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--- lat:homework [2009/11/27 22:50]
barbara.jobstmann
+++ lat:homework [2009/11/27 22:51]
barbara.jobstmann
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 ====== LAT Homework ======
-===== 02.10.2009: =====
+===== 02.10.2009 =====
  Consider the automata construction from proof of the Myhill-Nerode theorem for a language L. Prove
   * (a) the constructed automaton A is the minimal automaton (in the number of states) that recognizes L;
@@ Line 8: / Line 8: @@
-== 06.11.2009: ==
+===== 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.
@@ Line 15: / Line 15: @@
        - If f<sub>0</sub> is a winning strategy for Player 0 in the safety game for (G,W<sub>0</sub>), then f<sub>0</sub> 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,W<sub>0</sub>), then f_1 is a winning strategy in the Buchi game (G,F). {{hw1.gif|Figure 1}}
+       - if f_1 is a winning strategy for Player 1 in the guaranty game for (G,W<sub>0</sub>), then f_1 is a winning strategy in the Buchi game (G,F).
-== 13.11.2009: ==
+{{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
@@ Line 24: / Line 26: @@
      - 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 W<sub>0</sub> be the winning region of Player 0. For all s ∈ W<sub>0</sub> 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 ∈ W<sub>0</sub> there is a t ∈ W<sub>0</sub>, s.t. f(s) = f_t(s).
-== 27.11.2009: ==
+===== 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 s<sub>1</sub>s<sub>2</sub>s<sub>3</sub>s<sub>4</sub>, (1) add the current state at the front of the LAQ and delete the last state (2) move to the state t<sub>i</sub> whose number i is the number of different states in the current LAQ, e.g., for the sequence of states s<sub>1</sub>, s<sub>3</sub>, s<sub>3</sub>, s<sub>4</sub>, s<sub>1</sub>,... we obtain the following sequence of LAQs: s<sub>1</sub>s<sub>2</sub>s<sub>3</sub>s<sub>4</sub>, s<sub>1</sub>s<sub>1</sub>s<sub>2</sub>s<sub>3</sub>, s<sub>3</sub>s<sub>1</sub>s<sub>1</sub>s<sub>2</sub>, s<sub>3</sub>s<sub>3</sub>s<sub>1</sub>s<sub>1</sub>, s<sub>4</sub>s<sub>3</sub>s<sub>3</sub>s<sub>1</sub>,s<sub>1</sub>s<sub>4</sub>s<sub>3</sub>s<sub>3</sub>,...     Decide wheater the new LAQ strategy is a winning strategy for Player 0. Prove this or give a counter-example.