Mercurial > hg > Members > kono > Proof > automaton1
view nfa.agda @ 3:bf6d7843ba11
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 14 Nov 2020 09:58:56 +0900 |
parents | b4548639121e |
children | 2d8d29454b0f |
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module nfa where open import Level renaming ( suc to Suc ; zero to Zero ) open import Relation.Binary.PropositionalEquality open import Data.List open import Relation.Nullary open import automaton open import logic record NAutomaton {n : Level} ( Q : Set n ) ( Σ : Set ) : Set (Suc n) where field Nδ : Q → Σ → Q → Set NF : Q → Set open NAutomaton naccept : {n : Level} {Q : Set n} {Σ : Set } → (Nexists : (Q → Set) → Set) → NAutomaton Q Σ → (Q → Set) → List Σ → Set naccept {n} {Q} {Σ} Nexists nfa qs [] = Nexists (λ q → qs q ∧ NF nfa q ) naccept {n} {Q} {Σ} Nexists nfa qs (x ∷ input) = naccept Nexists nfa (λ q' → Nexists (λ q → qs q ∧ Nδ nfa q x q' )) input qlist : {n : Level} {Q : Set n} → (P : Q → Set ) → ((q : Q) → Dec ( P q)) → List Q → List Q qlist P dec [] = [] qlist P dec (q ∷ qs) with dec q ... | yes _ = q ∷ qlist P dec qs ... | no _ = qlist P dec qs ntrace : {n : Level} {Q : Set n} {Σ : Set } → (Nexists : (Q → Set) → Set) → (nfa : NAutomaton Q Σ) → (qs : Q → Set ) → (input : List Σ ) → naccept Nexists nfa qs input → ((qs : Q → Set) (q : Q) → Dec (qs q ∧ NF nfa q)) → ((qs : Q → Set) (x : Σ) (t : List Σ) (q : Q) → Dec (naccept Nexists nfa (λ q' → qs q ∧ Nδ nfa q x q') t)) → List Q → List (List Q) ntrace {n} {Q} {Σ} Nexists nfa qs [] a decNF decNδ L = qlist (λ q → qs q ∧ NF nfa q) (decNF qs) L ∷ [] ntrace {n} {Q} {Σ} Nexists nfa qs (x ∷ t) a decNF decNδ L = qlist (λ q → (naccept Nexists nfa (λ q' → qs q ∧ Nδ nfa q x q' ) t )) (decNδ qs x t) L ∷ ( ntrace Nexists nfa (λ q' → Nexists (λ q → qs q ∧ Nδ nfa q x q' )) t a decNF decNδ L ) data transition136 : Q3 → Σ2 → Q3 → Set where d0 : transition136 q₁ s1 q₂ d1 : transition136 q₁ s0 q₁ d2 : transition136 q₂ s0 q₂ d3 : transition136 q₂ s0 q₃ d4 : transition136 q₂ s1 q₃ d5 : transition136 q₃ s0 q₁ start136 : Q3 → Set start136 q = q ≡ q₁ nfa136 : NAutomaton Q3 Σ2 nfa136 = record { Nδ = transition136 ; NF = λ q → q ≡ q₁ } example136-1 = naccept exists-in-Q3 nfa136 start136 ( s0 ∷ s1 ∷ s0 ∷ s0 ∷ [] ) example136-0 = naccept exists-in-Q3 nfa136 start136 ( s0 ∷ [] ) example136-2 = naccept exists-in-Q3 nfa136 start136 ( s1 ∷ s0 ∷ s1 ∷ s0 ∷ s1 ∷ [] ) nfa136-FN : (q : Q3 ) → Dec ( NF nfa136 q ) nfa136-FN q₁ = yes refl nfa136-FN q₂ = no (λ ()) nfa136-FN q₃ = no (λ ()) Q3List : List Q3 Q3List = q₁ ∷ q₂ ∷ q₃ ∷ [] decFN136 : (qs qs' : Q3 → Set) → (x : List Σ2) → naccept exists-in-Q3 nfa136 qs x → (q : Q3) → Dec (qs' q ∧ NF nfa136 q) decFN136 qs qs' (_ ∷ _) _ _ = {!!} decFN136 qs qs' [] (qe1 x) q₁ = yes [ {!!} , refl ] decFN136 qs qs' [] _ q₂ = no (λ ()) decFN136 qs qs' [] _ q₃ = no (λ ()) decF̨δ136 : (qs qs' : Q3 → Set) (s : Σ2) (t x : List Σ2) → naccept exists-in-Q3 nfa136 qs x → (q : Q3) → Dec (naccept exists-in-Q3 nfa136 (λ q' → qs' q ∧ Nδ nfa136 q s q') t) decF̨δ136 qs qs' s t x a q₁ = {!!} decF̨δ136 qs qs' s t x a q₂ = {!!} decF̨δ136 qs qs' s t x a q₃ = {!!} nfa136trace : (qs : Q3 → Set) → (input : List Σ2 ) → naccept exists-in-Q3 nfa136 qs input → List (List Q3) nfa136trace qs x a = ntrace exists-in-Q3 nfa136 qs x a (λ qs' q → decFN136 qs qs' x a q ) (λ qs' s t q → decF̨δ136 qs qs' s t x a q ) Q3List subset-construction : {n : Level} { Q : Set n } { Σ : Set } → (Nexists : (Q → Set) → Set) → (NAutomaton Q Σ ) → Automaton {Suc Zero ⊔ n} (Q → Set) Σ subset-construction {n} {Q} { Σ} Nexists nfa = record { δ = λ qs x q' → Nexists (λ q → qs q ∧ Nδ nfa q x q' ) ; F = λ qs → Nexists ( λ q → NF nfa q ) } dfa136 : Automaton (Q3 → Set) Σ2 dfa136 = subset-construction exists-in-Q3 nfa136 t136 : accept dfa136 start136 (s0 ∷ s1 ∷ s0 ∷ s0 ∷ []) → List ( Q3 → Set ) t136 = trace dfa136 start136 ( s0 ∷ s1 ∷ s0 ∷ s0 ∷ [] ) subset-construction-lemma→ : { Q : Set } { Σ : Set } → (Nexists : (Q → Set) → Set) → (NFA : NAutomaton Q Σ ) → (astart : Q → Set ) → (x : List Σ) → naccept Nexists NFA ( λ q1 → astart q1) x → accept ( subset-construction Nexists NFA ) ( λ q1 → astart q1) x subset-construction-lemma→ = {!!} subset-construction-lemma← : { Q : Set } { Σ : Set } → (Nexists : (Q → Set) → Set) → (NFA : NAutomaton Q Σ ) → (astart : Q → Set ) → (x : List Σ) → accept ( subset-construction Nexists NFA ) ( λ q1 → astart q1) x → naccept Nexists NFA ( λ q1 → astart q1) x subset-construction-lemma← = {!!} open import regular-language open RegularLanguage open Automaton open import Data.Empty Union-Nexists : {n m : Level} → {A : Set n} → {B : Set m} → ( (A → Set) → Set )→ ( (B → Set) → Set ) → ( (A ∨ B → Set) → Set ) Union-Nexists {n} {m} {A} {B} PA PB P = PA (λ q → P (case1 q)) ∨ PB (λ q → P (case2 q)) Concat-NFA : {n : Level} {Σ : Set} → (A B : RegularLanguage {n} Σ ) → NAutomaton (states A ∨ states B) Σ Concat-NFA {n} {Σ} A B = record { Nδ = δnfa ; NF = nend } module Concat-NFA where data δnfa : states A ∨ states B → Σ → states A ∨ states B → Set where a-case : {q : states A} {i : Σ } → δnfa ( case1 q) i (case1 (δ (automaton A) q i)) ab-trans : {q : states A} {i : Σ } → F (automaton A) q → δnfa ( case1 q) i (case2 (δ (automaton B) (astart B) i)) b-case : {q : states B} {i : Σ } → δnfa ( case2 q) i (case2 (δ (automaton B) q i)) nend : states A ∨ states B → Set nend (case2 q) = F (automaton B) q nend (case1 q) = F (automaton A) q ∧ F (automaton B) (astart B) -- empty B case data state-is {n : Level} {Σ : Set } (A : RegularLanguage {n} Σ ) : (a : states A ) → Set where this : (a : states A) → state-is A a closed-in-concat : {n : Level} {Σ : Set } → (A B : RegularLanguage {n} Σ ) → ( x : List Σ ) (PA : (states A → Set) → Set) (PB : (states B → Set) → Set) → isRegular (Concat {n} (contain A) (contain B)) x record {states = states A ∨ states B → Set ; astart = λ q → state-is A (astart A) ; automaton = subset-construction (Union-Nexists PA PB) ( Concat-NFA A B )} closed-in-concat {Σ} A B x = {!!}