B uchi Complementation 2 ( n log n ) BA B BA B Automata Theory - PDF document

B¨ uchi Complementation 2 Θ( n log n ) BA B BA B Automata Theory Seminar BA: B¨ uchi Automaton B¨ uchi Complementation via AA: Alternating Alternating Automata Automaton AA A AA A Fabian Reiter Expensive: If B has n states, B has 2 Θ( n log n ) states in the worst case (Michel 1988, Safra 1988) . July 16, 2012 Complicated: Direct approaches are rather involved. Consider indirect approach: detour over alternating automata . 1 / 33 Transition Modes (1) Transition Modes (2) Existential : some run is accepting 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 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 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 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 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 e q 2 e q 3 e q 4 e q 5 e · · · Universal : every run is accepting q 0 q 1 f q 2 f q 3 f q 4 f q 5 f · · · · · · q 0 q 1 a q 2 a q 3 a q 4 a q 5 a · · · q 0 q 1 g q 2 g q 3 g q 4 g q 5 g · · · q 0 q 1 b q 2 b q 3 b q 4 b q 5 b q 0 q 1 h q 2 h q 3 h q 4 h q 5 h · · · · · · q 0 q 1 c q 2 c q 3 c q 4 c q 5 c · · · q 0 q 1 i q 2 i q 3 i q 4 i q 5 i · · · 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 · · · 2 / 33 3 / 33 Alternation and Complementation Outline Special case: A in existential mode 1 Weak Alternating Parity Automata A accepts iff ∃ run ρ : ρ fulfills acceptance condition of A Infinite Parity Games 2 A accepts iff ∀ run ρ : ¬ ( ρ fulfills acceptance condition of A ) iff ∀ run ρ : ρ fulfills dual acceptance condition of A 3 Proof of the Complementation Theorem ⇒ complementation � = dualization of: B¨ uchi Complementation Algorithm 4 transition mode acceptance condition Want acceptance condition that is closed under dualization . 4 / 33 5 / 33 Preview Outline Example ( ( b ∗ a ) ω ) 1 Weak Alternating Parity Automata B¨ uchi automaton B : Definitions and Examples b a Dual Automaton a q 0 q 1 2 Infinite Parity Games b Proof of the Complementation Theorem Equivalent WAPA A : 3 a , b a b 4 B¨ uchi Complementation Algorithm b q 0 q 1 a q 2 • 2 1 0 6 / 33 7 / 33

Weak Alternating Parity Automaton � Transitions � Example ( a ω ) Definition (Weak Alternating Parity Automaton) a a a A weak alternating parity automaton (WAPA) is a tuple q 1 • • δ : Q × Σ → B + ( Q ) 1 A := � Q , Σ , δ, q in , π � � q 0 , a � �→ q 0 ∨ ( q 1 ∧ q 2 ) q 0 • where 2 a � q 1 , a � �→ ( q 0 ∧ q 1 ) ∨ ( q 1 ∧ q 2 ) Q finite set of states q 2 � q 2 , a � �→ q 2 a 0 Σ finite alphabet δ : Q × Σ → B + ( Q ) transition function Definition (Minimal Models) Example q in initial state Mod ↓ ( θ ) ⊆ 2 Q : set of minimal models π : Q → N parity function Mod ↓ ( q 0 ∨ ( q 1 ∧ q 2 )) of θ ∈ B + ( Q ), i.e. the set of minimal = {{ q 0 } , { q 1 , q 2 }} (Thomas and L¨ oding, ∼ 2000) subsets M ⊆ Q s.t. θ is satisfied by � B + ( Q ): set of all positive Boolean formulae over Q if q ∈ M true q �→ (built only from elements in Q ∪ {∧ , ∨ , ⊤ , ⊥} ) false otherwise 8 / 33 9 / 33 � � Run Graph (1) Run Graph (2) Example ( a ω ) Definition (Run) a a a q 1 A run of a WAPA A = � Q , Σ , δ, q in , π � on a word a 0 a 1 a 2 . . . ∈ Σ ω • • 1 is a directed acyclic graph q 0 • a 2 R := � V , E � q 2 where a 0 V ⊆ Q × N with � q in , 0 � ∈ V Accepting run: V contains only vertices reachable from � q in , 0 � . q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · · � � E contains only edges of the form � p , i � , � q , i + 1 � . Rejecting run: For every vertex � p , i � ∈ V the set of successors is a minimal model of δ ( p , a i ) q 0 , 0 q 0 , 1 q 0 , 4 � � � � q ∈ Q | � p , i � , � q , i + 1 � ∈ E ∈ Mod ↓ ( δ ( p , a i )) q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · · 10 / 33 11 / 33 Acceptance � Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b Definition (Acceptance) Let A be a WAPA, w ∈ Σ ω and R = � V , E � a run of A on w . b q 0 q 1 q 2 a • 2 1 0 An infinite path ρ in R satisfies the acceptance condition of A iff the smallest occurring parity is even, i.e. Run on b ω : b b b b b b b min { π ( q ) | ∃ i ∈ N : � q , i � occurs in ρ } is even. 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 · · · R is an accepting run iff every infinite path ρ in R satisfies the acceptance condition. Run on ( ba ) ω : A accepts w iff there is some accepting run of A on w . 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 12 / 33 13 / 33 Dual Automaton (1) � 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 Definition (Dual Automaton) a b δ ( q 1 , a ) = q 2 The dual of a WAPA A = � Q , Σ , δ, q in , π � is b δ ( q 1 , b ) = q 1 q 0 q 1 a q 2 • 2 1 0 δ ( q 2 , a ) = q 2 A := � Q , Σ , δ, q in , π � δ ( q 2 , b ) = q 2 where δ ( q , a ) is obtained from δ ( q , a ) by exchanging ∧ , ∨ and ⊤ , ⊥ Dual A : δ ( q 0 , a ) = q 0 π ( q ) := π ( q ) + 1 δ ( q 0 , b ) = q 0 ∨ q 1 a , b a , b b for all q ∈ Q and a ∈ Σ 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 14 / 33 15 / 33

Complementation Theorem Outline Weak Alternating Parity Automata 1 Main statement of this talk: 2 Infinite Parity Games Theorem (Complementation) The dual A of a WAPA A accepts its complement, i.e. Proof of the Complementation Theorem 3 L ( A ) = Σ ω \ L ( A ) 4 B¨ uchi Complementation Algorithm (Thomas and L¨ oding, ∼ 2000) 16 / 33 17 / 33 Automaton vs. Pathfinder � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 c q 0 b • w = a ω A : a 2 q 2 a 0 Game G A , w : 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 · · · player A player P { q 1 , q 2 } , 0 find accepting run R find rejecting path in R q 2 , 1 { q 2 } , 1 q 2 , 2 · · · 18 / 33 19 / 33 � Playing a Game � Infinite Parity Game (2) Definition (Play) Definition (Game) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . 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 Definition (Winner) The winner of a play γ is G A , w := � V A ˙ ∪ V P , E � where player A iff the smallest parity of occurring V A -nodes is even V A := Q × N (decision nodes of player A ) player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd V P := 2 Q × N (decision nodes of player P ) X ∈ { A , P } : a player, X : its opponent E ⊆ ( V A × V P ) ∪ ( V P × V A ) Definition (Strategy) s.t. the only contained edges are • � � � q , i � , � M , i � iff M ∈ Mod ↓ ( δ ( q , a i )) A strategy f X : V X → V X for player X selects for every • � � � M , i � , � q , i + 1 � iff q ∈ M decision node of player X one of its successor nodes in G A , w . for q ∈ Q , M ⊆ Q , i ∈ N f X is a winning strategy iff player X wins every play γ that is played according to f X . (Thomas and L¨ oding, ∼ 2000) 20 / 33 21 / 33 Strategies Outline Example Winning strategy for player A (so far): parities 1 Weak Alternating Parity Automata q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · q 0 �→ 2 · · · { q 1 , q 2 } , 1 Infinite Parity Games 2 q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · q 1 �→ 1 3 Proof of the Complementation Theorem { q 1 , q 2 } , 0 Lemma 1 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · q 2 �→ 0 Lemma 2 Lemma 3 Not a winning strategy for player A : Sublemma q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · Putting it All Together · · · { q 1 , q 2 } , 1 B¨ uchi Complementation Algorithm q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · 4 { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · 22 / 33 23 / 33