CUADROS COMPARATIVOS:
IV. CONSUMO DE INSUMOS QUÍMICOS:
So far we recalled FO and MSO. In this section, we proceed by briefly intro- ducing the modal µ-calculus Lµ as well as its quantitative variant Qµ. As usual for modal logics, these logics are evaluated over Kripke structures. Definition 2.4.5. A Kripke structureK = (V, (Ea)a∈Σ,(Pi)i∈I) for some al-
phabet Σ and some index set I is a graph with Σ-labeled edges whose vertices are labeled by the monadic predicates Pi.
Syntactically, formulas of Lµ and Qµ are identical, and they are built according to the following grammar, where P is a monadic predicate, X is a monadic second-order variable, and a∈ Σ:
φ∶∶= P ∣ X ∣ ¬φ ∣ φ ∧ φ ∣ φ ∨ φ ∣ ◻aφ∣ ◇aφ∣ µX.φ ∣ νX.φ,
such that for subformulas µX.φ and νX.φ, X appears positively in φ (that is, under an even number of negations).
To speak about evaluations, let us first fix, given a Kripke structureK, the setFK= {0, 1}V of functions mapping vertices to truth values. This will
be used to evaluate second-order variables, and it can easily be adapted later to fit Qµ.
Definition 2.4.6. The semantics of Lµ, given a Kripke structure K and an evaluation β∶ V → FK of the second-order variables X, is a function
⟦⋅⟧K,β∶ V → {0, 1} that is defined recursively:
1. ⟦P⟧K,β∶ v ↦⎧⎪⎪ ⎨⎪⎪ ⎩ 1, v ∈ PK 0, otherwise. 2. ⟦X⟧K,β∶ v ↦ (β(X))(v). 3. ⟦¬φ⟧K,β(v) = 1 − ⟦φ⟧K,β(v). 4. ⟦φ ∧ ψ⟧K,β = min(⟦φ⟧K,β,⟦ψ⟧K,β). 5. ⟦φ ∨ ψ⟧K,β = max(⟦φ⟧K,β,⟦ψ⟧K,β).
6. ⟦◻aφ⟧K,β(v) = infv′∈vE
a⟦φ⟧K,β(v′).
7. ⟦◇aφ⟧K,β(v) = supv′∈vE
a⟦φ⟧K,β(v′).
8. ⟦µX.φ⟧K,β is the least fixed point of f ↦ ⟦φ⟧K,β[X↦f].
9. ⟦νX.ψ⟧K,β is the greatest fixed point of f ↦ ⟦φ⟧K,β[X↦f].
Note that the least and greatest fixed points always exist due to the Knaster-Tarski fixed point theorem, as the functions V → {0, 1} combined with the order≤ where f ≤ g if f(v) ≤ g(v) for all v ∈ V form a complete lattice, and the fixed-point operators are monotone.
A more accessible characterization of the semantics of the modal µ- calculus can be obtained via parity games: Indeed, it turns out that parity games are precisely the model-checking games for Lµ. The connection is especially tight in this case, as it is also possible to define the winning regions of parity games with a bounded number d of colors via a formula of Lµ using dalternations of least and greatest fixed points. For more details on this connection and on the modal µ-calculus in general, we refer to [13].
The semantics of the quantitative µ-calculus is defined similarly to that of Lµ, with a few differences: First of all, Qµ is evaluated on quantitative Kripke structures, which means that the monadic predicates are replaced with functions Pi∶ V → Z∞, that is, functions from the vertex set to the
integers together with∞ and −∞. Accordingly, FKis replaced with the set
of functions V → Z∞. When modifying the semantics of negation to−⟦φ⟧K,β
and changing the codomain of⟦⋅⟧A,βto Z∞, the semantics of Qµ is given.
Again, there is a close connection to quantitative parity games. For a more detailed overview, see [45] and Chapter 5. (Note that in general, Qµ also allows discounts on the edges, which is omitted here for the sake of simplicity.)
Counter Parity Games
In this chapter we introduce a class of games which combine a parity objective for infinite plays with a quantitative objective modeled via a finite set of counters for finite plays. Such games are related to other models of games with counters, such as for example cost games (see, e.g., [32,87]), but differ from these in that counters are used only for finite plays where the parity condition cannot be applied. The main motivation to study counter parity games comes from the field of quantitative logics, especially the quantitative µ-calculus, as is argued later.
The results of this chapter are joint work with Dietmar Berwanger and Łukasz Kaiser and appeared previously in [2, 3], although there they are presented using a different form of marks.
3.1
Definition
Counter parity games(CPGs) are parity games where quantitative outcomes of finite plays depend on a vector of counters. These counters are updated along the edges, while vertices are colored to determine the outcomes of infinite plays. In the following, we consider games with k counters, where k≥ 1 is a natural number. Thus, a counter evaluation can be represented as an element of Nk, and counter updates correspond to affine mappings of such
vectors. This leads to the following definition.
Definition 3.1.1. A counter parity game (CPG)G = (G, pay) with k counters is played on a finite arena G= (V, V0, V1, E, λ, Ω, τ), where
• Ω∶ V → [d] labels the vertices with colors,
• λ∶ E → E with E ∶= {f∶ Nk→ Nk, c↦ Ac + b ∣ A ∈ Nk×k, b∈ Nk} assigns
an affine mapping over k-dimensional vectors of naturals to each edge, 29
• and τ∶ T → {1, −1} × [k] assigns a sign and a counter index to every terminal position.
The payoff function of a CPG assigns∞ to infinite plays that satisfy the parity condition, and−∞ to infinite plays where the minimal color seen infinitely often is odd. For a finite play π= v0. . . vn with τ(vn) = (s, i), the payoff is
s⋅ cπ
i, where cπi is the i-th entry of
cπ ∶= λ(vn−1, vn)(λ(vn−2, vn−1)(. . . λ(v0, v1)(0k))).
Intuitively, this means that the game starts with counter evaluation 0k
and is played like a quantitative parity game, updating the counters on an edge from v to w from c to λ(v, w)(c). As usual, Pl. 0 wants to maximize the payoff, while Pl. 1 tries to minimize it. To avoid confusion regarding the roles of the players, we refer to Pl. 0 as Maximizer, and to Pl. 1 as Minimizer. Remark 3.1.2. Counter parity games are a subclass of infinite quantitative parity games. In fact, when considering the product of the arena of a CPG with Nkin such a way that((v, c), (w, c′)) is an edge if and only if (v, w) ∈ E
and λ(v, w)(c) = c′, Ω(v, c) = Ω(v) and τ(v, c) = s ⋅ ci if τ(v) = (s, i), then
the product is a quantitative parity game where there is an obvious one-to- one-mapping between plays in both games, and corresponding plays yield the same payoffs.
As quantitative parity games, even on infinite arenas, are determined, we obtain determinacy as in the following corollary.
Corollary 3.1.3. Counter parity games are determined. For every CPGG and every vertex v, valG(v) exists.
However, despite proving the existence of the value, the reduction to quantitative parity games does not yield a procedure to compute the value. We provide such an algorithm in the next sections that uses a different approach, namely a reduction to a win-lose game with imperfect recall, to compute a finite quantitative parity game with the same value. Before we do so, let us consider an example of a counter parity game.
Example 3.1.4. Consider the game depicted in Figure 3.1, where vertices of Maximizer are depicted as circles, while square vertices belong to Minimizer.
If the game starts at the middle vertex of Minimizer, then he may choose any number of repetitions of the right loop before moving the token to the vertex of Maximizer. Accordingly, the counter evaluation there is(10, n) for some n∈ N. At his turn, Maximizer may now choose to take the negative value of one of the two counters as payoff, or return to Minimizer’s vertex.
−c1 −c2 1 3 2 ×1,+1 ×1,×1 +1,×0 +10,×1
Figure 3.1: An example of a counter parity game
As the first counter is never decreased, but increases with every return to Minimizer’s vertex, while Minimizer can set the second counter to a value of his choice, it follows that the maximal outcome of finite plays that Minimizer allows is−10. As taking the right loop infinitely often is bad for Minimizer, he will move to the left loop eventually. As taking the loop is bad for Maximizer, Maximizer has to go to a terminal to avoid a payoff of−∞. Accordingly, the
value of the game is−10. ⊣
Applications Besides the theoretical interest in finitely representable infi- nite quantitative parity games, counter parity games have been applied to solve model-checking problems for variants of the quantitative µ-calculus.
In fact, the solvability result below improved the complexity of model- checking the quantitative µ-calculus on initialized linear hybrid systems from nonelementary to elementary complexity, as shown in [47].
Furthermore, the decidability result for the counting logic described in Chapter 5 on a generalization of pushdown systems is proved via a reduction from the infinite model-checking game to a finite counter parity game (see Chapter 5 and [63]).