Unit-6: Model-checking ω-regular properties

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Unit-6: Model-checking ω -regular properties

B. Srivathsan

Chennai Mathematical Institute

NPTEL-course

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Module 2:

ω -regular expressions to NBA

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Σ ={a,b}

Example 1: Infinite word consisting only ofa {aaaaaaaaaaaaaaaa... }

Example 4: Infinite words whereboccursonly finitely often {aaaaaaaaaaaaaaaa... , baaaaaaaaaa... , babbaaaaaaaaaaaa..., ...}

Example 3: a word inaaΣ^∗aafollowed by onlyb-s

{aaaabbbbbbb... , aababaabbbbbb... , aabbbbaabbbbbbb..., ...} Example 2: Infinite words containing onlyaor onlyb

{aaaaaaaaaaaaaaa... , bbbbbbbbbbbb... }

a

^ω

a

^ω ⁺

b

^ω

aa

^Σ^∗

aa · b

^ω

(

a

⁺

b

⁾^∗

· a

^ω

(4)

ω-regular expressions

G

⁼

E

₁

· F

₁^ω ⁺

E

₂

· F

₂^ω ⁺

· · ·

⁺

E

_n

· F

_n^ω

E

₁

, ..., E

_n

, F

₁

, ..., F

_n

are regular expressions and ε / ∈ L

⁽

F

_i⁾

for all 1 ≤ i ≤ n

L

⁽

F

^ω^{) =}

{ w

₁

w

₂

w

₃

... | each w

_i

∈ L

⁽

F

⁾

}

(5)

More examples

É (a+b)^ωset ofall infinite words

É a(a+b)^ωinfinite wordsstarting withana

É (a+bc+c)^ωwords where everybisimmediately followedbyc

É (a+b)^∗c(a+b)^ωwords with asingle occurrenceofc

É ((a+b)^∗c)^ωwords wherecoccurs infinitely often

(6)

ω-regular expressions

G

⁼

E

₁

· F

₁^ω ⁺

E

₂

· F

₂^ω ⁺

· · ·

⁺

E

_n

· F

_n^ω

Goal: Convert ω-regular expression to NBA

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Part 1: Given regular expression U, find NBA for U

^ω

(8)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀ q₁

q⁰₀

a

b a

a a

NBA forU^ω

(9)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀ q₁

q⁰₀

a

b a

a a

NBA forU^ω

(10)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀ q₁

q⁰₀

a

b a

a a

NBA forU^ω

(11)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀ q₁

q⁰₀

q₁ q⁰₀

a a a

NBA forU^ω

(12)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁ q⁰₀

a a a

NBA forU^ω

(13)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁ q⁰₀

a a a

NBA forU^ω

(14)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀ q₀

q⁰₀

a

b a

b

NBA forU^ω

(15)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀ q₀

q⁰₀

a

b a

b

NBA forU^ω

(16)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀ q₀

q⁰₀

a

b a

b

NBA forU^ω

(17)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀ q₁

q⁰₀

q₀ q⁰₀

a

b a

NBA forU^ω

(18)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀

q₀ q⁰₀

a

b a

NBA forU^ω

(19)

q₀ q₁ a

b

NFA forU

q₀ q₁

q⁰₀

a

b a

Standardized NFA

q₀

q₁

q⁰₀

q₀ q⁰₀

a

b a

NBA forU^ω

(20)

q₀

q₁

q₂ a

b a c

NFA forU

q₀

q₁

q₂ q⁰₀

a

b a c a

a

Standardized NFA

q₀

q₁

q₂ q⁰₀

q₁

q₂ q⁰₀

a

b a c a

a a a

NBA forU^ω

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q₀

q₁

q₂ a

b a c

NFA forU

q₀

q₁

q₂ q⁰₀

a

b a c a

a

Standardized NFA

q₀

q₁

q₂ q⁰₀

q₁

q₂ q⁰₀

a

b a c a

a a a

NBA forU^ω

(22)

q₀

q₁

q₂ a

b a c

NFA forU

q₀

q₁

q₂ q⁰₀

a

b a c a

a

Standardized NFA

q₀

q₁

q₂ q⁰₀

q₁

q₂ q⁰₀

a

b a c a

a a a

NBA forU^ω

(23)

q₀

q₁

q₂ a

b a c

NFA forU

q₀

q₁

q₂ q⁰₀

a

b a c a

a

Standardized NFA

q₀

q₁

q⁰₀

q₁

q₂ q⁰₀

a

b a a

a a

NBA forU^ω

(24)

q₀

q₁

q₂ a

b a c

NFA forU

q₀

q₁

q₂ q⁰₀

a

b a c a

a

Standardized NFA

q₀

q₁

q₂ q⁰₀

q₁

q⁰₀

a

b a a

a a

NBA forU^ω

(25)

q₀

q₁

q₂ a

b a c

NFA forU

q₀

q₁

q₂ q⁰₀

a

b a c a

a

Standardized NFA

q₀

q₁

q₂ q⁰₀

q₁

q⁰₀

a

b a a

a a

NBA forU^ω

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a

b b

a

11/20

NFA forU

a b

a

b b

a

11/20

Standardized NFA forU

a b

a

b b

a b a

11/20

NBA forU^ω

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a

b b

a

11/20

NFA forU

a b

a

b b

a

11/20

a b

a

b b

a b a

11/20

NBA forU^ω

(28)

a

b b

a

11/20

NFA forU

a b

a

b b

a

11/20

a b

a

b b

a b a

NBA forU^ω

11/22

(29)

ω-regular expressions

G

⁼

E

₁

· F

₁^ω ⁺

E

₂

· F

₂^ω ⁺

· · ·

⁺

E

_n

· F

_n^ω

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Part 1: Given regular expression U, find NBA for U

^ω

Done!

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Part 2:Given regular expressionUand NBA forVfind NBA forU·V

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q₀ a q₁ q₂ a,b

a

aΣ^∗a

p₀ p₁

a b

(a^∗b)^ω

q₀ a q₁ q₂ p₀ p₁

a,b a

a

b a

a b

aΣ^∗a·(a^∗b)^ω

(33)

a

aΣ^∗a

p₀ p₁

a b

(a^∗b)^ω

q₀ a q₁ q₂ p₀ p₁

a,b a

a

b a

a b

(34)

a

aΣ^∗a

p₀ p₁

a b

(a^∗b)^ω

q₀ a q₁ q₂ p₀ p₁

a,b a

a

b a

a b

(35)

q₀ q₁

q₂ q₃

a a,b

a,b b

aΣ^∗ + Σ^∗b

p₁ p₂

a b

(a^∗b)^ω

q₀ q₁

q₂ q₃

p₁ p₂

a a,b

a,b b

a b

a b a,b

a

b

(aΣ^∗+ Σ^∗b)·(a^∗b)^ω

(36)

q₀ q₁

q₂ q₃

a a,b

a,b b

aΣ^∗ + Σ^∗b

p₁ p₂

a b

(a^∗b)^ω

q₀ q₁

q₂ q₃

p₁ p₂

a a,b

b

a b

a b a,b

a (aΣ^∗+ Σ^∗b)·(a^∗b)^ω

(37)

U V

U·V

a

b b

a a

b b

a

b

b a

a

(38)

U V

U·V

a

b b

a a

b b

a

b

b a

a

(39)

U V

U·V

a

b b

a a

b b

a b

b

a

(40)

U V

U·V

a

b b

a a

b b

a b

b a

a

(41)

U V

U·V

a

b b

a a

b b

a b

b a

a

(42)

ω-regular expressions

G

⁼

E

₁

· F

₁^ω ⁺

E

₂

· F

₂^ω ⁺

· · ·

⁺

E

_n

· F

_n^ω

(43)

Part 1:Given regular expressionU, find NBA forU^ω

Done!

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Part 3:Given NBA forUand NBA forV find NBA forU + V

Union of NBA already seen in Unit 5

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Union of NBA already seen in Unit 5

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Part 1:Given regular expressionU, find NBA forU^ω

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Theorem

Every ω-regular expression can be converted to an NBA