Q2. What is the idea of the simulation of a succinct graph?
The idea is to simulate a traversal of the succinct graph in their MPG: if the authors make 2N valid steps without revisiting a vertex of the succinct graph then that guarantees a Hamiltonian cycle.
Q3. What is the upper bound on the memory required in the forcibly terminating game?
Note that an upper bound on the memory required is the number of states in the reachability game restricted to a winning strategy, and this is exponential in N , the bound obtained in Lemma 15.
Q4. What is the determinacy result of the statement that the limit inferior of a given sequence?
More precisely, the statement that the limit inferior of a given sequence (an)n∈N is non-negative is a Π03-statement (for every k there exists a t such that for all n ≥ t an ≥ −2−k).
Q5. What is the corresponding exponential increase in the complexity of the class membership problem?
Given the exponential blow-up in the construction of the game of limited observation, it is not surprising that there is a corresponding exponential increase in the complexity of the class membership problem.
Q6. What is the weight on the edge between the copy of state c+i and the copy?
As you can see, the weight on the edge between the copy of state c+i of observation s to the copy of this state in observation s′ is equal to +1, while the weight on the edge between the copy of state c−i of observation s to the copy of this state in observation s′ is equal to −1.
Q7. What is the simplest way to encode the value of a counter?
an abstract path, corresponding to the simulation of a run of the machine, will encode the value of counter i, at each step, as the weight of the shortest suffix from the initial pumping gadget to c+i .
Q8. What is the first class of games obtained when their cycle-forming game is restricted to simple cycles?
The first, forcibly first abstract cycle games (forcibly FAC games, for short), is the natural class of games obtained when their cycle-forming game is restricted to simple cycles.
Q9. How many copies of each observation are required?
The authors require two copies of each such observation since, in order to punish Adam or Eve (whoever plays the role of Simulator), existential and universal gadgets have to be set up in a different manner.
Q10. What is the sufficiency condition for si?
The sufficient condition for si will include the sufficiency condition for any runs with indices from Si having the same current vertex and current mean-payoff difference at most 2−i−1.
Q11. How does the Ramsey theorem prove that there are suitable collections?
The existence of suitable collections will be proven by iterative applications of Ramsey’s theorem:Theorem 3 (Infinite Ramsey’s theorem).