Games with perfect information

Games with perfect information

Exercise sheet 11

Sebastian Muskalla Summer term 2018

Out: June 27 Due: July 4

Submit your solutions on Wednesday, July 4, at the beginning of the lecture.

Please submit in groups of three persons.

Exercise 1: Counter machines

Show how to construct a counter machine of dimensiond⩾ 2 with two control statesq0,qfsuch that there is a transition sequence from(q0,n,m, . . .)that reachesqfif and only if

a) n⩾m, b) n< m,

c) nis divisible bym.

Hint:You may use an arbitrary constant number of additional counters.

Exercise 2: Primality testing

Show how to construct a counter machine of dimensiond ⩾ 1 with two statesq₀,q_fsuch that there isa transition sequence from(q₀,n, . . .)that reaches stateq_fif and only ifnisnota prime number. Explain your construction.

Hints: You may use an arbitrary constant number of additional counters. You can use non- determinism. You may split your construction into smaller parts (gadgets) and explain later how these should be combined.

Exercise 3: One-counter automata as pushdowns

Prove that one-counter automata can be simulated by pushdown systems.

Recall that a pushdown system is an automaton(Q,→)with memoryS^∗, whereSis some finite stack alphabet. The transition rules in→are of the shape

q −−−−−^push→^a p or q−−−−−^push→^a p for symbolsa∈S. There is a transition((q,m)→(p,m^′))∈Tif

• there is a ruleq−−−−−^push→^a pandm^′ =m.a, or

• there is a ruleq−−−−^pop→^a pandm =m^′.a.

(Here, we use the convention that the right end of the wordmencodes the top of stack.) Note that a popatransition is only enabled whenais indeed the top of stack.

Assume that some one-counter automatonA_OCA =(Q^′,→^′)with statesq₀,q_fis given. Show how to construct a pushdown systemA_PDS = (Q,→)withQ^′ ⊆ Qover a suitable stack alphabet such thatq_f is reachable inA_OCAfrom(q0,0)if and only ifq_f is reachable inA_PDS from some suitable initial configuration. Briefly argue that your construction is correct.

Exercise 4: Integer counter machines

Aninteger counter machineof dimensiondis defined similarly to a counter machine of dimen- siond. However, the counters can reach negative values, i.e. the memory isZ^d. A transition of typeq−−−^xi−−→pis enabled even if the value of counterx_iis zero.

a) LetAICM be an integer counter machine of dimensiond, andq0,qfcontrol states. Show how to construct a counter machine ACM with states q^′₀ and q^′_f such that qf is reachable from (q0,0, . . . ,0)inAICMif and only ifq^′_fis reachable from(q^′0,0, . . . ,0)inACM.

b) LetA_CM be a counter machine of dimensiond, andq^′₀,q^′_f control states. Show how to con- struct an integer counter machineAICM with statesq0andqf such thatqf is reachable from (q0,0, . . . ,0)inAICMif and only ifq^′_fis reachable from(q^′0,0, . . . ,0)inACM.

In both cases, argue briefly that your construction is correct.