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• • •
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• • • • • • •
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• : → • ∈ , ⊂ , ∀ , ∃ •
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• •
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• • •
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• • • •
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•
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$ perl -i -pe 's/foo/bar/g' input foo oaaahhhhhhhh! -> bar oaaahhhhhhhh! $ perl -i'*.bak' -pe 's/foo/bar/g' input
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• HT 0x09 ¥t LF 0x0a ¥n CR 0x0d ¥r LF ¥n CR ¥r CR+LF ¥r¥n
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$ perl -i -pe 's/^¥t*//g' input
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B -> B $ perl -i -pe 's/<(.*)>(.*)/$2/g' input BBBBBBBIAA -> BBBBBBBIAA ¥1
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• http://www.rexv.org/ • https://jex.im/regulex/
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• • https://speakerdeck.com/spark6251/
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• • • •
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• 0 ∈ ℕ
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•
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1 1 2
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• 1 1 1 1 1 2 2 2 1 2
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• ≠ Σ • • Σ • • • Σ Σ5 ∋ = = 5 • = 0 • Σ = , , , , , , , , ⋯
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def = , Σ, , 0 , Σ : × Σ → 0 ∈ ⊂
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1 = {1 , 2 } Σ = 0,1 : × Σ → 1 ∈ 2 ⊂ 0 1 1 2 1 2 2 2 1 2
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def =
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1 = 0 1 (.*)0(.*) 1 2
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1 2 1 2 2
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Σ = < RESET >, 0,1, ⋯ , • • < RESET >
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• = , Σ, , 0 , • Σ ∋ = 1 2 ⋯ ∀ ∈ Σ def ∃ ∈ s. t. 0 = 0 , +1 = +1 ∈
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def ∃: NFA s. t. recognize
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• , • ∪ ≔ ∈ or ∈ • ∘ ≔ ∈ and ∈ • ∗ ≔ 1 2 ⋯ ∈ ℕ, ∈ •
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• Σ ≔ , , ⋯ , • ≔ good, bad • ≔ start, end • ∪ = good, bad, start, end • ∘ = goodstart, goodend, badstart, badend • ∗ = , good, bad, goodgood, goodbad, badbad, ⋯
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1 , 2 : Reg. Lang. ⇒ 1 ∪ 2 : Reg. Lang. 1 , 2 : Reg. Lang. ⇒ 1 ∘ 2 : Reg. Lang. : Reg. Lang. ⇒ ∗: Reg. Lang.
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• 1 , 2 1 , 2 • 1 ∪ 2 • 1 , 2 • 1 or 2 • • 1 2 •
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• 1 = 1 , Σ, 1 , 1 , 1 1 • 2 = 2 , Σ, 2 , 2 , 2 2 1. ≔ 1 , 2 1 ∈ 1 and 2 ∈ 2 2. ∀ 1 , 2 ∈ : ∀ ∈ Σ: 1 , 2 , = 1 1 , , 2 2 , 3. 0 ≔ 1 , 2 4. ≔ 1 , 2 1 ∈ 1 or 2 ∈ 2 = 1 × 2 ∪ 2 × 1 = , Σ, , 0 ,
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1 1 2 1 2 1 2
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• = , Σ, , 0 , = 1 , 2 , 1 , 1 , 1 , 2 , 1 , 2 , 1 , 1 , 1 , 2 : 1 × 2 × Σ → 1 × 2 0 = 1 , 2 = 1 , 2 , 1 , 1 , 1 , 2 , 1 , 1 1 1 2 1 2 1 2
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• • •
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• • • • •
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• 2 1 3 4
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2 1 1 1 2 1 3 3 3 4 2 1 3 4
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def = , Σ, , 0 , Σ : × Σ → Σ ≔ Σ ∪ 0 ∈ ⊂
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= {1 , 2 , 3 , 4 } Σ = 0,1 : × Σ → 1 ∈ 2 ⊂ 0 1 ε 1 1 1 , 2 ∅ 2 3 ∅ 3 3 4 ∅ ∅ 4 ∅ ∅ ∅ 2 1 3 4
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• = , Σ, , 0 , • Σ ∋ = 1 2 ⋯ ∀ ∈ Σ def ∃ ∈ s. t. 0 = 0 ∀ ∈ ℕ, ≤ − 1: +1 ∈ , +1 ∈
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iff ∀1 : NFA: ∃2 : DFA s. t. 1 ≡ 2 ∀1 , 2 : FA 1 ≡ 2 def 1 2
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• = , Σ, , 0 , • ′ = ′, Σ, ′, 0 ′ , ′ • ′ • • • • card = ⇒ card = 2
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• = , Σ, , 0 , • ′ = ′, Σ, ′, 0 ′ , ′ 1. ′ ≔ 2. ∀ ∈ ′: ∀ ∈ Σ: ′ , = ∈ ∈ , , ∈ = , ∈ 3. 0 ′ ≔ 0 4. ′ ≔ ∈ ′ ⊂
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• ⊂ ′ ∈ ′ ∗ → • ⊂ ≔ ∗ → ′ , = , = ∈ ∈ , , ∈ 0 ′ = 0 ///
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∃: DFA s. t. recognize
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• • • ′ = ′ ′ 2 1 3
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• = , Σ, , 0 , • ′ = ′, Σ, ′, 0 ′ , ′ • 0 ′ = 0 = 1 , 2 • ′ = 3 , 1 , 3 , 2 , 3 , 1 , 2 , 3 2 1 3
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0 1 ε 1 1 1 , 2 ∅ 2 3 ∅ 3 3 ∅ ∅ ∅ ′ 0 1 1 1 , 2 , 3 1 , 2 2 3 ∅ 3 ∅ ∅ 2 1 3
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′ 0 1 1 1 , 2 , 3 1 , 2 2 3 ∅ 3 ∅ ∅ ′ 0 1 1 1 , 2 , 3 1 , 2 2 3 ∅ 3 ∅ ∅ 1 , 2 1 , 2 , 3 1 , 2 1 , 2 , 3 1 , 2 , 3 1 , 2 2 1 3
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1 , 2 , 3 1 , 2 3 2 1
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1 , 2 , 3 1 , 2
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′ = 1 , 2 , 1 , 2 , 3 0 ′ = 1 , 2 ′ = 1 , 2 , 3 1 , 2 , 3 1 , 2 ′ 0 1 1 , 2 1 , 2 , 3 1 , 2 1 , 2 , 3 1 , 2 , 3 1 , 2
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1 , 2 : Reg. Lang. ⇒ 1 ∪ 2 : Reg. Lang. 1 , 2 : Reg. Lang. ⇒ 1 ∘ 2 : Reg. Lang. : Reg. Lang. ⇒ ∗: Reg. Lang.
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1. ≔ 0 ∪ 1 ∪ 2 2. ∀ ∈ : ∀ ∈ Σ : , = 1 , ∈ 1 2 , ∈ 2 1 , 2 = 0 and = ∅ = 0 and ≠ 3. ≔ 1 ∪ 2 • = 0 and = • 1 ∘ 2 , ∗
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• • •
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• Σ = 0,1 • 0 ∪ 1 ∗ • 0 ∪ 1 ∗0 0 ∪ 1 ∗ ∘ 0 • 0Σ∗ ∪ 1Σ∗ ΣΣ∗
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• ∗, 1 ∘ 2 , 1 ∪ 2 • 1 ∗ ∪ 2 ∘ 3 ∪ 4 ∗ 1 ∗ ∪ 2 ∘ 3 ∪ 4 ∗
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def 1. ∈ Σ 2. 3. ∅ 4. 1 ∪ 2 5. 1 ∘ 2 6. 1 ∗ 1 , 2
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• + ≔ ∗ • + = ∪ • ≔ ⋯ ∈ ℕ • 1∗∅ ≔ ∅ • ∅∗ ≔ • ∗
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+∪−∪ + ∪ +. ∗ ∪ ∗. + +72, −5.2, 2. , −.6
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•
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• NFA DFA def ∃: NFA s. t. recognize
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⇐ ⇒ ⇒
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⇐ 1. ∈ Σ = = = 1 , 2 , Σ, , 1 , 2 2. = = = 1 , Σ, , 1 , 1 a 1 2 1
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⇐ 3. = ∅ = ∅ = 1 , Σ, , 1 , ∅ 4. = 1 ∪ 2 5. = 1 ∘ 2 6. = 1 ∗ 1
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⇒
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• • • • •
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• = , Σ, , 0 , • ′ = ′, Σ, ′, , • → ′ 1. 0 → 2. ∈ : → ∅
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+ 2 + 1 2
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1 2 2 3 3 1 1 3 1 2 ∗3
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def = , Σ, , , Σ : − × − → ℛ ℛ ∈ ∈ 1 , 2 = 1 2
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• = , Σ, , , • Σ ∋ = 1 2 ⋯ ∀ ∈ Σ∗ def ∃ ∈ s. t. 0 = = ∀ ∈ ℕ: = −1 , ⇒ ∈
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⇒ • CONVERT
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⇒ CONVERT 5. 6. = 2 →
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⇒ CONVERT 7. > 2 , ≠ rip ∈ ′ = ′, Σ, ′, , ′ = − rip ∀ ∈ ′ − , ∀ ∈ ′ − : ′ , = 1 2 ∗ 3 ∪ 4 1 = , rip , 2 = rip , rip , 3 = rip , , 4 = , 8. CONVERT ′ = 2
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⇒ CONVERT CONVERT CONVERT CONVERT G′ CONVERT ′ CONVERT ′′ CONVERT −2
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⇒ ∀: GNFA: CONVERT ≡ ′ = CONVERT ∀1 , 2 : FA 1 ≡ 2 def 1 2
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⇒ ⇒ = 2 • •
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⇒ ⇒ − 1 ′ , 1 , 2 , ⋯ , ≠ rip ∀ ′
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⇒ ⇒ rip → rip → ′
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⇒ ⇐ ′ rip ′
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⇒ ′ − 1 ′ ≡ ≡
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2 1 3
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1 ∪ 0 ∪ 1 2 1 3 0
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1 ∪ 0 ∪ 1 2 1 0
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1 ∪ 0 0 ∪ 1 1
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0 ∪ 1 ∗ 1 ∪ 0 [01]*1?0
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• 0∗1∗ • 01 • • •
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•
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• •