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Complexity of solving games with combination of objectives using separating automata Highlights 2020 Ashwani Anand, Chennai Mathematical Institute, India This is a joint work with Nathanal Fijalkow and Jrme Leroux LaBRI, France We will

  1. Complexity of solving games with combination of objectives using separating automata Highlights 2020 Ashwani Anand, Chennai Mathematical Institute, India This is a joint work with Nathanaël Fijalkow and Jérôme Leroux LaBRI, France

  2. We will consider two variants: MP , with of averages, and MP , with of the averages. • Parity ( P ): Eve wins the game, if the maximum of the infjnitely many times occuring colours is even. Adam wins, otherwise. • Mean-Payoff ( MP ): Eve wins the game, if the average limit of the infjnite sequence is non-negative. Adam wins, otherwise. Defjnitions and notations We consider games on labelled graphs played between two players, Adam and Eve, with certain winning objectives on the infjnite sequences of labels generated by playing. Some of the popular objectives are: 1

  3. We will consider two variants: MP , with of averages, and MP , with of the averages. • Mean-Payoff ( MP ): Eve wins the game, if the average limit of the infjnite sequence is non-negative. Adam wins, otherwise. Defjnitions and notations We consider games on labelled graphs played between two players, Adam and Eve, with certain winning objectives on the infjnite sequences of labels generated by playing. Some of the popular objectives are: • Parity ( P ): Eve wins the game, if the maximum of the infjnitely many times occuring colours is even. Adam wins, otherwise. 1

  4. We will consider two variants: MP , with of averages, and MP , with of the averages. Defjnitions and notations We consider games on labelled graphs played between two players, Adam and Eve, with certain winning objectives on the infjnite sequences of labels generated by playing. Some of the popular objectives are: • Parity ( P ): Eve wins the game, if the maximum of the infjnitely many times occuring colours is even. Adam wins, otherwise. • Mean-Payoff ( MP ): Eve wins the game, if the average limit of the infjnite sequence is non-negative. Adam wins, otherwise. 1

  5. Defjnitions and notations We consider games on labelled graphs played between two players, Adam and Eve, with certain winning objectives on the infjnite sequences of labels generated by playing. Some of the popular objectives are: • Parity ( P ): Eve wins the game, if the maximum of the infjnitely many times occuring colours is even. Adam wins, otherwise. • Mean-Payoff ( MP ): Eve wins the game, if the average limit of the infjnite sequence is non-negative. Adam wins, otherwise. We will consider two variants: MP , with lim sup of averages, and MP , with lim inf of the averages. 1

  6. • Denoted as W 1 W 2 , in two dimension. • Eve wins W 1 W 2 , if projection of the infjnite sequence on fjrst coordinate satisfjes W 1 , or that on second coordinate satisfjes W 2 . • We give the algorithms for solving the games with combination of objectives by constructing separating automata for them, combining those for the individual objectives as black boxes. Games with combination of objectives • Games with multi-dimensional labels. 2

  7. • Eve wins W 1 W 2 , if projection of the infjnite sequence on fjrst coordinate satisfjes W 1 , or that on second coordinate satisfjes W 2 . • We give the algorithms for solving the games with combination of objectives by constructing separating automata for them, combining those for the individual objectives as black boxes. Games with combination of objectives • Games with multi-dimensional labels. • Denoted as W 1 ∨ W 2 , in two dimension. 2

  8. • We give the algorithms for solving the games with combination of objectives by constructing separating automata for them, combining those for the individual objectives as black boxes. Games with combination of objectives • Games with multi-dimensional labels. • Denoted as W 1 ∨ W 2 , in two dimension. • Eve wins W 1 ∨ W 2 , if projection of the infjnite sequence on fjrst coordinate satisfjes W 1 , or that on second coordinate satisfjes W 2 . 2

  9. Games with combination of objectives • Games with multi-dimensional labels. • Denoted as W 1 ∨ W 2 , in two dimension. • Eve wins W 1 ∨ W 2 , if projection of the infjnite sequence on fjrst coordinate satisfjes W 1 , or that on second coordinate satisfjes W 2 . • We give the algorithms for solving the games with combination of objectives by constructing separating automata for them, combining those for the individual objectives as black boxes. 2

  10. 3 1 3 1 3 1 3 1 P MP 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b Eve ( 1 , − 3 ) ( 3 , 1 ) 3

  11. 3 1 3 1 3 1 3 1 P MP 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a b b Eve ( 1 , − 3 ) ( 3 , 1 ) ( 3 , 1 ) 3

  12. 3 1 3 1 3 1 3 1 P MP 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b Eve ( 1 , − 3 ) ( 3 , 1 ) ( 3 , 1 )( 3 , 1 ) 3

  13. 3 1 3 1 3 1 3 1 P MP 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a b b Eve ( 1 , − 3 ) ( 3 , 1 ) ( 3 , 1 )( 3 , 1 )( 3 , 1 ) 3

  14. 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP 3

  15. 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP 3

  16. 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a b b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP ( 2 , 3 ) 3

  17. 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP ( 2 , 3 )( 1 , − 3 ) 3

  18. 2 3 1 3 2 3 1 3 P MP 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a b b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP ( 2 , 3 )( 1 , − 3 )( 2 , 3 ) 3

  19. 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP (( 2 , 3 )( 1 , − 3 )( 2 , 3 )( 1 , − 3 )) ω | = P ∨ MP 3

  20. 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP (( 2 , 3 )( 1 , − 3 )( 2 , 3 )( 1 , − 3 )) ω | = P ∨ MP 3

  21. 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a b b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP (( 2 , 3 )( 1 , − 3 )( 2 , 3 )( 1 , − 3 )) ω | = P ∨ MP ( 3 , 1 ) 3

  22. 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP (( 2 , 3 )( 1 , − 3 )( 2 , 3 )( 1 , − 3 )) ω | = P ∨ MP ( 3 , 1 )( 1 , − 3 ) 3

  23. 3 1 1 3 3 1 1 3 P MP Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a b b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP (( 2 , 3 )( 1 , − 3 )( 2 , 3 )( 1 , − 3 )) ω | = P ∨ MP ( 3 , 1 )( 1 , − 3 )( 3 , 1 ) 3

  24. Example: P ∨ MP P in 1 st coordinate MP in 2 nd coordinate ( 3 , 1 ) ( 2 , 3 ) Adam a a b b Eve ( 1 , − 3 ) ( 3 , 1 ) (( 3 , 1 )( 3 , 1 )( 3 , 1 )( 3 , 1 )) ω | = P ∨ MP (( 2 , 3 )( 1 , − 3 )( 2 , 3 )( 1 , − 3 )) ω | = P ∨ MP (( 3 , 1 )( 1 , − 3 )( 3 , 1 )( 1 , − 3 )) ω ̸| = P ∨ MP 3

  25. • P may represent qualitative constraints like reachability of a good behaviour, and MP may represent quantative constraints like power consumption. Why do we care? • Synthesis of systems satisfying multiple constraints, qualitative or quantative 4

  26. Why do we care? • Synthesis of systems satisfying multiple constraints, qualitative or quantative • P may represent qualitative constraints like reachability of a good behaviour, and MP may represent quantative constraints like power consumption. 4

  27. W W – For all n -sized graphs satisfying W , accepts all paths in the graph W n – rejects all paths not satisfying W Theorem (Colcombet, Fijalkow 2019) Let G be a game of size n with positional objective W and be a n W −separating automaton. Then Eve has a strategy ensuring W if and only if she has a strategy winning the safety game G . Separating automata for a winning condition W * • Automaton A with safety acceptance condition such that * Notion of Separating automata was introduced by Bojańczyk and Czerwiński, and this defjntion was given by Colcombet and Fijalkow. 5

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