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[[:lat|Back to course homepage.]] ====== LAT Homework ====== ==== 02.10.2009: ==== Consider the automata construction from proof of the Myhill-Nerode theorem for a language L. Prove that: * (a) the constructed automaton A is the minimal automaton (in the number of states) that recognizes L; * (b) any minimal automaton recognizing L is isomorphic to A. - 06.11.2009: - Given a reachability game (G,F), give an algorithm that computes the 0-Attractor(F) set in time O(|E|). - Consider the game graph shown in below and the following winning conditions: a) Occ(ρ) ∩ {1} ≠ ∅ and b) Occ(ρ) ⊆ {1,2,3,4,5,6} and c) Inf(ρ) ∩ {4,5} ≠ ∅. Compute the winning regions and corresponding winning strategies showing the intermediate steps (i.e., the Attractor and Recurrence sets) of the computation. - Given a game graph G=(S,S₀,T) and a set F ⊆ S. Let W₀ and W_1 be the winning regions of Player 0 and Player 1, respectively, in the Buchi game (G,F). Prove or disprove: - The winning set of Player 0 in the safety game for (G,W₀) is W₀, - If f₀ is a winning strategy for Player 0 in the safety game for (G,W₀), then f₀ is also a winning strategy for Player 0 in the Buchi game for (G,F), - the winning set of Player 1 in the guaranty game for (G,W_1) is W_1, and - if f_1 is a winning strategy for Player 1 in the guaranty game for (G,W₀), then f_1 is a winning strategy in the Buchi game (G,F). {{hw1.gif|Figure 1}} - 13.11.2009: - Consider the game graph shown below. Let the winning condition for Player 0 be Occ(ρ)={1,2,3,4,5,6,7} - Find the winning region for Player 0 and describe a winning strategy - Show that there is no positional winning strategy for Player 0 in this game.\\ {{hw3.gif|Figure 2}} - Compute the winning regions and the corresponding positional winning strategies for Player 0 and 1 in the weak-parity game shown below. {{hw2.gif|Figure 3}} - A winning strategy is called "uniform" if it is a winning strategy from every winning state in the game. Let (G,p) be a weak parity game and let W₀ be the winning region of Player 0. For all s ∈ W₀ let f_s be a positional winning strategy from s for Player 0. Construct a uniform winning strategy f from the strategies f_s meaning that for every s ∈ W₀ there is a t ∈ W₀, s.t. f(s) = f_t(s). - 27.11.2009: - Consider the game graph below and the Muller condition F={{2,4,5,7},{1,2,3,4,5,6,7}}. Find an automaton winning strategy for Player 0 in the Muller game with as few states as possible and show that any other automaton strategy with less states is not winning for Player 0. {{hw4.gif}} - Consider the game shown on page 6 of {{slide10.pdf}}. We propose the following "latest appearance queue" (LAQ) strategy for Player 0, initializing the queue with s₁s₂s₃s₄, (1) add the current state at the front of the LAQ and delete the last state (2) move to the state t_i whose number i is the number of different states in the current LAQ, e.g., for the sequence of states s₁, s₃, s₃, s₄, s₁,... we obtain the following sequence of LAQs: s₁s₂s₃s₄, s₁s₁s₂s₃, s₃s₁s₁s₂, s₃s₃s₁s₁, s₄s₃s₃s₁,s₁s₄s₃s₃,... Decide wheater the new LAQ strategy is a winning strategy for Player 0. Prove this or give a counter-example. - Let A=(S,s₀,T,p) be a parity automaton with p:S -> {1,..,k}. Construct a Streett automaton B=(S,s₀,T,F) equivalent to A that has the same transition structure as A. (i.e., find a suitable set of Streett pairs (F_i,E_i))