b, ba, ba2, ba3, ba4, ...} Design a Finite automaton for ab∗ ∪ ba∗ q0 q1 q2 a b a b Non-deterministic automaton

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} ba∗ = { b, ba, ba2, ba3, ba4, ...} Design a Finite automaton for ab∗ ∪ ba∗

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} ba∗ = { b, ba, ba2, ba3, ba4, ...} Design a Finite automaton for ab∗ ∪ ba∗ q0 q1 q3 q2 a b b a

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Alphabet: { a,b } ab∗ = { a, ab, ab2, ab3, ab4, ...} ba∗ = { b, ba, ba2, ba3, ba4, ...} Design a Finite automaton for ab∗ ∪ ba∗ q0 q1 q3 q2 a b b a Multiple initial states: non-deterministic automaton

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What is the language of the following automaton? q0 q1 q2 a a,b a

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What is the language of the following automaton? q0 q1 q2 a a,b a Answer: a Σ∗a words starting and ending with a

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What is the language of the following automaton? q0 q1 q2 a,b a b a,b

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What is the language of the following automaton? q0 q1 q2 a,b a b a,b Answer: Σ∗ab Σ∗ words containing ab

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What is the language of the following automaton? q0 q1 q2 a,b a b a,b a,b

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What is the language of the following automaton? q0 q1 q2 a,b a b a,b a,b Answer: Σ∗a Σ∗ b Σ∗ words where there exists an a followed by a b after sometime

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What is the language of the following automaton? q0 q1 q2 a,b,c a c a,b,c b

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What is the language of the following automaton? q0 q1 q2 a,b,c a c a,b,c b Answer: Σ∗a b∗ c Σ∗ (Σ = { a, b, c })

words where there exists an a followed by only b’s and after sometime a c occurs

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Alphabet: { a,b } L = { ε, ab, aabb, aaabbb, ..., aibi, ...} Can we design a Finite automaton for L?

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Alphabet: { a,b } L = { ε, ab, aabb, aaabbb, ..., aibi, ...} Can we design a Finite automaton for L? Need infinitely many states to remember the number of a’s

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Alphabet: { a,b } L = { ε, ab, aabb, aaabbb, ..., aibi, ...} Can we design a Finite automaton for L? Need infinitely many states to remember the number of a’s Cannot construct finite automaton for this language

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Regular languages

Languages { anbn | n ≥ 0} Regular languages

Σ∗, a∗, ab∗, etc.

L′ L

Definition

A language is called regular if it can be accepted by a finite automaton

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Words Languages Finite Automata

Deterministic (DFA) Non-deterministic (NFA) Regular languages

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Words Languages Finite Automata

Deterministic (DFA) Non-deterministic (NFA) Regular languages Next module: Are DFA and NFA equivalent?

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