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

Automata Theory Seminar B¨ uchi Complementation via Alternating Automata Fabian Reiter July 16, 2012

B¨ uchi Complementation 2 Θ( n log n ) BA B BA B BA: B¨ uchi Automaton AA: Alternating Automaton AA A AA A Expensive: If B has n states, B has 2 Θ( n log n ) states in the worst case (Michel 1988, Safra 1988) . Complicated: Direct approaches are rather involved. Consider indirect approach: detour over alternating automata . 1 / 33

Transition Modes (1) Existential : some 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 · · · Universal : 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 · · · 2 / 33

Transition Modes (2) 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 3 / 33

Alternation and Complementation Special case: A in existential mode A accepts iff ∃ run ρ : ρ fulfills acceptance condition of A A accepts iff ∀ run ρ : ¬ ( ρ fulfills acceptance condition of A ) iff ∀ run ρ : ρ fulfills dual acceptance condition of A ⇒ complementation � = dualization of: transition mode acceptance condition Want acceptance condition that is closed under dualization . 4 / 33

Outline Weak Alternating Parity Automata 1 Infinite Parity Games 2 Proof of the Complementation Theorem 3 B¨ uchi Complementation Algorithm 4 5 / 33

Outline Weak Alternating Parity Automata 1 Definitions and Examples Dual Automaton Infinite Parity Games 2 Proof of the Complementation Theorem 3 B¨ uchi Complementation Algorithm 4 6 / 33

Preview Example ( ( b ∗ a ) ω ) B¨ uchi automaton B : b a a q 0 q 1 b Equivalent WAPA A : a , b a b b q 0 q 1 a q 2 • 2 1 0 7 / 33

Weak Alternating Parity Automaton � Definition (Weak Alternating Parity Automaton) A weak alternating parity automaton (WAPA) is a tuple A := � Q , Σ , δ, q in , π � where Q finite set of states Σ finite alphabet δ : Q × Σ → B + ( Q ) transition function q in initial state π : Q → N parity function (Thomas and L¨ oding, ∼ 2000) B + ( Q ): set of all positive Boolean formulae over Q (built only from elements in Q ∪ {∧ , ∨ , ⊤ , ⊥} ) 8 / 33

Transitions � Example ( a ω ) a a a q 1 δ : Q × Σ → B + ( Q ) • • 1 � q 0 , a � �→ q 0 ∨ ( q 1 ∧ q 2 ) q 0 • a � q 1 , a � �→ ( q 0 ∧ q 1 ) ∨ ( q 1 ∧ q 2 ) 2 q 2 � q 2 , a � �→ q 2 a 0 Definition (Minimal Models) Example Mod ↓ ( θ ) ⊆ 2 Q : set of minimal models Mod ↓ ( q 0 ∨ ( q 1 ∧ q 2 )) of θ ∈ B + ( Q ), i.e. the set of minimal = {{ q 0 } , { q 1 , q 2 }} subsets M ⊆ Q s.t. θ is satisfied by � if q ∈ M true q �→ false otherwise 9 / 33

� Run Graph (1) 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 · · · 10 / 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. 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

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 � 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 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

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 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 � 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 � . 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 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 . 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