Parameterized complexity of games with monotonically ordered omegaregular objectives
C O N C U R
Parameterized complexity of games with monotonically ordered ωregular objectives
Véronique Bruyère Département d'informatique 0
Université de Mons (UMONS) 0
Belgium 0
Quentin Hautem 0
Département d'informatique 0
Université de Mons (UMONS) 0
Belgium 0
JeanFrançois Raskin 0
0 Département d'informatique, Université Libre de Bruxelles (ULB) , Brussels , Belgium
In recent years, twoplayer zerosum games with multiple objectives have received a lot of interest as a model for the synthesis of complex reactive systems. In this framework, Player 1 wins if he can ensure that all objectives are satisfied against any behavior of Player 2. When this is not possible to satisfy all the objectives at once, an alternative is to use some preorder on the objectives according to which subset of objectives Player 1 wants to satisfy. For example, it is often natural to provide more significance to one objective over another, a situation that can be modelled with lexicographically ordered objectives for instance. Inspired by recent work on concurrent games with multiple ωregular objectives by Bouyer et al., we investigate in detail turnedbased games with monotonically ordered and ωregular objectives. We study the threshold problem which asks whether player 1 can ensure a payoff greater than or equal to a given threshold w.r.t. a given monotonic preorder. As the number of objectives is usually much smaller than the size of the game graph, we provide a parametric complexity analysis and we show that our threshold problem is in FPT for all monotonic preorders and all classical types of ωregular objectives. We also provide polynomial time algorithms for Büchi, coBüchi and explicit Muller objectives for a large subclass of monotonic preorders that includes among others the lexicographic preorder. In the particular case of lexicographic preorder, we also study the complexity of computing the values and the memory requirements of optimal strategies. 2012 ACM Subject Classification Theory of computation → Fixed parameter tractability, Theory of computation → Algorithmic game theory 1 Quentin Hautem is supported by a FRIA fellowship. 2 JeanFrançois Raskin is supported by the ERC Starting Grant inVEST (279499), by the ARC project “NonZero Sum Game Graphs: Applications to Reactive Synthesis and Beyond” (Fédération WallonieBruxelles), and by the EOS project “Verifying Learning Artificial Intelligence Systems” (FNRSFWO), and he is Professeur Francqui de Recherche funded by the Francqui foundation.
and phrases twoplayer zerosum games played on graphs; ωregular objectives; ordered objectives; parameterized complexity

Bruyère and JeanFrançois Raskin are both supported by the FNRS PDR project “Subgame
perfection in graph games” (T.0088.18).
Acknowledgements We would like to thank Antonia Lechner for useful discussions.
1
Introduction
Twoplayer zerosum games played on directed graphs form an adequate framework for the
synthesis of reactive systems facing an uncontrollable environment [21]. To model properties
to be enforced by the reactive system within its environment, games with Boolean objectives
and games with quantitative objectives have been studied, for example games with ωregular
objectives [15] and meanpayoff games [23].
Recently, games with multiple objectives have received a lot of attention since in practice,
a system must usually satisfy several properties. In this framework, the system wins if it
can ensure that all objectives are satisfied no matter how the environment behaves. For
instance, generalized parity games are studied in [11], multimeanpayoff games in [22], and
multidimensional games with heterogeneous ωregular objectives in [7].
When multiple objectives are conflicting or if there does not exist a strategy that can
enforce all of them at the same time, it is natural to consider tradeoffs. A general framework
for defining tradeoffs between n (Boolean) objectives Ω1, . . . , Ωn consists in assigning to
each infinite path π of the game a payoff v ∈ {0, 1}n such that v(i) = 1 iff π satisfies Ωi,
and then to equip {0, 1}n with a preorder  to define a preference between pairs of payoffs:
v  v0 whenever payoff v0 is preferred to payoff v. Because the ideal situation would be
to satisfy all the objectives together, it is natural to assume that the preorder  has the
following monotonicity property: if v0 is such that whenever v(i) = 1 then v0(i) = 1, then it
should be the case that v0 is preferred to v.
As an illustration, let us consider a game in which Player 1 strives to enforce three
objectives: Ω1, Ω2, and Ω3. Assume also that Player 1 has no strategy ensuring all three
objectives at the same time, that is, Player 1 cannot ensure the objective Ω1 ∩ Ω2 ∩ Ω3. Then
several options can be considered, see e.g. [6]. First, we could be interested in a strategy of
Player 1 ensuring a maximal subset of the three objectives. Indeed, a strategy that enforces
both Ω1 and Ω3 should be preferred to a strategy that enforces Ω3 only. This preference is
usually called the subset preorder. Now, if Ω1 is considered more important than Ω2 itself
considered more important than Ω3, then a strategy that ensures the most important possible
objective should be considered as the most desirable. This preference is called the maximize
preorder. Finally, we could also translate the relative importance of the different objectives
into a lexicographic preorder on the payoffs: satisfying Ω1 and Ω2 would be considered as
more desirable than satisfying Ω1 and Ω3 but not Ω2. Those three examples are all monotonic
preorders.
In this paper, we consider the following threshold problem: given a game graph G, a set
of ωregular objectives3 Ω1, . . . , Ωn, a monotonic preorder  on the set {0, 1}n of payoffs,
and a threshold μ, decide whether Player 1 has a strategy such that for all strategies of
Player 2, the outcome of the game has payoff v greater than or equal to μ (for the specified
preorder), i.e. μ  v. As the number n of objectives is typically much smaller than the size
of the game graph G, it is natural to consider a parametric analysis of the complexity of
the threshold problem in which the number of objectives and their size are considered to be
fixed parameters of the problem. Our main results are as follows.
3 We cover all classical ωregular objectives: reachability, safety, Büchi, coBüchi, parity, Rabin, Streett,
explicit Muller, or Muller.
Contributions
First, we provide fixed parameter tractable solutions to the threshold problem for all monotonic
preorders and for all classical types of ωregular objectives. Our solutions rely on the following
ingredients:
1. We show that solving the threshold problem is equivalent to solve a game with a
single objective Ω that is a union of intersections of objectives taken among Ω1, . . . , Ωn
(Theorem 3). This is possible by embedding the monotonic preorder  in the subset
preorder and by translating the threshold μ in preorder  into an antichain of thresholds
in the subset preorder. A threshold in the subset preorder is naturally associated with
a conjunction of objectives, and an antichain of thresholds leads to a union of such
conjunctions.
2. We provide a fixed parameter tractable algorithm to solve games with a single objective
Ω as described previously for all types of ωregular objectives Ω1, . . . , Ωn, leading to a
fixed parameter algorithm for the threshold problem (Theorem 4). Those results build
on the recent breakthrough of Calude et al. that provides a quasipolynomial time
algorithm for parity games as well as their fixed parameter tractability [9], and on the
fixed parameter tractability of games with an objective defined by a Boolean combination
of Büchi objectives (Proposition 5).
Second, we consider games with a preorder  having a compact embedding, with the main
condition that the antichain of thresholds resulting from the embedding in the subset preorder
is of polynomial size. The maximize preorder, the subset preorder, and the lexicographic
preorder, given as examples above, all possess this property. For games with a compact
embedding, we go beyond fixed parameter tractability as we are able to provide deterministic
polynomial time solutions for Büchi, coBüchi, and explicit Muller objectives (Theorem 6).
Polynomial time solutions are not possible for the other types of ωregular objectives as
we show that the threshold problem for the lexicographic preorder with reachability, safety,
parity, Rabin, Streett, and Muller objectives cannot be solved in polynomial time unless
P = PSPACE (Theorem 7). Finally, we present a full picture of the study of the lexicographic
preorder for each studied objective. We give the exact complexity class of the threshold
problem, show that we can obtain the values from the threshold problem (which thus yields
a polynomial algorithm for Büchi, coBüchi and Explicit Muller objectives, and an FPT
algorithm for the other objectives) and provide tight memory requirements for the optimal
and winning strategies (Table 2).
Related Work
In [6], Bouyer et al. investigate concurrent games with multiple objectives leading to payoffs
in {0, 1}n which are ordered using Boolean circuits. While their threshold problem is slightly
more general than ours, their games being concurrent and their preorders being not necessarily
monotonic, the algorithms that they provide are nondeterministic and guess witnesses whose
size depends polynomially not only in the number of objectives but also in the size of the
game graph. Their algorithms are sufficient to establish membership to PSPACE for all
classical types of ωregular objectives but they do not provide a basis for the parametric
complexity analysis of the threshold problem. In stark contrast, we provide deterministic
algorithms whose complexity only depends polynomially in the size of the game graph. Our
new deterministic algorithms are thus instrumental to a finer complexity analysis that leads
to fixed parameter tractability for all monotonic preorders and all ωregular objectives. We
also provide tighter lowerbounds for the important special case of lexicographic preorder, in
particular for parity objectives.
The particular class of games with multiple Büchi objectives ordered with the maximize
preorder has been considered in [2]. The interested reader will find in that paper clear practical
motivations for considering multiple objectives and ordering them. The lexicographic ordering
of objectives has also been considered in the context of quantitative games: lexicographic
meanpayoff games in [5], some special cases of lexicographic quantitative games in [8, 16],
and lexicographically ordered energy objectives in [12].
In [1] and [19], the authors investigate partially (or totally) ordered specifications expressed
in LTL. None of their complexity results leads to the results of this paper since the complexity
is de facto much higher with objectives expressed in LTL. Moreover no FPT result is provided
in those references.
Structure of the paper
In Section 2, we present all the useful notions about games with monotonically ordered
ωregular objectives. In Section 3, we show that solving the threshold problem is equivalent to
solve a game with a single objective that is a union of intersections of objectives (Theorem 3),
and we establish the main result of this paper: the fixed parameter complexity of the threshold
problem (Theorem 4). Section 4 is devoted to games with a compact embedding and in
particular to the threshold problem for lexicographic games. The last section is dedicated to
the study of computing the values and memory requirements of optimal strategies in the
case of lexicographic games (Table 2). Full paper is available on arXiv.4
2
Monotonically ordered ωregular games
We consider zerosum turnbased games played by two players, P1 and P2, on a finite directed
graph. Given several objectives, we associate with each play of this game a vector of bits
called payoff, the components of which indicate the objectives that are satisfied. The set of
all payoffs being equipped with a preorder, P1 wants to ensure a payoff greater than or equal
to a given threshold against any behavior of P2. In this section we give all the useful notions
and the studied problem.
Preorders
Given some nonempty set P , a preorder over P is a binary relation  ⊆ P × P that is
reflexive and transitive. The equivalence relation ∼ associated with  is defined such that
x ∼ y if and only if x  y and y  x. The strict partial order ≺ associated with  is then
defined such that x ≺ y if and only if x  y and x 6∼ y. A preorder  is total if x  y
or y  x for all x, y ∈ P . A set S ⊆ P is upperclosed if for all x ∈ S, y ∈ P , if x  y,
then y ∈ S. An antichain is a set S ⊆ P of pairwise incomparable elements, that is, for all
x, y ∈ S, if x 6= y, then x 6 y and y 6 x.
Game structures and strategies
A game structure is a tuple G = (V1, V2, E) where (V, E) is a finite directed graph, with
V = V1 ∪ V2 the set of vertices and E ⊆ V × V the set of edges such that for each v ∈ V ,
there exists (v, v0) ∈ E for some v0 ∈ V (no deadlock), and (V1, V2) forms a partition of V
such that Vi is the set of vertices controlled by player Pi with i ∈ {1, 2}.
A play of G is an infinite sequence of vertices π = v0v1 . . . ∈ V ω such that (vk, vk+1) ∈ E
for all k ∈ N. We denote by Plays(G) the set of plays in G. Histories of G are finite
sequences ρ = v0 . . . vk ∈ V + defined in the same way. Given a play π = v0v1 . . ., the
set Occ(π) denotes the set of vertices that occur in π, and the set Inf(π) denotes the set
of vertices visited infinitely often along π, i.e., Occ(π) = {v ∈ V  ∃k ≥ 0, vk = v} and
Inf(π) = {v ∈ V  ∀k ≥ 0, ∃l ≥ k, vl = v}. Given a set Ω ⊆ V ω, we denote by Ω the set
V ω \ Ω.
A strategy σi for Pi is a function σi : V ∗Vi → V assigning to each history ρv ∈ V ∗Vi
a vertex v0 = σi(ρv) such that (v, v0) ∈ E. It is memoryless if σi(ρv) = σi(ρ0v) for all
histories ρv, ρ0v ending with the same vertex v, that is, if σi is a function σi : Vi → V . It is
finitememory if σi(ρv) only needs finite memory of the history ρv (recorded by a Moore
machine). The size of σi is the size of its Moore machine.
The set of all strategies of Pi is denoted by Σi. Given a strategy σi of Pi, a play
π = v0v1 . . . of G is consistent with σi if vk+1 = σi(v0 . . . vk) for all k ∈ N such that vk ∈ Vi.
Given an initial vertex v0, and a strategy σi of each player Pi, we have a unique play
consistent with both strategies σ1, σ2, called outcome and denoted by Out(v0, σ1, σ2).
Single objectives and ordered objectives
An objective for P1 is a set of plays Ω ⊆ Plays(G). A game (G, Ω) is composed of a game
structure G and an objective Ω. A play π is winning for P1 if π ∈ Ω, and losing otherwise.
As the studied games are zerosum, P2 has the opposite objective Ω, meaning that a play π
is winning for P1 if and only if it is losing for P2. Given a game (G, Ω) and an initial vertex
v0, a strategy σ1 for P1 is winning from v0 if Out(v0, σ1, σ2) ∈ Ω for all strategies σ2 of P2.
Vertex v0 is thus called winning for P1. We also say that P1 is winning from v0 or that he
can ensure Ω from v0. Similarly the winning vertices of P2 are those from which P2 can
ensure his objective Ω.
A game (G, Ω) is determined if each of its vertices is either winning for P1 or winning for
P2. Martin’s theorem [20] states that all games with Borel objectives are determined. The
problem of solving a game (G, Ω) means to decide, given an initial vertex v0, whether P1 is
winning from v0 (or dually whether P2 is winning from v0 when the game is determined).
Instead of a single objective Ω, one can consider several objectives Ω1, . . . , Ωn that are
ordered with respect to a preorder  over {0, 1}n in the following way. We first define
the payoff of a play as a vector5 of bits the components of which indicate the objectives
that are satisfied. Formally, given n objectives Ω1, . . . , Ωn ⊆ Plays(G), the payoff function
Payoff : Plays(G) → {0, 1}n assigns a vector of bits to each play π ∈ Plays(G), where for all
k ∈ {1, . . . , n}, Payoffk(π) = 1 if π ∈ Ωk and 0 otherwise.
Given the preorder  over {0, 1}n, P1 prefers a play π to a play π0 whenever Payoff(π0)
Payoff(π). We call ordered game the tuple (G, Ω1, . . . , Ωn, ), the payoff function of which is
defined w.r.t. the objectives Ω1, . . . , Ωn and its values are ordered with . In this context,
we are interested in the following problem.
I Problem 1. The threshold problem for ordered games (G, Ω1, . . . , Ωn, ) asks, given a
threshold μ ∈ {0, 1}n and an initial vertex v0 ∈ V , to decide whether P1 (resp. P2) has
a strategy to ensure the objective Ω = {π ∈ Plays(G)  Payoff(π) % μ} from v0 (resp.
Ω = {π ∈ Plays(G)  Payoff(π) 6% μ}).6
5 Note that in the sequel, we often manipulate equivalently vectors in {0, 1}n and sequences of n bits.
6 Note that when n = 1 and  is the usual order ≤ over {0, 1}, we recover the notion of single objective
with the threshold μ = 1.
In case P1 (resp. P2) has such a winning strategy, we also say that he can ensure (resp.
avoid) a payoff % μ.
Classical examples of preorders are the following ones [6]. Let x, y ∈ {0, 1}n.
Counting: x  y if and only if {j  xj = 1} ≤ {j  yj = 1}. The aim of P1 is to
maximize the number of satisfied objectives.
Subset: x  y if and only if {j  xj = 1} ⊆ {j  yj = 1}. The aim of P1 is to maximize
the subset of satisfied objectives with respect to the inclusion.
Maximise: x  y if and only if max{j  xj = 1} ≤ max{j  yj = 1}. The aim of P1 is to
maximize the higher index of the satisfied objectives.
Lexicographic: x  y if and only if either x = y or ∃j ∈ {1, . . . , n} such that xj < yj and
∀k ∈ {1, . . . , j − 1}, xk = yk. The objectives are ranked according to their importance.
The aim of P1 is to maximise the payoff with respect to the induced lexicographic order.
In this article, we focus on monotonic preorders. A preorder  is monotonic if it is
compatible with the subset preorder, i.e. if {i  xi = 1} ⊆ {i  yi = 1} implies x  y. Hence
a preorder is monotonic if satisfying more objectives never results in a lower payoff value.
This is a natural property shared by all the examples of preorders given previously.
I Example 2. Consider the game structure G depicted on Figure 1, where circle vertices
belong to P1 and square vertices belong to P2. We consider the ordered game (G, Ω1, Ω2, )
with Ωi = {π ∈ Plays(G)  vi ∈ Inf(π)} for i = 1, 2 and the lexicographic preorder .
Therefore the function Payoff assigns value 1 to each play π on the first (resp. second) bit
if and only if π visits infinitely often vertex v1 (resp. v2). In this ordered game, P1 has a
strategy to ensure a payoff % 01 from v0. Indeed, consider the memoryless strategy σ1 that
loops in v1 and in v2. Then, from v0, P2 decides to go either to v1 leading to the payoff 10,
or to v2 leading to the payoff 01. As 10 % 01, this shows that any play π consistent with σ1
satisfies Payoff(π) % 01. Notice that while P1 can ensure a payoff % 01 from v0, he has no
strategy to enforce the single objective Ω1 and similarly no strategy to enforce Ω2.
Homogeneous ωregular objectives
In this article, given a monotonically ordered game (G, Ω1, . . . , Ωn, ), we want to study the
threshold problem described in Problem 1 for homogeneous ωregular objectives, in the sense
that all the objectives Ω1, . . . , Ωn are of the same type, and taken in the following list of
wellknown ωregular objectives.
Given a game structure G = (V1, V2, E) and a subset U of V called target set:
The reachability objective asks to visit a vertex of U at least once, i.e. Reach(U ) = {π ∈
Plays(G)  Occ(π) ∩ U 6= ∅}.
The safety objective asks to always stay in the set U , i.e. Safe(U ) = {π ∈ Plays(G) 
Occ(π) ⊆ U }.
The Büchi objective asks to visit infinitely often a vertex of U , i.e. Buchi(U ) = {π ∈
Plays(G)  Inf(π) ∩ U 6= ∅}.
The coBüchi objective asks to eventually always stay in the set U , i.e. CoBuchi(U ) =
{π ∈ Plays(G)  Inf(π) ⊆ U }.
Given a family F = (Fi)ik=1 of sets Fi ⊆ V , and a family of pairs ((Ei, Fi)ik=1), with
Ei, Fi ⊆ V :
The explicit Muller objective asks that the set of vertices seen infinitely often is one
among the sets of F , i.e. ExplMuller(F ) = {π ∈ Plays(G)  ∃i ∈ {1, . . . , k}, Inf(π) = Fi}.
The Rabin objective asks that there exists a pair (Ei, Fi) such that a vertex of Fi is visited
k
infinitely often while no vertex of Ei is visited infinitely often, i.e. Rabin((Ei, Fi)i=1) =
{π ∈ Plays(G)  ∃i ∈ {1, . . . , k}, Inf(π) ∩ Ei = ∅ and Inf(π) ∩ Fi 6= ∅}.
The Streett objective asks that for each pair (Ei, Fi), a vertex of Ei is visited infinitely often
k
or no vertex of Fi is visited infinitely often, i.e. Streett((Ei, Fi)i=1) = {π ∈ Plays(G) 
∀i ∈ {1, . . . , k}, Inf(π) ∩ Ei 6= ∅ or Inf(π) ∩ Fi = ∅}.
Given a coloring function p : V → {0, . . . , d} that associates with each vertex a color, and
F = (Fi)ik=1 a family of subsets Fi of p(V ):
The parity objective asks that the minimum color seen infinitely often is even, i.e.
Parity(p) = {π ∈ Plays(G)  minv∈Inf(π) p(v) is even}.
The Muller objective asks that the set of colors seen infinitely often is one among the sets
of F , i.e. Muller(p, F ) = {π ∈ Plays(G)  ∃i ∈ {1, . . . , k}, p(Inf(π)) = Fi}.
In the sequel, we make the assumption that the considered preorders are monotonic, and
by ordered game, we always mean monotonically ordered games. When the objectives of an
ordered game are of kind X, we speak of an ordered X game, or of a  X game if we want
to specify the used preorder . As already mentioned, when n = 1, an ordered game (with
equal to ≤) resumes to a game (G, Ω) with a single objective Ω, that is traditionally called an
Ω game. For instance, an ordered game (G, Ω1, . . . , Ωn, ) where Ω1, . . . , Ωn are reachability
objectives and  is the lexicographic preorder is called a lexicographic reachability game,
and when n = 1 (G, Ω1) is called a reachability game.
Note that given an ordered game with n nonhomogeneous ωregular objectives Ωi, we
can always construct a new equivalent ordered parity game, since each objective Ωi can be
translated into a parity objective [15].
Monotonic preorders embedded in the subset preorder
We here show that solving the threshold problem for an ordered game (G, Ω1, . . . , Ωn, )
is equivalent to solving a game (G, Ω) with a single objective Ω equal to the union of
intersections of objectives taken in {Ω1, . . . , Ωn}. The arguments are the following ones. (1)
We consider the set {0, 1}n of payoffs ordered with  as well as ordered with the subset
preorder ⊆ (see the example of Figure 2 where  is the lexicographic preorder). To any
payoff ν ∈ {0, 1}n, we associate the set δν = {i ∈ {1, . . . , n}  νi = 1} containing all indices i
such that objective Ωi is satisfied. (2) Consider the set of payoffs ν % μ embedded in the
set {0, 1}n ordered with ⊆. By monotonicity of , we obtain an upperclosed set S that
can be represented by the antichain of its minimal elements (with respect to ⊆), that we
denote by M(μ). (3) P1 can ensure a payoff % μ if and only if he has a strategy such that
any consistent outcome π has a payoff ν∗ ⊇ ν for some ν ∈ M(μ), equivalently such that π
satisfies (at least) the conjunction of the objectives Ωi such that νi = 1. (4) The objective Ω
of P1 is thus a disjunction (over ν ∈ M(μ)) of conjunctions (over i ∈ δν ) of objectives Ωi.
This statement is formulated in the next theorem (see again Figure 2).
000
001
010
011
100
101
110
111
111
101
010
000
011
001
110
100
I Theorem 3. Let (G, Ω1, . . . , Ωn, ) be an ordered game, μ ∈ {0, 1}n be some threshold,
and v0 be an initial vertex. Then, P1 can ensure a payoff % μ from v0 in (G, Ω1, . . . , Ωn, )
if and only if P1 has a winning strategy from v0 in the game (G, Ω) with the objective
We end this section by giving some additional notations and terminology. Thanks to
Theorem 3, we will prove in Section 3 that the threshold problem is fixed parameter tractable.
The proof of this result uses two sizes depending on the number n of objectives:
the size s(n) of M(μ). It is upper bounded by 2n (an antichain of maximum size in the
subset preorder over {0, 1}n is of exponential size nn ).
b 2 c
the size s0(n) defined as follows. In case of Büchi objectives Ωi, we need to rewrite the
objective ∪ν∈M(μ) ∩i∈δν Ωi in conjunctive normal form ∩k ∪l Ω0k,l with Ω0k,l ∈ {Ω1, . . . , Ωn}.
We denote by s0(n) the size of this conjunction. It is bounded by 22n .
In Section 4 we will show that, for several objectives, we can go beyond fixed parameter
tractability by providing polynomial time algorithms when the sizes s(n) and s0(n) are
polynomial in n. An ordered game (G, Ω1, . . . , Ωn, ) is said to have a compact embedding
(in the subset preorder) if both sizes s(n) and s0(n) are polynomial in n. We will also show
that lexicographic games have a compact embedding.
3
Fixed parameter complexity of ordered ωregular games
Parameterized complexity
A parameterized language L is a subset of Σ∗ × N, where Σ is a finite alphabet, the second
component being the parameter of the language. It is called fixed parameter tractable (FPT)
if there is an algorithm that determines whether (x, t) ∈ L in f (t) · xc time, where c is a
constant independent of the parameter t and f is a computable function depending on t only.
We also say that L belongs to (the class) FPT. Intuitively, a language is FPT if there is an
algorithm running in polynomial time w.r.t the input size times some computable function
on the parameter. In this framework, we do not rely on classical polynomial reductions but
rather use so called FPTreductions. An FPTreduction between two parameterized languages
L ⊆ Σ∗ × N and L0 ⊆ Σ0∗ × N is a function R : L → L0 such that
(x, t) ∈ L if and only if (x0, t0) = R(x, t) ∈ L0,
R is computable by an algorithm that takes f (t) · xc time where c is a constant, and
t0 ≤ g(t) for some computable function g.
Moreover, if L0 is in FPT, then L is also in FPT. We refer the interested reader to [13] for
more details on parameterized complexity.
Our main result states that the threshold problem is in FPT for all the ordered games of
this article. Parameterized complexities are given in Table 1.
Table 1 Fixed parameter tractability of ordered games (G, Ω1, . . . , Ωn, ): for i ∈ {1, . . . , n},
ki/di denotes the number of pairs/colors of each Rabin/Streett/Muller objective Ωi. Sizes s(n) and
s0(n) are resp. upper bounded by 2n and 22n . For j ∈ {1, 2, 3}, Mj =22mj where m1 = Pin=1 2 · ki,
m2 = m3 = Pin=1 di, and N1 = s(n) · Pin=1 2 · ki, N2 = s(n) · Pin=1 d2i , N3 = s(n) · Pin=1 2di · di.
Objectives
Reachability, Safety
Büchi
coBüchi
Explicit Muller
Rabin, Streett
Parity
Muller
Parameters
n
n
n
n
n, k1, . . . , kn
n, d1, . . . , dn
n, d1, . . . , dn
Threshold problem
O(s(n) · n + 2n · (V  + E))
O(s(n) · n + s0(n) · V 2)
O(s(n) · n + s(n) · V 2)
O(s(n) · n + (s(n) · maxi Fi)3 · V 2 · E2)
O((2M1 · N1 + M1M1 ) · V 5)
O((2M2 · N2 + M2M2 ) · V 5)
O((2M3 · N3 + M3M3 ) · V 5)
I Theorem 4. The threshold problem is in FPT for ordered reachability, safety, Büchi,
coBüchi, explicit Muller, Rabin, Streett, parity, and Muller games.
The proof of this theorem needs to introduce additional kinds of games (G, Ω) with a
single ωregular objective Ω, like the Boolean Büchi games. It also needs to show that solving
the latter games is in FPT.
Parameterized complexity of Boolean Büchi games
Let G be a game structure and U1, . . . , Um be m target sets. Let φ be a Boolean formula over
variables x1, . . . , xm. We say that a play π satisfies (φ, U1, . . . , Um) if the truth assignment
{xi = 1 if and only if Inf(π) ∩ Ui 6= ∅, and xi = 0 otherwise} satisfies φ. An objective Ω is a
Boolean combination of Büchi objectives, or shortly a Boolean Büchi objective, if Ω = {π ∈
Plays(G)  π satisfies (φ, U1, . . . , Um)}. It is denoted by BooleanBuchi(φ, U1, . . . , Um).
All operators ∨, ∧, ¬ are allowed in Boolean Büchi objectives. However we denote by φ
the size of φ equal to the number of disjunctions and conjunctions inside φ, and we say that
the Boolean Büchi objective BooleanBuchi(φ, U1, . . . , Um) is of size φ and with m variables.
The definition of φ is not the classical one that usually counts the number of operators
∨, ∧, ¬ and variables. This is not a restriction since one can transform any Boolean formula
φ into one such that negations only apply on variables.
We need to introduce some other kinds of ωregular objectives with Boolean combinations
of objectives that are limited to
intersections of objectives: like a generalized reachability objective or a generalized Büchi
objective denoted respectively by GenReach(U1, . . . , Um) and GenBuchi(U1, . . . , Um),
unions of intersections (UI) of objectives: like a UI reachability objective, a UI safety
objective, or a UI Büchi objective.
I Proposition 5. Solving Boolean Büchi games (G, Ω) is in FPT, with an algorithm in
O(2M · φ + (M M · V )5) time with M = 2m such that m is the number of variables of φ in
the Boolean Büchi objective Ω.
Proof. Let us show the existence of an FPTreduction from Boolean Büchi games to Muller
games. For this purpose, consider a Boolean Büchi game (G, Ω) with the objective Ω =
BooleanBuchi(φ, U1, . . . , Um), where φ is a Boolean formula over variables x1, . . . , xm, and
m is seen as a parameter. We build an adequate Muller game (G, Muller(p, F )) on the same
game structure and parameterized by the number of colors. The coloring function p and the
family F are constructed as follows.
To any vertex v ∈ V , we associate a color p(v) = μ which is a subset of {1, . . . , m} in the
following way: i ∈ μ if and only if v ∈ Ui.7 Intuitively, we keep track for all i, whether a
vertex belongs to Ui or not. The total number M of colors is thus equal to 2m. One can
notice that (∗) a play π visits a vertex v ∈ Ui if and only if π visits a color μ that contains i.
To any subset F of p(V ), we associate the truth assignment χ(F ) ∈ {0, 1}m of variables
x1, . . . , xm such that for all i, χ(F )i = 1 if there exists μ ∈ F such that i ∈ μ, and 0
otherwise. The idea (by (∗)) is that the set F of colors visited infinitely often by a play π
corresponds to the set Inf(π) of vertices visited infinitely often such that χ(F )i = 1 if and
only if Inf(π) ∩ Ui 6= ∅. We then define F = {F ⊆ p(V )  χ(F ) = φ}, that is, F corresponds
to the set of all truth assignments satisfying φ.
In this way we have the desired FPTreduction: first, parameter M = 2m only depends on
parameter m. Second, we have that P1 is winning in (G, BooleanBuchi(φ, U1, . . . , Um)) from
an initial vertex v0 if and only if he is winning in (G, Muller(p, F )) from v0. Indeed, a play π
satisfies (φ, U1, . . . , Um) if and only if the truth assignment ( xi = 1 if and only Inf(π)∩Ui 6= ∅,
and xi = 0 otherwise ) satisfies φ. This is equivalent to have that F = p(Inf(π)) belongs to
F (by definition of χ(F )), that is, π belongs to Muller(p, F ). Third, the construction of the
Muller game is in O(22m
O(m · V ) time for the c·olφor)intgimfeunscintcioenitpr,eaqnudireOs(O22(mV  + E) time for the game structure,
· φ) time for the family F .
From this FPTreduction and as solving Muller games is in O((dd · V )5)) time where
d is the number of colors [9], we have an algorithm solving the Boolean Büchi game in
O(2M · φ + (M M · V )5) time, where M = 2m. J
Proof of FPT membership for ordered games
Thanks to Theorem 3, we provide a proof of Theorem 4 with the parameterized complexities
given in Table 1.
Proof of Theorem 4. By Theorem 3, solving the threshold problem for an ordered game
(G, Ω1, . . . , Ωn, ) is equivalent to solving a classical game (G, Ω) with Ω = ∪ν∈M(μ) ∩i∈δν Ωi.
We have M(μ) = s(n) and δν  ≤ n ∀ν ∈ M(μ). Recall that s(n) ≤ 2n and s0(n) ≤ 22n .
We first show that the threshold problem for ordered reachability, safety, Büchi, coBüchi,
and explicit Muller games is in FPT with parameter n. The reduction provided in Theorem 3
is an FPTreduction as the number of disjunctions/conjunctions in Ω only depends on n.
Moreover the construction of the game (G, Ω) is in O(V  + E + s(n) · n) time. In the
following items we describe a second FPTreduction to add to the first one. The sum of the
complexities of both FPTreductions leads to the complexities given in Table 1, rows 25.
If each Ωi is a reachability (resp. safety) objective, then (G, Ω) is a UI reachability (resp.
safety) game that can be reduced to a reachability (resp. safety) game over a game
structure of size 2n · V  [14].8 The latter is solved in O(2n · (V  + E)) time.
If Ω is a union of intersections of Büchi objectives, then it can be rewritten as the
intersection of unions of Büchi objectives which is a generalized Büchi objective with at
most s0(n) target sets. The latter game is solved in O(s0(n) · V 2) time by [
10
]. The
7 Our definition of color requires μ to be an integer. It suffices to associate with v a vector μv ∈ {0, 1}m
stuhcaht athssaotcμiaivte=s w1iitfhve∈acUhivaenrtdex0 votthheerwciosleo,rapn(dv)tosudcehfintheatthietscoblionrainryg feunnccotdioinngpi:s Veq→ual{t0o, .μ.v. ., 2m − 1}
8 This result does not appear explicitly in [14] but can be easily adapted to the case of UI reachability
(resp. safety) objectives.
union of intersections of coBüchi objectives is the complementary of a generalized Büchi
objective with at most s(n) target sets, leading to an algorithm in O(s(n) · V 2) time.
If each Ωi is an explicit Muller objective ExplMuller(Fi) then Ω is also an explicit Muller
objective. Indeed the intersection (resp. union) of explicit Muller objectives is an
explicit Muller objective such that ∩iExplMuller(Fi) = ExplMuller(F ) with F = ∩iFi
(resp. ∪iExplMuller(Fi) = ExplMuller(F ) with F = ∪iFi). Thus Ω can be here rewritten
as ExplMuller(F ) for some set F such that F  ≤ Pν∈M(μ) minj∈δν Fj . The latter game
is solved in O(F  · (V  · E + F )2) = O((s(n) · maxi Fi)3 · V 2 · E2) time by [17].
We now show that the threshold problem for ordered parity, Rabin, Streett, and Muller
games is in FPT thanks to Proposition 5.
Let us show that the threshold problem for ordered parity games is in FPT with parameters
n, d1, . . . , dn. If each Ωi is a parity objective with di colors, then each Ωi is a Boolean
2
Büchi objective of size at most d2i and using di variables. Indeed, as a play is winning
for Ωi if and only there exists an even priority seen infinitely often along the play and
no lower priority seen infinitely often. Therefore, Ω is a Boolean Büchi objective Ω0 of
2
size φ ≤ s(n) · Pin=1 d2i , and with m = Pin=1 di variables as ∪ν∈M(μ){Ωi  i ∈ δν } ⊆
{Ω1, . . . , Ωn}. We thus have an FPTreduction to the game (G, Ω0) depending on the
parameters n, d1, . . . , dn and with an algorithm in O(V +E+φ) time. By Proposition 5,
solving the game (G, Ω0) is in FPT with an algorithm in O(2M · φ + (M M · V )5) time
with M = 2m. Thus the threshold problem is in FPT with parameters n, d1, . . . , dn, with
an overall algorithm in O((2M · N + M M ) · V 5) time where N = 2n · Pin=1 d2i2 .
The arguments are similar for ordered Rabin, Streett, and Muller games. The only
differences are the upper bound on size φ and the number m of variables of the related
formula φ. J
4
Ordered games with a compact embedding
In the previous section, we have shown that solving the threshold problem for ordered
ωregular games is in FPT. This result depends on sizes s(n) and s0(n) which vary with the
number n of objectives. In this section, we study ordered games with a compact embedding,
that is, such that these sizes are polynomial in n.
Beyond fixed parameter tractability
While the threshold problem is in FPT for ordered Büchi, coBüchi, and explicit Muller
games, it becomes polynomial as soon as their preorder has a compact embedding. This is a
direct consequence of Table 1, rows 24.
I Theorem 6. The threshold problem is solved in polynomial time for ordered Büchi, coBüchi,
and explicit Muller games with a compact embedding.
One can easily prove that ordered games using the subset or the maximize preorder have
a compact embedding. We will later prove that this also holds for the lexicographic preorder.
Nevertheless it is not the case for the counting preorder. Indeed solving the threshold problem
for counting Büchi games is coNPcomplete [6].
Recall that solving the threshold problem for ordered Büchi games reduces to solving
some UI Büchi game (by Theorem 3). Whereas solving the latter games is coNPcomplete [4],
solving the threshold problem for ordered Büchi games is only polynomial when they have a
compact embedding (see Theorem 6).
There is no hope to extend Theorem 6 to the other ωregular objectives studied in this
article, unless P = PSPACE. Indeed, we have PSPACEhardness of the threshold problem for
the following lexicographic games.
I Theorem 7. (1) Lexicographic games have a compact embedding and (2) the threshold
problem is PSPACEhard for lexicographic reachability, safety, Rabin, Streett, parity, and
Muller games.
The rest of this section is devoted to the proof of Theorem 7.
Lexicographic games
We now focus on the lexicographic preorder . Let us first provide several useful terminology
and comments on this preorder. Recall that the lexicographic preorder is monotonic. It is
also total, hence x ∼ y if and only if x = y, and x ≺ y if and only if ¬(y  x). Given a vector
x ∈ {0, 1}n, we denote by x the complement of x, i.e. xi = 1 − xi, for all i ∈ {1, . . . , n}.
We denote by x − 1 the predecessor of x 6= 0n, that is, the greatest vector which is strictly
smaller than x. We define the successor x + 1 of x similarly. In the sequel, as the threshold
problem is trivial for x = 0n, we do not consider this threshold. By abuse of notation, we
keep writing x ∈ {0, 1}n without mentioning that x 6= 0n. We denote by Last1(x) the last
index i of x such that xi = 1, i.e. Last1(x) = max{i ∈ {1, . . . , n}  xi = 1}. Note that P1 can
ensure a payoff % x 6= 0n if and only if he can ensure a payoff x − 1, and when P2 can
avoid a payoff % x, we rather say that P2 can ensure a payoff ≺ x.
We now prove that the lexicographic games have a compact embedding (Part (1) of
Theorem 7): we first show that s(n) is polynomial in Proposition 8, and we then show that
s0(n) is also polynomial in Proposition 10.
I Proposition 8. Let x ∈ {0, 1}n. Then the set M(x) is equal to {x} ∪ {yj ∈ {0, 1}n  xj =
0 ∧ j < Last1(x)}, where for all j ∈ {1, . . . , Last1(x) − 1}, we define the vector yj ∈ {0, 1}n
as equal to x1 . . . xj−110n−j (x and yj share the same (possibly empty) prefix x1 . . . xj−1).
Moreover, s(n) = M(x) ≤ n.
I Example 9. Consider the vector x = 0010100 such that Last1(x) = 5. Then, the set M(x)
is equal to {x} ∪ {1000000, 0100000, 0011000}.
Proof of Proposition 8. We recall that M(x) is the set of minimal elements (with respect to
the subset preorder ⊆) of the set of payoffs y % x embedded in the set {0, 1}n ordered with
⊆. Let us show both inclusions between M(x) and M = {x} ∪ {yj ∈ {0, 1}n  xj = 0 ∧ j <
Last1(x)}.
Let y ∈ M(x). If y = x, then trivially y ∈ M . Otherwise, assume y x and let j be the
first index such that yj = 1 and xj = 0. Note that x1 . . . xj−1 = y1 . . . yj−1 since y x. We
associate with y the vector yj = y1 . . . yj−110n−j. Note that yj x. By minimality of y and
by construction of yj, we obtain y = yj showing that y ∈ M .
For the second inclusion, as the lexicographic preorder is monotonic, we have x ∈ M(x).
Now, consider some yj ∈ M such that xj = 0 and j < Last1(x). Let us show that yj
belongs to M(x), that is, yj % x and there is no y % x, y 6= yj, such that y ⊂ yj (i.e.
{i  yi = 1} ⊂ {i  yij = 1}). First, we clearly have yj % x since yj = x1 . . . xj−110n−j and
xj = 0. Towards a contradiction, assume now that there exists some y % x, y 6= yj, such
that y ⊂ yj. Let i be the first index such that yi = 0 and yij = 1. As y ⊂ yj, we have i ≤ j.
If i < j, then y has x1 . . . xi−10 as prefix, yij = xi = 1, showing that y ≺ x in contradiction
with y % x. If i = j, then y = x1 . . . xj−10n−j+1, and again y ≺ x since j < Last1(x) by
construction of yj. J
I Proposition 10. Let (G, Ω1, . . . , Ωn, ) be a lexicographic Büchi game and μ ∈ {0, 1}n.
Then, Ω = ∪ν∈M(μ) ∩i∈δν Ωi can be rewritten in conjunctive normal form with a conjunction
of size s0(n) ≤ n.
Proof. The proof uses the property that given a lexicographic game (G, Ω1, . . . , Ωn, ) and
a threshold μ ∈ {0, 1}n, P1 can ensure a payoff % μ in (G, Ω1, . . . , Ωn, ) if and only if P1
can ensure a payoff  μ in the lexicographic game (G, Ω1, . . . , Ωn, ). By Theorem 3 and
Martin’s theorem [20], equivalently, P2 cannot satisfy the objective ∪ν∈M(μ+1) ∩i∈δν Ωi. This
is equivalent to say that P1 can satisfy the complement of the latter objective, that is, the
objective ∩ν∈M(μ+1) ∪i∈δν Ωi. We have M(μ + 1) ≤ n by Proposition 8. J
We finally prove Part (2) of Theorem 7.
Proof of Theorem 7, Part (2). Let us study the complexity lower bounds.
The PSPACEhardness of the threshold problem for lexicographic reachability (resp. safety)
games is obtained thanks to a polynomial reduction from solving generalized reachability
games which is PSPACEcomplete [14]. Let (G, Ω) be a generalized reachability game
with Ω = GenReach(U1, . . . , Un). Let (G, Ω1, . . . , Ωn, ) be the lexicographic reachability
(resp. safety) game with Ωi = Reach(Ui) (resp. Ωi = Safe(V \ Ui)) ∀i.
Reachability: We have that P1 is winning in (G, Ω) from v0 if and only if P1 can ensure
a payoff % μ = 1n from v0 in the lexicographic reachability game (G, Ω1, . . . , Ωn, ).
Safety: We claim that P1 is winning in (G, Ω) from v0 if and only if P1 can ensure a
payoff % μ = 0n−11 from v0 in the lexicographic safety game. This follows from the
determinacy of generalized reachability games, and from the fact that P1 can ensure a
payoff % μ from v0 in the lexicographic safety game if and only if P2 is losing in the
generalized reachability game (G, Ω) from v0.
The hardness of the threshold problem for lexicographic parity games is obtained thanks
to a polynomial reduction from solving games (G, Ω) the objective Ω of which is a union
of a Rabin objective and a Streett objective, which is known to be PSPACEcomplete [3].
Let Ω = Rabin((Ei, Fi)in=11) ∪ Streett((Ei, Fi)in=n1+1). As any Rabin (resp. Streett)
objective is the union (resp. intersection) of parity objectives [11], we can rewrite Ω
as Ω = ∪in=11(Parity(pi)) ∪ (∩in=n1+1Parity(pi)), where all pi are coloring functions. Let
(G, Ω1, . . . , Ωn, ) be the lexicographic parity game where Ωi = Parity(pi) for all i. We
claim that P1 is winning in the game (G, Ω) from v0 if and only if P1 can ensure a payoff
% μ from v0 in the lexicographic parity game (G, Ω1, . . . , Ωn, ) where μ = 0n1 1n−n1 .
Indeed, if a play π satisfies Payoff(π) % μ then either Payoff(π) = μ in which case
π ∈ ∩in=n1+1Parity(pi), i.e. π satisfies the Streett objective, or Payoff(π) μ in which case
there exists 1 ∈ {1, . . . , n1} such that π ∈ Parity(pi), i.e. π satisfies the Rabin objective.
Conversely, if a play π satisfies the Streett or the Rabin objective then Payoff(π) % μ
since Payoff(π) % μ (resp. μ) as soon as π satisfies the Streett (resp. Rabin) objective.
As parity objectives are a special case of Rabin (Streett) objectives, the lower bound
follows (from the previous item) for both lexicographic Rabin and Streett games.
Lexicographic Muller games with n = 1 and μ = 1 are a special case of Muller games and
solving the latter games is PSPACEcomplete [18].
This completes the proof. J
5
Values and optimal strategies for lexicographic games
In this section, we first recall the notion of values and optimal strategies. We then show how
to compute the values in lexicographic games, and what are the memory requirements for the
related optimal strategies. This yields a full picture of the study of lexicographic games, see
In a lexicographic game, one can define the best reward that P1 can ensure from a given
vertex, that is, the highest threshold μ for which P1 can ensure a payoff % μ. Dually, we can
also define the worst reward that P2 can ensure.
In the following definition, the infimum and supremum functions are applied with .
I Definition 11. Given a lexicographic game (G, Ω1, . . . , Ωn, ), for every vertex v ∈ V , the
upper value Val(v) and the lower value Val(v) are defined as:
Val(v) = inf sup Payoff(Out(v, σ1, σ2)) and Val(v) = sup inf Payoff(Out(v, σ1, σ2)).
σ2∈Σ2 σ1∈Σ1 σ1∈Σ1 σ2∈Σ2
The lexicographic game (G, Ω1, . . . , Ωn, ) is valuedetermined if Val(v) = Val(v) ∀v ∈ V .
In this case, we write Val(v) = Val(v) = Val(v) and we call Val(v) the value of v. Note that
the inequality Val(v)  Val(v) always holds. If P1 (resp. P2) can ensure a payoff % Val(v)
(resp.  Val(v)) from v, his related winning strategy σ1∗ (resp. σ2∗) is called optimal from v.
Notice that for all lexicographic games such that the objectives Ω1, . . . , Ωn are Borel sets,
we have that these games are valuedetermined and have optimal strategies by Theorem 3 and
Martin’s theorem [20]. In the following theorem, we go further by giving time complexities
and memory sizes of the optimal strategies.
I Theorem 12. (1) The value of each vertex in lexicographic Büchi, coBüchi, and explicit
Muller games can be computed with a polynomial time algorithm, and with an exponential
time and an FPT algorithm for lexicographic reachability, safety, parity, Rabin, Streett, and
Muller games.
(2) The following assertions hold for both winning strategies of the threshold problem
and optimal strategies. Linear memory strategies are necessary and sufficient for P1 (resp.
P2) while memoryless strategies are sufficient for P2 (resp. P1) in lexicographic Büchi (resp.
coBüchi) games. Exponential memory strategies are both necessary and sufficient for both
players in lexicographic reachability, safety, explicit Muller, parity, Rabin, Streett, and Muller
games.
We only give a sketch of the proof.
First, for the considered lexicographic games, the values can be obtained by solving n
times wellchosen threshold problems. Therefore, results of Part (1) of Theorem 12 follows
from the second column of Table 2. In addition to give the exact value of a vertex, this
procedure also shows that optimal strategies correspond to winning strategies for specific
threshold problems. Therefore, we just have to analyze memory requirements of winning
strategies for the threshold problem in lexicographic games to obtain those of optimal
strategies. Upper bounds on memory sizes of winning strategies are obtained by analyzing
the several reductions done in the proof of Theorem 4 in the case of a preorder with a
compact embedding. Lower bounds for lexicographic Büchi and coBüchi games are obtained
thanks to a reduction from generalized Büchi games, and for the other lexicographic games
thanks to the reductions proposed in the proof of Theorem 7, Part (2). This yields results of
Part (2) of Theorem 12.
1
2
3
4
5
6
7
8
9
10
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