B uchi Complementation via Alternating Automata Fabian Reiter - PowerPoint PPT Presentation

� Run Graph (2) Definition (Run) A run of a WAPA A = � Q , Σ , δ, q in , π � on a word a 0 a 1 a 2 . . . ∈ Σ ω is a directed acyclic graph R := � V , E � where V ⊆ Q × N with � q in , 0 � ∈ V V contains only vertices reachable from � q in , 0 � . 11 / 33

� Run Graph (2) Definition (Run) A run of a WAPA A = � Q , Σ , δ, q in , π � on a word a 0 a 1 a 2 . . . ∈ Σ ω is a directed acyclic graph R := � V , E � where V ⊆ Q × N with � q in , 0 � ∈ V V contains only vertices reachable from � q in , 0 � . � � � p , i � , � q , i + 1 � E contains only edges of the form . 11 / 33

� Run Graph (2) Definition (Run) A run of a WAPA A = � Q , Σ , δ, q in , π � on a word a 0 a 1 a 2 . . . ∈ Σ ω is a directed acyclic graph R := � V , E � where V ⊆ Q × N with � q in , 0 � ∈ V V contains only vertices reachable from � q in , 0 � . � � � p , i � , � q , i + 1 � E contains only edges of the form . For every vertex � p , i � ∈ V the set of successors is a minimal model of δ ( p , a i ) � � � � q ∈ Q | � p , i � , � q , i + 1 � ∈ E ∈ Mod ↓ ( δ ( p , a i )) 11 / 33

Acceptance � Definition (Acceptance) Let A be a WAPA, w ∈ Σ ω and R = � V , E � a run of A on w . An infinite path ρ in R satisfies the acceptance condition of A iff the smallest occurring parity is even, i.e. min { π ( q ) | ∃ i ∈ N : � q , i � occurs in ρ } is even. 12 / 33

Acceptance � Definition (Acceptance) Let A be a WAPA, w ∈ Σ ω and R = � V , E � a run of A on w . An infinite path ρ in R satisfies the acceptance condition of A iff the smallest occurring parity is even, i.e. min { π ( q ) | ∃ i ∈ N : � q , i � occurs in ρ } is even. R is an accepting run iff every infinite path ρ in R satisfies the acceptance condition. 12 / 33

Acceptance � Definition (Acceptance) Let A be a WAPA, w ∈ Σ ω and R = � V , E � a run of A on w . An infinite path ρ in R satisfies the acceptance condition of A iff the smallest occurring parity is even, i.e. min { π ( q ) | ∃ i ∈ N : � q , i � occurs in ρ } is even. R is an accepting run iff every infinite path ρ in R satisfies the acceptance condition. A accepts w iff there is some accepting run of A on w . 12 / 33

Acceptance � Example ( a ω ) a a a q 1 • • 1 q 0 • 2 a q 2 a 0 q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · · q 0 , 0 q 0 , 1 q 0 , 4 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · · 12 / 33

Acceptance � Example ( a ω ) a a a q 1 • • 1 q 0 • 2 a q 2 a 0 Accepting run: q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · · q 0 , 0 q 0 , 1 q 0 , 4 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · · 12 / 33

Acceptance � Example ( a ω ) a a a q 1 • • 1 q 0 • 2 a q 2 a 0 Accepting run: q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · · Rejecting run: q 0 , 0 q 0 , 1 q 0 , 4 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · · 12 / 33

Acceptance Alternating : in some set of runs every run is accepting · · · q 0 q 1 a q 2 a q 3 a q 4 a q 5 a q 0 q 1 b q 2 b q 3 b q 4 b q 5 b · · · q 0 q 1 c q 2 c q 3 c q 4 c q 5 c · · · · · · q 0 q 1 d q 2 d q 3 d q 4 d q 5 d · · · q 0 q 1 e q 2 e q 3 e q 4 e q 5 e · · · q 0 q 1 f q 2 f q 3 f q 4 f q 5 f · · · q 0 q 1 g q 2 g q 3 g q 4 g q 5 g · · · q 0 q 1 h q 2 h q 3 h q 4 h q 5 h · · · q 0 q 1 i q 2 i q 3 i q 4 i q 5 i 12 / 33

Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 Run on ( ba ) ω : b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

� Dual Automaton (1) Definition (Dual Automaton) The dual of a WAPA A = � Q , Σ , δ, q in , π � is A := � Q , Σ , δ, q in , π � 14 / 33

� Dual Automaton (1) Definition (Dual Automaton) The dual of a WAPA A = � Q , Σ , δ, q in , π � is A := � Q , Σ , δ, q in , π � where δ ( q , a ) is obtained from δ ( q , a ) by exchanging ∧ , ∨ and ⊤ , ⊥ for all q ∈ Q and a ∈ Σ 14 / 33

� Dual Automaton (1) Definition (Dual Automaton) The dual of a WAPA A = � Q , Σ , δ, q in , π � is A := � Q , Σ , δ, q in , π � where δ ( q , a ) is obtained from δ ( q , a ) by exchanging ∧ , ∨ and ⊤ , ⊥ π ( q ) := π ( q ) + 1 for all q ∈ Q and a ∈ Σ 14 / 33

Dual Automaton (2) Example ( ( b ∗ a ) ω ) WAPA A : δ ( q 0 , a ) = q 0 δ ( q 0 , b ) = q 0 ∧ q 1 a , b a b δ ( q 1 , a ) = q 2 b δ ( q 1 , b ) = q 1 q 0 q 1 q 2 a • 2 1 0 δ ( q 2 , a ) = q 2 δ ( q 2 , b ) = q 2 15 / 33

Dual Automaton (2) Example ( ( b ∗ a ) ω ) WAPA A : δ ( q 0 , a ) = q 0 δ ( q 0 , b ) = q 0 ∧ q 1 a , b a b δ ( q 1 , a ) = q 2 b δ ( q 1 , b ) = q 1 q 0 q 1 q 2 a • 2 1 0 δ ( q 2 , a ) = q 2 δ ( q 2 , b ) = q 2 Dual A : δ ( q 0 , a ) = q 0 δ ( q 0 , b ) = q 0 ∨ q 1 a , b a , b b δ ( q 1 , a ) = q 2 δ ( q 1 , b ) = q 1 q 0 b q 1 a q 2 3 2 1 δ ( q 2 , a ) = q 2 δ ( q 2 , b ) = q 2 15 / 33

Complementation Theorem Main statement of this talk: Theorem (Complementation) The dual A of a WAPA A accepts its complement, i.e. L ( A ) = Σ ω \ L ( A ) (Thomas and L¨ oding, ∼ 2000) 16 / 33

Outline Weak Alternating Parity Automata 1 Infinite Parity Games 2 Proof of the Complementation Theorem 3 B¨ uchi Complementation Algorithm 4 17 / 33

c b a Automaton vs. Pathfinder 18 / 33

Automaton vs. Pathfinder c b a player A 18 / 33

Automaton vs. Pathfinder c b a player A find accepting run R 18 / 33

Automaton vs. Pathfinder c b a player A player P find accepting run R 18 / 33

Automaton vs. Pathfinder c b a player A player P find accepting run R find rejecting path in R 18 / 33

� Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 19 / 33

� Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

� Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � (Thomas and L¨ oding, ∼ 2000) 20 / 33

� Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) (Thomas and L¨ oding, ∼ 2000) 20 / 33

� Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) (Thomas and L¨ oding, ∼ 2000) 20 / 33

� Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) E ⊆ ( V A × V P ) ∪ ( V P × V A ) (Thomas and L¨ oding, ∼ 2000) 20 / 33

� Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) E ⊆ ( V A × V P ) ∪ ( V P × V A ) s.t. the only contained edges are • � � � q , i � , � M , i � M ∈ Mod ↓ ( δ ( q , a i )) iff for q ∈ Q , M ⊆ Q , i ∈ N (Thomas and L¨ oding, ∼ 2000) 20 / 33

� Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) E ⊆ ( V A × V P ) ∪ ( V P × V A ) s.t. the only contained edges are • � � � q , i � , � M , i � M ∈ Mod ↓ ( δ ( q , a i )) iff • � � � M , i � , � q , i + 1 � iff q ∈ M for q ∈ Q , M ⊆ Q , i ∈ N (Thomas and L¨ oding, ∼ 2000) 20 / 33

Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Example q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 21 / 33

Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Definition (Winner) The winner of a play γ is player A iff the smallest parity of occurring V A -nodes is even player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd Example q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 21 / 33

Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Definition (Winner) The winner of a play γ is player A iff the smallest parity of occurring V A -nodes is even player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd X ∈ { A , P } : a player, X : its opponent Definition (Strategy) A strategy f X : V X → V X for player X selects for every decision node of player X one of its successor nodes in G A , w . 21 / 33

Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Definition (Winner) The winner of a play γ is player A iff the smallest parity of occurring V A -nodes is even player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd X ∈ { A , P } : a player, X : its opponent Definition (Strategy) A strategy f X : V X → V X for player X selects for every decision node of player X one of its successor nodes in G A , w . f X is a winning strategy iff player X wins every play γ that is played according to f X . 21 / 33

Strategies Example parities q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · q 0 �→ 2 { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · q 1 �→ 1 { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · q 2 �→ 0 22 / 33

Strategies Example Winning strategy for player A (so far): parities q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · q 0 �→ 2 { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · q 1 �→ 1 { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · q 2 �→ 0 22 / 33

Strategies Example Winning strategy for player A (so far): parities q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · q 0 �→ 2 { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · q 1 �→ 1 { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · q 2 �→ 0 Not a winning strategy for player A : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · 22 / 33

Outline 1 Weak Alternating Parity Automata Infinite Parity Games 2 Proof of the Complementation Theorem 3 Lemma 1 Lemma 2 Lemma 3 Sublemma Putting it All Together B¨ uchi Complementation Algorithm 4 23 / 33

Lemma 1 Let A be a WAPA and w ∈ Σ ω . Lemma 1 Player A has a winning strategy in G A , w iff A accepts w . 24 / 33

Lemma 1 Let A be a WAPA and w ∈ Σ ω . Lemma 1 Player A has a winning strategy in G A , w iff A accepts w . Explanation (oral) : Player A wins every play γ played according to f A . G A , w : · · · q , i + 1 { q , q ′ , q ′′ } , i q ′ , i + 1 p , i · · · q ′′ , i + 1 24 / 33

Lemma 1 Let A be a WAPA and w ∈ Σ ω . Lemma 1 Player A has a winning strategy in G A , w iff A accepts w . Explanation (oral) : Player A wins every play γ There is a run graph R in which played according to f A . every path ρ is accepting. G A , w : · · · q , i + 1 q , i + 1 { q , q ′ , q ′′ } , i q ′ , i + 1 q ′ , i + 1 p , i p , i R : · · · q ′′ , i + 1 q ′′ , i + 1 24 / 33

Lemma 2 Let A be a WAPA and w ∈ Σ ω . Lemma 2 Player P has a winning strategy in G A , w iff A does not accept w . (pointed out by Jan Leike) 25 / 33

Lemma 2 Let A be a WAPA and w ∈ Σ ω . Lemma 2 Player P has a winning strategy in G A , w iff A does not accept w . (pointed out by Jan Leike) Explanation (oral) : Player P wins every play γ played according to f P . · · · G A , w : { .., q , .. } , i q , i + 1 · · · · · · { .., q ′ , .. } , i q ′ , i + 1 p , i · · · · · · { .., q ′′ , .. } , i q ′′ , i + 1 · · · 25 / 33

Lemma 2 Let A be a WAPA and w ∈ Σ ω . Lemma 2 Player P has a winning strategy in G A , w iff A does not accept w . (pointed out by Jan Leike) Explanation (oral) : Player P wins every play γ Every run graph R contains a played according to f P . rejecting path ρ . · · · · · · G A , w : { .., q , .. } , i p , i q , i + 1 q , i + 1 R : · · · · · · · · · · · · { .., q ′ , .. } , i q ′ , i + 1 q ′ , i + 1 p , i R ′ : p , i · · · · · · · · · · · · q ′′ , i + 1 R ′′ : p , i { .., q ′′ , .. } , i q ′′ , i + 1 · · · · · · 25 / 33

Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff 26 / 33

Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff Proof: W.l.o.g. θ is in DNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M 26 / 33

Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff Proof: W.l.o.g. θ is in DNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M Then θ is in CNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M 26 / 33

Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff Proof: W.l.o.g. θ is in DNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M Then θ is in CNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M Thus S ⊆ Q is a model of θ iff it contains at least one element from each disjunct of θ . 26 / 33

B uchi Complementation via Alternating Automata Fabian Reiter - PowerPoint PPT Presentation

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