A Simple Algorithm for Solving Qualitative Probabilistic Parity - PowerPoint PPT Presentation

A Simple Algorithm for Solving Qualitative Probabilistic Parity Games Sven Schewe University of Liverpool joint work with Ernst Moritz Hahn, Andrea Turrini & Lijun Zhang

motivation reachability games & attractors parity games Probabilistic Parity Games R 1 zone 1 zone 2 Good abstraction for systems with choices under our control choices under environment control randomised choices zone 3 zone 4 Suitable for linear time properties R 0 � � R 0 � �P ≥ 1 [ GF ( zone 1 ∧ F zone 2 )] Our goal is to win the game with probability 1. 1 / 15

motivation reachability games & attractors parity games Reachability Games a c b g e f d Reachability Game R = � V 0 , V 1 , E , F � V 0 , and V 1 are disjoint finite sets of game positions E ⊆ ( V 0 ∪ V 1 ) × ( V 0 ∪ V 1 ) is a set of edges, and F ⊆ ( V 0 ∪ V 1 ) is a set of final / accepting game positions Played by placing a pebble on the arena – on V 0 player 0 chooses a successor, on V 1 player 1 ⇒ infinite play π S = ( V 0 ∪ V 1 ) � F ∈ S ω ❀ player 0 wins, π ∈ S ω ❀ player 1 wins π / 2 / 15

motivation reachability games & attractors parity games Solving Reachability Games arena F Algorithm – for R = � V 0 , V 1 , E , F � start with the final states F set W 0 to 0-attractor( F ) set W 1 to V � W 0 σ -attractor( X ): fix-point of X union for player σ : positions in V σ that can reach X for player 1 − σ : positions in V 1 − σ that can only reach X 3 / 15

motivation reachability games & attractors parity games Solving Reachability Games arena W 0 Algorithm – for R = � V 0 , V 1 , E , F � start with the final states F set W 0 to 0-attractor( F ) set W 1 to V � W 0 σ -attractor( X ): fix-point of X union for player σ : positions in V σ that can reach X for player 1 − σ : positions in V 1 − σ that can only reach X 3 / 15

motivation reachability games & attractors parity games Solving Reachability Games arena W 0 W 1 Algorithm – for R = � V 0 , V 1 , E , F � start with the final states F set W 0 to 0-attractor( F ) set W 1 to V � W 0 σ -attractor( X ): fix-point of X union for player σ : positions in V σ that can reach X for player 1 − σ : positions in V 1 − σ that can only reach X 3 / 15

motivation reachability games & attractors parity games Parity Games 3 1 3 2 2 1 4 Parity Game P = � V 0 , V 1 , E , α � V 0 , and V 1 are disjoint finite sets of game positions E ⊆ ( V 0 ∪ V 1 ) × ( V 0 ∪ V 1 ) is a set of edges, and α : ( V 0 ∪ V 1 ) → N is a priority function Played by placing a pebble on the arena – on V 0 player 0 chooses a successor, on V 1 player 1 ⇒ infinite play, lowest priority occurring infinite often even ❀ player 0 wins, odd ❀ player 1 wins 4 / 15

motivation reachability games & attractors parity games State of the Art # priorities 3 4 5 6 7 8 O ( m n 2 ) O ( m n 3 ) O ( m n 4 ) O ( m n 5 ) O ( m n 6 ) O ( m n 7 ) McNaughton O ( m n 3 ) O ( m n 3 ) O ( m n 4 ) O ( m n 4 ) O ( m n 5 ) O ( m n 5 ) Browne & al. O ( m n 2 ) O ( m n 2 ) O ( m n 3 ) O ( m n 3 ) O ( m n 4 ) O ( m n 4 ) Jurdzi´ nski O ( m n 2 ) O ( m n 3 ) w.o. strategy / [GW15] O ( m n ) O ( m n 1 1 O ( m n 2 1 O ( m n 2 3 O ( m n 3 1 2 ) O ( m n 2 ) 3 ) 4 ) 16 ) Big Steps [S07] O ( m n ) O ( n 3 1 O ( n 3 3 O ( n 4 1 O ( n 4 9 O ( n 2 . 5 ) O ( n 3 ) 3 ) 4 ) 16 ) 20 ) [CHL15] but if you don’t tell the games that they are hard McNaughton is by far the fastest 5 / 15

motivation reachability games & attractors parity games McNaughton’s Algorithm arena α − 1 ( c ) McNaughton’s Algorithm – for P = � V 0 , V 1 , E , α � set c to the minimal priority, σ to c modulo 2, and σ to 1 − σ α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 6 / 15

motivation reachability games & attractors parity games McNaughton’s Algorithm arena A McNaughton’s Algorithm – for P = � V 0 , V 1 , E , α � set c to the minimal priority, σ to c modulo 2, and σ to 1 − σ α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 6 / 15

motivation reachability games & attractors parity games McNaughton’s Algorithm arena U σ A U σ McNaughton’s Algorithm – for P = � V 0 , V 1 , E , α � set c to the minimal priority, σ to c modulo 2, and σ to 1 − σ α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 6 / 15

motivation reachability games & attractors parity games McNaughton’s Algorithm arena W σ McNaughton’s Algorithm – for P = � V 0 , V 1 , E , α � set c to the minimal priority, σ to c modulo 2, and σ to 1 − σ α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 6 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Probabilistic Parity Games 3 1 3 2 2 1 4 Probabilistic Parity Game P = � V 0 , V 1 , V R , E , α � V 0 , V 1 , and V R are disjoint finite sets of game positions E ⊆ ( V 0 ∪ V 1 ∪ V R ) × ( V 0 ∪ V 1 ∪ V R ) is a set of edges, and α : ( V 0 ∪ V 1 ∪ V R ) → N is a priority function Played by placing a pebble on the arena – on V 0 player 0 chooses a successor, on V 1 player 1 – on V R a successor is chosen randomly ⇒ infinite play, lowest priority occurring infinite often even ❀ player 0 wins, odd ❀ player 1 wins 7 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Removing the Random Player: Gadget Construction random vertex with priority 0 0 0 0 0 8 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Removing the Random Player: Gadget Construction random vertex with priority 1 1 1 1 1 0 1 8 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Removing the Random Player: Gadget Construction random vertex with priority 2 2 2 2 2 0 1 2 8 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Removing the Random Player: Gadget Construction random vertex with priority 3 3 3 3 3 3 0 1 2 3 8 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Removing the Random Player: Gadget Construction random vertex with priority 4 4 4 4 4 4 0 1 2 3 4 8 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Different Attractors Weak attractors reach goal almost surely Strong attractors reach goal with positive probability ⇒ treat probabilistic nodes as “yours” Note: weak attractor( G ) ⊆ strong attractor( G ) 9 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Probabilistic McNaughton arena α − 1 ( c ) Prob-McNaughton’s Algorithm – for P = � V 0 , V 1 , V R , E , α � set c to the minimal priority and σ to c modulo 2 treat random nodes as player σ vertices α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to Prob-McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to Prob-McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 10 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Probabilistic McNaughton arena A Prob-McNaughton’s Algorithm – for P = � V 0 , V 1 , V R , E , α � set c to the minimal priority and σ to c modulo 2 treat random nodes as player σ vertices α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to Prob-McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to Prob-McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 10 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Probabilistic McNaughton arena U σ A U σ Prob-McNaughton’s Algorithm – for P = � V 0 , V 1 , V R , E , α � set c to the minimal priority and σ to c modulo 2 treat random nodes as player σ vertices α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to Prob-McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to Prob-McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 10 / 15

probabilistic parity games gadgets attractors probabilistic McNaughton Probabilistic McNaughton – σ = 0 , σ = 1 arena W σ Prob-McNaughton’s Algorithm – for P = � V 0 , V 1 , V R , E , α � set c to the minimal priority and σ to c modulo 2 treat random nodes as player σ vertices α − 1 ( c ) � � set A to σ -attractor set ( U 0 , U 1 ) to Prob-McNaughton( P � A ) set W σ to σ -attractor( U σ ), and set W σ to ∅ set ( U 0 , U 1 ) to Prob-McNaughton( P � W σ ) return ( W 0 ˙ ∪ U 0 , W 1 ˙ ∪ U 1 ) 10 / 15