Deciding Maxmin Reachability in Half-Blind Stochastic Games Edon Kelmendi Hugo Gimbert LaBRI, Université de Bordeaux Highlights Brussels, September 2016 1 α I β 1 2 I Max vs Min for reachability Max has zero information I i I f a I 1 2 a I Picks finite words Min has perfect information Picks behavioral strategies Probabilistic finite automata with an opponent 2 Question For all > 0 there exists a finite word for Max such that for all strategies for Min, the chance of reaching the state f is larger than 1 − ? 2/9 Problem (Maxmin reachability problem) Given a game decide whether ,τ val(s) = sup inf Pw s (F ) = 1. w ∈Σ1 τ ∈Σ2 Maxmin reachability problem is undecidable. Theorem Maxmin reachability problem is decidable for leaktight half-blind games. 3/9 The notion of leaks comes from: Fijalkow, Gimbert, Oualhadj, 2012. Deciding the Value 1 Problem for Probabilistic Leaktight Automata I Captures tightly the complications responsible for the undecidability of the value 1 problem I Gives a robust class of PFAs with decidable value 1 problem The leaktight class is robust: I Fijalkow, 2015. Profinite techniques for probabilistic automata and the optimality of the markov monoid algorithm. I Fijalkow, Gimbert, K, Oualhadj, 2015. Deciding the value 1 problem for probabilistic leaktight automata. 4/9 What are leaks? I + R0 R1 R0 R1 R0 R1 iterate II 5/9 A game G effectively A finite structure: the belief monoid has reachability witness does not have reachability witness val(s) = 1 val(s) < 1 For both implications the leaktight hypothesis is used. 6/9 I I Elements of the belief monoid: sets of binary matrices abstracting the outcomes of the game Two operations: I I Product, abstracting concatenation of two strategies for Max Iteration, abstracting the possibility of iterating a word many times 7/9 (a, 12 ) b s1 t1 a > F b (a, s2 α 1 2) b b t2 β ⊥ a s a 8/9 Thank you for your attention. 9/9
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