In this chapter we presented counter parity games, which are a special case of infinite quantitative parity games and which are useful in the field of quantita- tive logics. In fact, they have been used to model-check both the quantitative µ-calculus on initialized linear hybrid systems as well as a counting variant of this µ-calculus on a class of pushdown systems. We reduced the problem of solving such games to solving a game with both imperfect information and imperfect recall, for which we proved that finite-memory strategies suffice for one player. The overall proof yields an algorithm which is in 4Exptime for a fixed number of counters, and in 6Exptime otherwise.
Note that the aforementioned application to solve a counting variant of Qµ is discussed in Chapter 5.
Mean Counter Games
In the previous chapter, we discussed counter parity games, a class of quan- titative games where the quantitative aspect is modeled via a finite set of counters. However, in these games the payoffs of infinite plays, thus of models of infinite behavior, depend only on the qualitative parity condition, while the counters are only used to give payoffs to finite plays. In the current chapter, we discuss a variant where the payoff of infinite plays also depends on counter values. As it is known that parity games can be modeled as mean- payoff games, and solving these is also in Np∩ coNp, our approach is based on the mean-payoff idea: Instead, however, of adding weights to the edges, we consider the average value of a designated register. This register cannot be accessed directly, but counter values can be copied there, which allows a manipulation of counters to be of effect only several steps later.
Part of the work in this chapter has been done jointly with Dietmar Berwanger and Marcus Gelderie, in particular the ideas to remove cycles and to search for a convenient notion of a pattern were developed together.
4.1
Definition and Examples
In contrast to counter parity games, we use a simpler class of counter manipu- lations for mean counter games: as in many models of automata with counters (e.g., B- and S-automata, see Chapter 6.3), counters may be increased, reset, or checked.
Definition 4.1.1. A mean counter game (MCG)G = (G, pay) with k counters is played on a finite arena G= (V, V0, V1, E, λ) without terminal positions,
where λ∶ E → {ι , c, r, n}k labels edges with k counter update actions,
such that λ(e) contains at most one c for every e∈ E.
v w
ι
c
r
Figure 4.1: An example of a mean counter game
Given an infinite path α= v0v1. . ., this path induces a sequence of counter
evaluations c= c0c1⋅ ⋅ ⋅ ∈ (Nk)ω, where c0 = 0k, and
cij+1∶=⎧⎪⎪⎪⎪⎨⎪⎪⎪ ⎪⎩ ci j+ 1, λ(vi, vi+1)j = ι, 0, λ(vi, vi+1)j = r. ci j, λ(vi, vi+1)j ∈ {c, n}.
Based on c, we define the sequence ρ= r0r1⋅ ⋅ ⋅ ∈ Nω of register values by
r0∶= 0, ri+1∶=⎧⎪⎪⎨⎪⎪
⎩
ri, c /∈ λ(vi, vi+ 1),
ci
j, λ(vi, vi+1)j = c.
The payoff function pay computes the limit inferior of the averages of prefixes of this sequence:
pay(α) ∶= lim inf
n→∞
∑n i=0ri
n+ 1 . As before, we call Pl. 0 Maximizer and Pl. 1 Minimizer.
Example 4.1.2. Before we discuss determinacy of MCGs, we consider a brief example. Consider the arena for one counter depicted in Figure 4.1. Assume first that all vertices belong to Minimizer. In this case, it is optimal to never choose the ι-loop. Thus, all counters always stay 0, and the sequence ρ of
register values is also constantly 0. Accordingly, the value is 0.
Assume now that all positions belong to Maximizer. Clearly, never taking the ι-edge is not reasonable. On the other hand, always taking it is neither,
as no value is checked in this case. We further remark that it is of no use for Maximizer to take the c r-loop twice in a row. Thus consider a play that
is of the form(ι nc r)ωfor some 1< n ∈ N. From some point onward, the
sequence ρ of register values will also be constant, but will always be n. As this can be achieved for every n, it follows that the value of the game is∞. In the present case, there are also optimal strategies, but they require infinite memory: for example, Maximizer could increase the number of ι -cycle
v w
ι n
c
Figure 4.2: Maximizer does not have an optimal strategy
Nevertheless, optimal strategies for Maximizer need not exist in general. One can easily construct a single player mean counter game played only by Maximizer where no strategy is optimal: There are two vertices, v and w, such that v has a self-loop along which the sole counter is increased, and an edge to w labeled with c. The vertex w only has an n-labeled self-loop.
This game, depicted in Figure 4.2, can be viewed as Maximizer choosing an arbitrary natural number which is then awarded to him as payoff. Of course, the supremum over all his strategies is unbounded, hence the value is∞. But every fixed strategy yields only a finite payoff: 0 if w is not reached, and some n∈ N otherwise.
Despite the fact that optimal strategies may not exist, mean counter games are always determined, which follows from Martin’s theorem about Borel determinacy [74].
Theorem 4.1.3. LetG be a mean counter game. G is determined.
Proof. We implicitly considerG from some fixed initial vertex v. Payoffs of mean counter games are always elements of R≥0∪ {∞}. Fix thus an arbitrary
threshold value δ∈ R≥0∪ {∞}. We prove that for every such δ, the win-lose
game where Maximizer wins a play α if and only if pay(α) ≥ δ is a Borel game, and hence determined. Since the rationals are dense in Q, this means that ρ satisfies ∀(ϵ ∈ Q>0)∃(n 0∈ N)∀(n > n0) ( 1 n n−1 ∑ i=0 ri > δ − ϵ) .
Fixing ϵ, n0 and n, the set of paths where the average of the first n values of
the respective ρ exceeds δ− ϵ is an open set x ⋅ Vω∩ Paths(G). (In fact, it is a
clopenset, as the complement is also open by the same argument.) It follows that the set of paths such that Maximizer wins is a Borel set, and contained in Π0
4as a countable intersection of a countable union of a countable intersection
of clopen sets. By [74], the win-lose game is determined for every threshold value δ.
Furthermore, if σ0 is a winning strategy for Maximizer for threshold
value δ, then it is also winning for every δ′< δ. Similarly, a strategy σ1 for
v w u
ι
c
r c
Figure 4.3: Maximizer needs infinite memory despite the value being finite now S0be the set of all δ such that Maximizer has a winning strategy, and let
S1 be the analogous set for Minimizer. Clearly, sup S0= inf S1= valG.
We already argued that Maximizer may not have optimal strategies, and that even if he does, he may require infinite memory. Note that even if the value of a game is finite, he may not have finite-memory strategies, as shown in the following example.
Example 4.1.4. Consider the single player Maximizer game in Figure 4.3. In this game, Maximizer can increase the counter while seeing register value 0, and then see the large counter for two steps until it is reset and the process starts all over again. Clearly, choosing larger counter values is advantageous for Maximizer; if he always chooses value N, the payoff is
2N N + 3 <2.
If the sequence of the respective N is monotonically increasing in every pass through v (which requires infinite memory), then the lim inf of the average
value of ρ is indeed 2. ⊣
As the register values are always positive, and discrete, it seems that the situation for Minimizer is different. Still, in contrast to classical mean payoff games, positional strategies do not suffice:
Example 4.1.5. Consider a game where Maximizer can reach vertex v via either setting counter c0 to 2 or counter c1 to 3, such that the other counter
remains 0. Minimizer may then, at v, either choose c0 or c1 as the payoff of
the play. Of course, this cannot be done positionally, as information about
which counter is 0 is required. ⊣
We further remark that if the value is∞, then every strategy of Minimizer is optimal. The findings above are summarized as follows.
2. If Maximizer has optimal strategies, they may require infinite memory. 3. Optimal strategies of Minimizer may require memory.
We conjecture further that finite memory is sufficient for Minimizer, and that Maximizer has optimal strategies whenever the value is less than∞. In the following, we focus on subclasses of mean counter games and prove that these can be solved. We study solitary one-counter games, both for Minimizer and Maximizer, which can also be viewed as special kinds of automata. We then prove that boundedness of the value of two-player one-counter games can be decided. Furthermore, we introduce a generalized version of mean