After having discussed the single player cases of one-counter MCGs above, we now turn to mean counter games with one-counter where both players have nontrivial moves.
4.4.1
Bounds for the Value
Using an approach similar to the one for solitary Maximizer games, we prove that it is decidable whether or not the value of a game from a given starting position is finite. Therefore, we first give a criterion to determine a set of positions from which Maximizer can achieve arbitrarily high payoffs, and argue later that these are indeed all positions for which the value is∞. 4.4.1.1 Positions with an Unbounded Value
As we are considering two-player games now,∞-arenas do not suffice any- more: Minimizer could simply try to leave the arena. To overcome this problem, we construct automata to determine, via simple games, positions which correspond to starting positions of∞-arenas.
To simplify reasoning about these automata and games, we first construct the arena ˇGon the incidence graph of the arena G= (V, V0, V1, E, λ): This
arena ˇG= (V ∪E, V0,(V ∪E)∖V0, E′, Ω) has as vertex set the set V enriched
by an additional vertex for every edge e∈ E. The partitioning is adapted such that all new edge vertices belong to Minimizer. The new edge relation E′is
lifted in such a way that(v, e) ∈ E′ if and only if e= (v, w) for some w, and
(e, w) ∈ E′ if and only if e= (v, w) for some v. The former edge labeling is
now used to label the edge vertices, while the remaining vertices receive some dummy symbol as label. Note that edge vertices have unique incoming and outgoing edges, thus paths (and also strategies) translate directly between G and ˇG. As above, let A= {ι, c, r, n} be the set of edge labels.
We first determine the set S ⊆ ˇV of positions from where Maximizer can ensure that the current register value is not decreased in the future, assuming that the counter value is currently not smaller than the register value. Therefore, let AS = (QS, A, δS, qS, F) be the following automaton:
QS= {qS, qr , qf}, F = {qS, qr }, and δ(qS, a) =⎧⎪⎪⎨⎪⎪ ⎩ qS, a≠ r qr, a= r , δ(qr , a) =⎧⎪⎪⎨⎪⎪ ⎩ qr , a≠ c qf, a= c , δ(qf, a) = qf.
Thus,AS is an automaton with a safety condition that states that whenever
a reset occurs, no check may occur later. Taking the product of ˇGandAS
with the winning condition fromAS, the set S is the set of positions from
which Maximizer (i.e., Pl. 0) wins, which is obviously computable.
Let Sc ⊆ S ∩ E be the positions e in S for which Ω(e) = λ(e) = c.
As a next step, we are interested in the positions L from which Maximizer can enforce the play to reach Sc without a previous reset. These can be
Let Y be a set of vertices, let σ ∈ {0, 1}, and let Λ be a set of labels (of either vertices or edges). The(Λ, σ)-attractor AttrΛ
σ(Y ) of Y is the set of
positions from where P l. σ has a positional strategy to reach a position in Y seeing only labels from Λ along the way.
Note that attractors can be computed using a least fixed-point induction, or an automaton with a reachability condition. L is now precisely the set L= AttrAMaximizer∖{ r } (Sc), where we ignore the dummy labels on V -vertices.
At last, we define a game with a Büchi winning condition to determine the positions from where Maximizer can guarantee that the counter is increased arbitrarily high, then checked and no smaller value is ever checked again. We do so once more by constructing an appropriate automaton, and taking the product with the arena afterwards. LetAL
I = (QLI, V ∪ E, δLI, q¬ ι , F) with QL I = {q¬ ι , q ι , qf}, F = {qι }, δLI(q¬ ι , a) = δIL(qι , a) = ⎧⎪⎪⎪⎪ ⎪⎪⎪ ⎨⎪⎪⎪ ⎪⎪⎪⎪ ⎩ qf, a/∈ L qf, a∈ L, Ω(a) = r qι , a∈ L, Ω(a) = ι q¬ ι , otherwise and δL
I(qf, a) = qf. Let W′be the winning region of Maximizer in the game
on the product ofAL
I and ˇG, where Maximizer takes the role of Pl. 0. Let
now W = AttrMaximizer(W′) be the unrestricted attractor (i.e., ignoring inter-
mediate labels) of W′.
Lemma 4.4.1. For every v∈ W ∩ V it holds that valG(v) = ∞.
Proof. For every v∈ W ∩ V , Maximizer has a (positional) attractor strategy to reach W′. From every w∈ W′∩ V , Maximizer wins the respective Büchi
game, that is, he has a strategy to remain within L that sees increment edges infinitely often, but no reset. As the play remains in L, Maximizer can choose—at any time—to reach a position with a previous check and no intermediate reset which is furthermore contained in S (as L is the attractor of Sc without reset, and edge positions in ˇGhave unique successors). From
this position, however, he can ensure that either no future reset occurs, or after the reset no more checks are played. As at the position after the check, counter and register coincide, this means that the register is never again smaller than the value at this respective position.
In summary, for every given n ∈ N, Maximizer can play a strategy σn
that reaches W′, follows the Büchi strategy until at least n increments have
safety strategy afterwards. It follows that valG(v) ≥ sup n∈N valσnG(v) ≥ sup n∈N n= ∞.
In other words, at positions in W , for every n ∈ N Maximizer has a strategy to increase the counter to at least n and keep this number in the register (or possibly larger values). As a next step, we use this to show that the value from positions not in W is in fact bounded.
4.4.1.2 Positions with a Bounded Value
Assume that a position is not in W′. Thus, there exists some n∗∈ N such that
for all n≥ n∗, Maximizer has no strategy to keep values of at least n in the
register from some point onward. As the games are determined, Minimizer has a strategy to ensure that whenever a value larger than n∗is checked, a
value of less than n∗ is checked later. Clearly, as the only way to check a
smaller value is via a reset after the previous check, and enforcing a check after such a reset is an attractor condition, Minimizer has a strategy to assure that a smaller value is checked after at most 2N ∶= 2∣V ∣ steps, and that this value is less than N. In total, there exists some n∗such that Minimizer can
play in such way that whenever a value of at least n∗ is checked, after at
most 2N steps the register value is again below N. We prove next a lemma that abstractly states that all plays satisfying this condition yield a bounded payoff.
Lemma 4.4.2. Let n∗, N ∈ N with n∗≥ N be two numbers. There exists a constant Bn∗ ∈ N such that for all sequences (ai)i∈N of natural numbers that
satisfy the conditions 1.-3., 1. a0= 0.
2. If ai < aj where i < j and ai−1 < ai as well as ak = ai for all i< k < j,
then aj≤ ai+ (j − i).
3. If ai ≥ n∗and ai−1< n∗, then there exists a j≤ 2N such that ai+j< N.
it holds that lim infn→∞∑
n i=0ai
n+1 ≤ Bn∗.
Proof. If only finitely many indices i exist with ai≥ n∗, then from some point
onward, the sequence is always below n∗, thus the limit average is also at
most n∗. Hence we assume that the sequence reaches n∗infinitely often.
Let(si)i∈N, si < si+1 be the sequence of indices s such that as ≥ n∗ and
be a function that maps s∈ (si) to the minimal t > s such that at< n∗(thus,
f(s) ≤ s + 2N), and is arbitrary otherwise. Obviously,
si+1−1
∑
j=f(si)
aj ≤ n∗⋅ ((si+1) − f(si))
for all i, and ∑s0−1
j=0 aj ≤ n∗s0. Furthermore, we have for all i f(si)−1 ∑ j=si aj ≤ (f(si) − si) ⋅ (N + (si− f(si−1))) + ( f(si) − si)2 2 ≤ 2N2+ 2N(s i− f(si−1)) + 4N2 2 ≤ 4(n∗)2+ 2n∗(s i− f(si−1)), as n∗≥ N, af(s
i−1)< N and the growth of the sequence respects 2.
Consider now the subsequence from position f(si) to position f(si+1).
It follows that f(si+1) ∑ i=f(si) ai≤ 4(n∗)2+ 3n∗(si+1− f(si)) ≤ (4(n∗)2+ 3n∗)(f(s i+1) − f(si)).
Accordingly, for all i∈ N,
f(si)
∑
j=0
aj ≤ f(si) ⋅ (4(n∗)2+ 3n∗) .
Clearly, lim infn→∞∑
n i=0ai
n+1 ≤ 4(n∗)2+ 3n∗=∶ Bn∗.
Lemma 4.4.3. Let v∈ V ∖ W. Then valG(v) < ∞.
Proof. As v /∈ W, there exists an n∗ such that Minimizer can ensure that whenever a value of at least n∗is checked, a value of at most N is checked in
the subsequent 2N steps. Clearly, the sequence of register values of any such play meets the requirements of Lemma 4.4.2, hence the value is bounded. Corollary 4.4.4. valG(v) = ∞ if and only if v ∈ W. Furthermore, this can be decided effectively.
In the above result, no information about the size of the possible bound is given. Although we conjecture that possible bounds on the value are rather small, we have not proved this so far. In fact, it seems plausible that
a bound linear in the size of the arena suffices, as whenever counters grow larger, vertices are repeated. It might be that Minimizer could then adapt his strategies for large values to play against smaller values (which are still larger than the size of the arena). However, further insight into the shape of strategies of both Maximizer and Minimizer seems required to prove or disprove the use of such adaptions.