Abstract
Parity games play a central role in model checking and satisfiability checking. Solving parity games is computationally expensive, among others due to the size of the games, which, for model checking problems, can easily contain \(10^9\) vertices or beyond. Equivalence relations can be used to reduce the size of a parity game, thereby potentially alleviating part of the computational burden. We reconsider (governed) bisimulation and (governed) stuttering bisimulation, and we give detailed proofs that these relations are equivalences, have unique quotients and they approximate the winning regions of parity games. Furthermore, we present gamebased characterisations of these relations. Using these characterisations our equivalences are compared to relations for parity games that can be found in the literature, such as direct simulation equivalence and delayed simulation equivalence. To complete the overview we develop coinductive characterisations of direct and delayed simulation equivalence and we establish a lattice of equivalences for parity games.
Introduction
We study preorders and equivalences defined on parity games. Such games are turnbased graph games between two players taking turns pushing a token along the vertices of a finitely coloured graph. These players, called \(\textit{even}\,\)and \(\textit{odd\,}\), strive to optimise the parity of the dominating colour occurring infinitely often in a play. Parity games appear in the core of various foundational results such as Rabin’s proof of the decidability of a monadic secondorder theory. Solving parity games is a computationally expensive but key step in many model checking algorithms [18, 50, 51] and synthesis and supervisory control algorithms [1, 2, 21].
Parity game solving enjoys a special status among combinatorial optimisation problems, being one of the rare problems in the intersection of the UP and coUP classes [35] that is not known to be in P. Despite the continued research effort directed to it, resulting in numerous algorithms for solving parity games, see, e.g., [5, 6, 36, 37, 44, 48,49,50, 53, 56], no polynomial time algorithm has yet been found.
Orthogonally to the algorithmic improvements, heuristics and static analyses have been devised that may speed up solving, or fully solve parity games that occur in practice [20, 32, 33]. Such heuristics work particularly well for verification problems, which give rise to games with only few different priorities. In a similar vein, heuristics based on the intimate ties between temporal logics and bisimulation relations are often exploited to speedup model checking. First minimising a state space by computing the equivalence quotient and only then analysing this quotient can be an effective strategy, see e.g. [38].
Given the close connection between parity game solving and model checking, a promising heuristic in this setting is to reduce (minimise) a game prior to solving it. Of course, this requires that the winning regions of the original game can be recovered cheaply from the winning regions of the minimised game. Moreover, minimisation makes sense only for equivalence relations that strike a favourable balance between their power to compress the game graph and the computational complexity of quotienting with respect to the equivalence relation. Indeed, in [13, 42] we showed that quotienting using standard strong bisimilarity and stuttering equivalence allow to solve parity games that could not be solved otherwise. Despite the immense reductions that can be obtained for parity games encoding natural decision problems, the results were mixed and, apart from a number of cases that become solvable, there was on average no clear gain from using such relations. It should be noted that the stuttering equivalence experiments in [13, 14] were conducted on the benchmarks from [40] using the GrooteVaandrager algorithm [30] which runs in \(\mathcal {O}(m n)\), where m is the number of edges and n is the number of states. A recent improvement on this algorithm, described in [31], may very well mean the scale tips in favour of using stuttering equivalence minimisation prior to solving a parity game, as experiments using this \(\mathcal {O}(m \log {n})\) algorithm have shown speedups of several orders of magnitude compared to the \(\mathcal {O}(m n)\) algorithm.
Similar observations can be made for governed bisimilarity [39] (also known as idempotenceidentifying bisimilarity in [42]) and governed stuttering bisimilarity [14], which weaken strong bisimilarity and stuttering equivalence, respectively, by taking the potentials of players into account. Quotienting for these relations relies on the claim that the relations are equivalences.
As a sidenote, simulation and bisimulation relations, tailored to parity games, may lead to insights into the core of the parity game solving problem. Indeed, in e.g. [34], Janin relies on different types of simulations to provide uniform proofs when showing the existence of winning strategies; at the same time he suggests simulation relations may ultimately be used to solve games efficiently.
Contributions. In this paper, we revisit the notions of (governed) bisimilarity and (governed) stuttering bisimilarity for parity games from [13, 14, 39, 42]. Parts of this paper have appeared earlier in ibid. In this paper we extend upon these works as follows.
We give formal proofs showing that they are indeed equivalence relations and, equally important, that they approximate the winning regions of a parity game, substantiating our claims in the aforementioned papers. Showing that the relations are indeed equivalence relations is technically rather involved, and slight oversights are easily made, see e.g. [3]. The added complexity of working in a setting with two players complicates matters significantly compared to, e.g., the setting of labelled transition systems studied in [3].
We furthermore study how our equivalence relations are related to two other notions that have been studied in the context of parity games, viz. direct simulation and delayed simulation [23] and the latter’s even and oddbiased versions. A complicating factor is the fact that these relations have only gamebased definitions, whereas our equivalences are defined coinductively. We mend this by providing alternative coinductive definitions for direct simulation and delayed simulation, inspired by [45], and we show that these coincide with their gamebased definitions. Likewise, we give gamebased definitions for our coinductively defined relations, drawing inspiration from [8, 55], thereby offering a more operational view on our relations.
Finally, we show that, contrary to (even and oddbiased) delayed simulation equivalence, direct simulation equivalence, governed bisimilarity and governed stuttering bisimilarity have unique quotients, and, in fact, unique smallest quotients. As a result, one can reduce a parity game to the point that it contains no redundancy (from the perspective of the equivalence relation).
In this paper, proofs that are straightforward have been sketched or omitted. Detailed versions of these proofs can be found in the preprint [15]. The proofs of Propositions 7 and 8 are nontrivial, but for the sake of readability, the full details of these proofs have been deferred to the “Appendix”.
Related work. In logic, bisimulation has been used to characterise the subfamily of firstorder logic that is definable in modal logic [4], and the fragment of monadic secondorder logic that is captured by the modal \(\mu \)calculus. Bisimulation and simulationlike relations, called consistent correlations [54] and consistent consequence [25] for PBESs, a fixpointlogic based framework which is closely related to parity games, were imperative to prove the soundness of the syntaxbased static analysis techniques described in [12, 41, 46, 47]. Various simulation relations have been used successfully for minimising Büchi automata, see e.g. [10, 19, 43].
In the context of process theory, there is an abundance of different simulation and bisimulation relations, allowing to reason about the powers of different types of observers of a system’s behaviour, see [27, 28]. Coinductive definitions of weak behavioural equivalences such as stuttering equivalence (which is, essentially, the same as branching bisimulation for labelled transition systems) are commonplace, see [28] for a comprehensive overview. Typically, these definitions rely on the transitive closure of the transition relation. As argued by Namjoshi [45], local reasoning, using single steps instead of resorting to transitive closure, typically leads to simpler arguments. He therefore introduced wellfounded bisimulation, a notion equivalent to stuttering bisimulation which solely relies on local reasoning by introducing a wellfounded order into the relation. Still, at its basis, wellfounded bisimulation only serves to show the reachability of some pair of related vertices. In our coinductive characterisation of the delayed simulation of [23], we use Namjoshi’s ideas. However, we need to factor in that in delayed simulation each step on one side must be matched by exactly one step on the simulating side.
There are only a few documented attempts that provide gamebased definitions for weak behavioural equivalences. Yin et al. [55] describe branching bisimulation games for normed process algebra. A gamebased characterisation of divergenceblind stuttering bisimulation was provided by Bulychev et al. [8]. Neither of these definitions is easily extended to the setting of governed stuttering bisimulation for parity games. In particular, the latter definition is only sound for transition systems that are free of divergences and requires a separate preprocessing step to deal with these. Recently, in [16] we developed a gamebased characterisation of branching bisimulation (with explicit divergence) that does not need such a preprocessing step. Our gamebased characterisation of governed stuttering bisimulation is based on the same ideas, but requires extensions to the twoplayer setting of parity games.
Structure of the paper. Parity games are introduced in Sect. 2. In Sect. 3, we introduce notation that facilitates us to define preorders and equivalences on parity games and we state several basic results concerning this notation. A technical overview of the relations studied in the remainder of the paper, and how these are related is presented in Sect. 4. In Sect. 5 we study direct simulation, delayed simulation and its biased versions and in Sect. 6 we study governed bisimulation and governed stuttering bisimulation. Quotienting, for all involved equivalences that admit unique quotients, is discussed in Sect. 7 and in Sect. 8 we return to, and substantiate, the overview we presented in Sect. 4. We wrap up with conclusions and an outlook for future work in Sect. 9.
Parity games
A parity game is a twoplayer graph game, played by two players \(\textit{even}\,\)and \(\textit{odd\,}\) (denoted and ) on a total directed graph in which the vertices are partitioned into two sets, one for each player, and in which a natural priority is assigned to every vertex. The game is played by placing a token on some initial vertex, and if the token is on a vertex owned by player \(\textit{even}\,\), then she moves the token to a successor of the current vertex (likewise for vertices owned by \(\textit{odd\,}\)). The infinite play that results from playing such moves indefinitely is won by player \(\textit{even}\,\)if and only if the least priority on play has even parity. The game is formally defined as follows.
Definition 1
(Parity game) A parity game is a directed graph \((V, \rightarrow , \varOmega , \mathcal {P})\), where

V is a finite set of vertices,

\({\rightarrow } \subseteq V \times V\) is a total edge relation (i.e., for each \(v \in V\) there is at least one \(w \in V\) such that \((v,w) \in {\rightarrow }\)),

\(\varOmega \mathop {:}V \rightarrow \mathbb {N}\) is a priority function that assigns priorities to vertices,

is a function assigning vertices to players.
Instead of \((v,w) \in \rightarrow \) we typically write \(v \rightarrow w\), and we write \(v^\bullet \) for the set \(\{w \in V ~~ v \rightarrow w\}\). If \( i _{}\) is a player, then \(\lnot i _{}\) denotes the opponent of \( i _{}\), i.e., and . The function \(\mathcal {P}\) induces a partitioning of V into a set of vertices owned by player \(\textit{even}\,\)and a set of vertices owned by player \(\textit{odd\,}\); we use \(\mathcal {P}\) and interchangeably. The reward order [53] on natural numbers is defined such that \(n \preccurlyeq m\) if n is even and m is odd; or n and m are even and \(n \le m\), or n and m are odd and \(m \le n\). Note that \(n \prec m\) means that n is better than m for player \(\textit{even}\,\). Notions \(\min \) and \(\max \) are always used with respect to the standard ordering on natural numbers. Finally, we remark that the assumption that the edge relation is total only serves to simplify the theory described in this paper. All results can be generalised to deal with the situation in which one of the players is unable to move.
Paths. A sequence of vertices \(v_0 \ldots v_n\) for which \(v_m \rightarrow v_{m+1}\) for all \(m < n\) is a path. The concatenation \(p_{1} p_{2}\) of paths \(p_{1}\) and \(p_{2}\) is again a path, provided there is a step from the last vertex in \(p_{1}\) to the first vertex in \(p_{2}\). Infinite paths are defined in a similar manner. We use \({p_{}}[j]\) to denote the jth vertex in a path \(p_{}\), counting from 0. The set of paths of length n starting in v is defined inductively for \(n \ge 1\) as follows:
The set of infinite paths starting in v is denoted , and the set of both finite and infinite paths starting in v is defined as follows:
Plays and their winners. A game starts by placing a token on some vertex \(v \in V\). Players move the token indefinitely according to the following simple rule: if the token is on some vertex v, player \(\mathcal {P}(v)\) moves the token to some vertex w such that \(v \rightarrow w\). The result is an infinite path \(p_{}\) in the game graph; we refer to this infinite path as a play. The parity of the lowest priority that occurs infinitely often on \(p_{}\) defines the winner of the play. If this priority is even, then player wins, otherwise player wins.
Strategies. A strategy for player \( i _{}\) is a partial function \(\sigma \mathop {:}V^{*\!\!}\rightarrow V\), that is defined only for paths ending in a vertex owned by player \( i _{}\) and determines the next vertex to be played onto. The set of strategies for player \( i _{}\) in a game \(\mathcal {G}\) is denoted \(\mathbb {S}_{\mathcal {G}, i _{}}^*\), or simply \(\mathbb {S}_{ i _{}}^*\) if \(\mathcal {G}\) is clear from the context. If a strategy yields the same vertex for every pair of paths that end in the same vertex, then the strategy is said to be memoryless. The set of memoryless strategies for player \( i _{}\) in a game \(\mathcal {G}\) is denoted \(\mathbb {S}_{\mathcal {G}, i _{}}\), abbreviated to \(\mathbb {S}_{ i _{}}\) when \(\mathcal {G}\) is clear from the context. A memoryless strategy is usually given as a partial function \(\sigma \mathop {:}V\rightarrow V\!\).
A strategy \(\sigma \in \mathbb {S}_{ i _{}}^*\) allows a path \(p_{}\) of length n, denoted \(\sigma \Vdash p_{}\), if and only if for all \(j < n1\) it is the case that if \(\sigma \) is defined for , then . The definition of allows is extended to infinite paths in the obvious manner. We generalise the definition of to paths allowed by a strategy \(\sigma \); formally, we define:
The definition for infinite paths is generalised in the same way and denoted . By we denote , i.e., the set of all finite and infinite paths starting in v and allowed by \(\sigma \).
A strategy \(\sigma \in \mathbb {S}_{ i _{}}^*\) is said to be a winning strategy from a vertex v if and only if \( i _{}\) is the winner of every infinite path allowed by \(\sigma \). A vertex is won by player \( i _{}\) if \( i _{}\) has a winning strategy from that vertex. Likewise, a strategy \(\sigma \in \mathbb {S}_{ i _{}}^*\) is a winning strategy from a set of vertices \(W \subseteq V\) if \(\sigma \) is winning from all \(v \in W\).
Solving parity games. It is wellknown that parity games are determined, i.e. that each vertex in a game is won by exactly one player, and if a winning strategy for a player exists from a vertex, then also a memoryless strategy exists. This is summarised in the following theorem.
Theorem 1
(Memoryless determinacy [17]) For every parity game there is a unique partition such that winning strategies from and from exist. Furthermore, for , if \(\sigma _{ i _{}} \in \mathbb {S}_{ i _{}}^*\) is winning from \(W_{ i _{}}\) a memoryless strategy \(\psi _{ i _{}} \in \mathbb {S}_{ i _{}}\) winning from \(W_{ i _{}}\) exists.
The problem of solving a parity game is defined as the problem of computing the winning partition of a parity game.
Notation
In the remainder of this paper we frequently need to reason about the concept of a player being able to force the play towards a set of vertices. We introduce notation that facilitates such reasoning and we provide some lemmata that express basic properties of parity games in terms of this extended notation. Throughout this section, we fix a parity game \((V, \rightarrow , \varOmega , \mathcal {P})\). Furthermore, we let \(T, U \subseteq V\) be subsets of vertices in the game.
Given a memoryless strategy \(\sigma \), we introduce a singlestep relation \(\mathop {\,\!_{\sigma }\smash \rightarrow } \subseteq \rightarrow \) that contains only those edges allowed by \(\sigma \):
In other words, \(\mathop {\,\!_{\sigma }\smash \rightarrow }\) contains those edges (v, u) from \(\rightarrow \) for which \(\sigma (v)\) is undefined, or \(\sigma (v)\) is defined and \(u = \sigma (v)\).
In line with \(v \rightarrow u\), we write \(v \mathop {\,\!_{\sigma }\smash \rightarrow } u\) if \((v, u) \in \mathop {\,\!_{\sigma }\smash \rightarrow }\). Abstracting from the specific strategy, we write \(v \mathop {\,\!_{ i _{}}\smash \rightarrow } u\) iff player \( i _{}\) has a memoryless strategy \(\sigma \) such that \(v \mathop {\,\!_{\sigma }\smash \rightarrow } u\).
We introduce special notation to express which parts of the graph can be reached from a certain node. We use \(v \mathop {\,\!_{}\smash \mapsto ^{}_{U}} T\) to denote that there is a finite path , for some n, such that \(v = v_0\), \(v_n \in T\) and for all \(j < n\), \(v_j \in U\). Conversely, \(v \mathop {\,\!_{}\smash \mapsto ^{}_{U}}\) denotes the existence of an infinite path for which \(v = v_0\) and for all j, \(v_j \in U\).
We extend this notation to restrict the reachability analysis to plays that can be enforced by a specific player. We say that strategy \(\sigma \) forces the play from v to T via U, denoted , if and only if for all plays p starting in v such that \(\sigma \Vdash p\), there exists an n such that \({p}[n] \in T\) and \({p}[j] \in U\) for all \(j<n\). Note that, in particular, if \(v \in T\). Similarly, strategy \(\sigma \) forces the play to diverge in U from v, denoted \(v \mathop {\,\!_{\sigma }\smash \mapsto ^{}_{U}}\), if and only if for all such plays p, \({p}[j]\in U\) for all j.
Finally, if we are not interested in a particular strategy, but only in the existence of a strategy for a player \( i _{}\) via which certain parts of the graph are reachable from v, we replace \(\sigma \) by \( i _{}\) in our notation to denote an existential quantification over memoryless strategies:
In the following lemmas and definitions, let \(v\in V\) be a vertex, \(U,T,T'\subseteq V\) be sets of vertices, and \( i _{}\) a player. The lemma below shows that rather than using memoryless strategies, one may, if needed, use arbitrary strategies when reasoning about .
Lemma 1
iff .
Proof
Observe that the implication from left to right holds by definition. So assume that for some \(\sigma \in \mathbb {S}_{ i _{}}^*\), we have . Note that iff . The truth value of the latter predicate does not depend on priorities of the vertices and only depends on the edges that originate in \(U \setminus T\). Therefore, the truth value of this predicate will not change if we apply the following transformation to our graph:

for all \(u \not \in U\), replace all outgoing edges by a single edge \(u \rightarrow u\).

set the priorities for all \(u \in T\) to 0 iff and the priorities of all other vertices to 1 iff .
Since , vertex v is won by \( i _{}\) in the resulting graph. As parity games are memoryless determined (Theorem 1), \( i _{}\) must have a memoryless strategy to move from \(U \setminus T\) to T in the resulting graph. Hence there is some \(\sigma ' \in \mathbb {S}_{ i _{}}\) such that in the resulting graph, but then also in the original graph, and hence also the required . \(\square \)
The complement of these relations is denoted by a slashed version of the corresponding arrow, e.g., can be written . We extend the transition relation of the parity game to sets and to sets of sets in the usual way, i.e., if T is a set of vertices, and \(\mathcal {U}\) is a set of vertex sets, then
All other arrow notation is extended in the same way; if a set of sets \(\mathcal {U}\) is given as a parameter, it is interpreted as the union of \(\mathcal {U}\).
The notation is closely related to the notion of attractor sets [44]. In essence, the attractor set \({_{U}}\! Attr ^{}_{ i _{}}(T)\) captures the subset of \(U \mathop {\cup } T\) from which \( i _{}\) can force the game to \(T \subseteq V\), by staying within U until T is reached.
Definition 2
(Attractor set) We define \({_{U}}\! Attr ^{}_{ i _{}}(T)\) as where:
The attractor set as defined in [44] is obtained for \(U = V\). Note that \( Attr {}\) is a monotone operator; i.e. for \(T \subseteq T'\) we have \({_{U}}\! Attr ^{}_{ i _{}}(T) \subseteq {_{U}}\! Attr ^{}_{ i _{}}(T')\). The correspondence between our ‘forcing’ arrow notation and the (generalised) attractor is given by the following lemma.
Lemma 2
Let \(U,T \subseteq V\). Then iff \(v \in {_{U}}\! Attr ^{}_{ i _{}}(T)\).
Proof
We first introduce some additional notation. Let denote that there is a \(\sigma \in \mathbb {S}_{ i _{}}\) such that for all , there is some \(m \le n\) such that \({p_{}}[m] \in T\), and \({p_{}}[j] \in U\) for all \(j < m\). Note that iff there is some n such that . Using induction on n, it follows that iff \(u \in {_{U}}\! Attr ^{n}_{ i _{}}(T)\) for all u. The required property is then a direct consequence. \(\square \)
We are now ready to formalise some intuitions using our notation. The first lemma is essentially about avoiding sets of vertices: it states that if one player can force divergence within a set, then this is the same as saying that the opponent cannot force the play outside this set.
Lemma 3
Proof
Follows using a similar argument as the proof of Lemma 1. \(\square \)
Next, we formalise the idea that if a player can force the play to a first set of vertices, and from there he can force the play to a second set of vertices, then he must be able to force the play to that second set.
Lemma 4
Proof
By Lemma 2, implies \(T \subseteq {_{U}}\! Attr ^{}_{ i _{}}(T')\). By monotonicity of \( Attr {}\) we have \({_{U}}\! Attr ^{}_{ i _{}}(T) \subseteq {_{U}}\! Attr ^{}_{ i _{}}({_{U}}\! Attr ^{}_{ i _{}}(T'))\). Since \({_{U}}\! Attr ^{}_{ i _{}}({_{U}}\! Attr ^{}_{ i _{}}(T')) = {_{U}}\! Attr ^{}_{ i _{}}(T')\), we thus have \({_{U}}\! Attr ^{}_{ i _{}}(T) \subseteq {_{U}}\! Attr ^{}_{ i _{}}(T')\). By the same token, from we find \(v \in {_{U}}\! Attr ^{}_{ i _{}}(T)\). Combined, we find \(v \in {_{U}}\! Attr ^{}_{ i _{}}(T')\) which, by Lemma 2, yields the desired . \(\square \)
Finally, we state two results that relate a player’s capabilities to reach a set of vertices T to the capabilities of the vertices that are able to leave a set of vertices U in a single step.
First, we show that if some player \( i _{}\) can force the play from v to T via U, then either \( i _{}\) own a vertex in U with a transition to T, or there is vertex owned by \(\lnot i _{}\) of which all transitions end up in T.
Lemma 5
Let \(S = \{ u \in U ~~ u^\bullet \cap T \ne \emptyset \}\) and \(v \notin T\). Then implies \(V_{ i _{}} \cap S \ne \emptyset \) or for some \(u \in S\), \(u^\bullet \subseteq T\).
Proof
Assume and suppose \(V_{ i _{}} \cap S = \emptyset \). Assume that for all \(u \in S\), not \(u^\bullet \subseteq T\). Then \({_{U}}\! Attr ^{}_{ i _{}}(T) = T\) follows from \(S \subseteq V_{\lnot i _{}}\). Since \(v \notin T\) this implies \(v \notin {_{U}}\! Attr ^{}_{ i _{}}(T)\). By Lemma 2 we then have . Contradiction. So there is some \(u \in S\) such that \(u^\bullet \subseteq T\). \(\square \)
If player \( i _{}\) can force the play from v to T via U, U and T do not overlap, and there are no transitions from U to \(T \setminus T'\), then player \( i _{}\) can also for the play from v to \(T'\) via U.
Lemma 6
For \(v \in U\), implies whenever \(T' \subseteq T\), \(U \cap T = \emptyset \), and \(w^\bullet \subseteq U \cup T'\) for all \(w \in U\).
Proof
Choose \(\sigma \in \mathbb {S}_{ i _{}}\) such that , and let \(p_{}\) be a path for which \({p_{}}[0] = v \in U\) and \(\sigma \Vdash p_{}\). Then for some \(j > 0\), \({p_{}}[j] \in T\) and \({p_{}}[k] \in U\) (and since \(U \cap T = \emptyset \), \({p_{}}[k] \not \in T\)) for all \(k < j\). Since \({p_{}}[j1]^\bullet \subseteq U \cup T'\), also \({p_{}}[j] \in T'\) and therefore . Thus . \(\square \)
A lattice of parity game relations
In the rest of this paper, we study relations on parity games. We forego a formal treatment and present an overview of the studied relations and how these are related in this section. Since we study both relational (or coinductive) and gamebased characterisations, we first introduce relations and (bi)simulation games.
Relations
Let \(\mathrel {R^{}}\) be a relation over a set V, i.e. \({\mathrel {R^{}}} \subseteq {V \times V}\). For \(v, w \in V\) we write \(v \mathrel {R^{}}w\) to denote \((v, w) \in {\mathrel {R^{}}}\). For a relation \(\mathrel {R^{}}\) and vertex \(v \in V\) we define \({v \mathrel {R^{}}} \mathop {=}\limits ^{\varDelta }\{ w \in V \mid v \mathrel {R^{}}w \}\), and likewise \(\mathrel {R^{}}v \mathop {=}\limits ^{\varDelta }\{ w \in V \mid w \mathrel {R^{}}v \}\). We also generalise this notation to sets of vertices such that, for \(U \subseteq V\), , and .
A relation \(\mathrel {R^{}}\) is a preorder if it is reflexive and transitive. If, in addition, \(\mathrel {R^{}}\) is symmetric, then it is an equivalence relation. Note that for an equivalence relation \(\mathrel {R^{}}\), and vertex \(v \in V\), we have \({v \mathrel {R^{}}} = {\mathrel {R^{}}v}\). In this case we also write \([v]_{\mathrel {R^{}}} \mathop {=}\limits ^{\varDelta }\{ v \in V \mid v \mathrel {R^{}}w \}\), and call this the equivalence class of v under \(\mathrel {R^{}}\). By abuse of notation, for a subset \(U \subseteq V\), we write \([U]_{\mathrel {R^{}}}\) for the set of equivalence classes with respect to V, i.e. the set \(\{[v]_{\mathrel {R^{}}} ~~ v \in U \}\). The set of equivalence classes of V under \(\mathrel {R^{}}\) is denoted \({V}_{/\mathrel {R^{}}}\), and defined as \(\{ [v]_{\mathrel {R^{}}} \mid v \in V \}\).
(Bi)simulation games
Some of the relations we consider in this paper are defined using twoplayer games. The games we consider are, essentially, EhrenfeuchtFraïssé games. For an extensive review of their applications in computer science, the interested reader is referred to [52].
Like parity games, the twoplayer games we use to characterise relations are instances of twoplayer, infiniteduration games with \(\omega \)regular winning conditions, that are played on game arenas that can be represented by graphs. Each vertex in these games is assigned to one of two players, Spoiler and duplicator. In the game, tokens are passed from one vertex to the next in the following fashion. The player that owns the vertex on which the token currently resides pushes the token to an adjacent vertex. These moves continue as long as possible, possibly forever. The path consisting of the sequences of vertices visited by the token is called a play. The winner of the play is decided depending on the predetermined winning criterion, which differs between the games. We say that a player can win from a given vertex if she has a strategy such that any play with the token initially at that vertex will be won by her.
The twoplayer games are also memoryless determined, i.e., every vertex is won by exactly one player, who also has positional winning strategy. These winning strategies can be efficiently computed while solving the game. We refer to [29] for an indepth treatment of the underlying theory.
Introducing a lattice of equivalences
Preorders for parity games are particularly (and perhaps only) interesting if they allow one to approximate the winning regions of a game. A preorder \(\mathrel {R^{}}\) approximates the winning region of a game if, whenever \(v \mathrel {R^{}}w\), and player \(\textit{even}\,\)has a winning strategy from v, then she also has a winning strategy from w. For equivalence relations, this requirement is stronger and often more useful: we require that if \(v \mathrel {R^{}}w\), then \(\textit{even}\,\)has a winning strategy from v if and only if she has a winning strategy from w. The finest natural equivalence relation on V is graph isomorphism, denoted \(\cong \).
Definition 3
(Isomorphism) Let \((V, \rightarrow , \varOmega , \mathcal {P})\) be a parity game. Vertices \(v,w \in V\) are isomorphic, denoted \(v \cong w\) iff \(\phi (v) = w\) for some bijection \(\phi :V \rightarrow V\) that satisfies, for all \(\bar{v} \in V\):

\(\varOmega (\bar{v}) = \varOmega (\phi (\bar{v}))\),

\(\mathcal {P}(\bar{v}) = \mathcal {P}(\phi (\bar{v}))\), and

\(\bar{v} \rightarrow \bar{v}'\) if and only if \(\phi (\bar{v}) \rightarrow \phi (\bar{v}')\).
The coarsest sensible equivalence on parity games is the equivalence induced by the determinacy of parity games, viz. the equivalence that exactly relates those vertices won by the same player.
Definition 4
(Winner equivalence) Let \((V, \rightarrow , \varOmega , \mathcal {P})\) be a parity game. Vertices \(v, w \in V\) are winner equivalent, denoted \(v \sim _ww\) iff v and w are won by the same player.
Deciding winner equivalence of parity games is equivalent to partitioning the vertices in the parity game into winning sets.
Winner equivalence and isomorphism are the extreme points in the lattice of equivalence relations shown in Fig. 1. Between the extremal points in the lattice of Figure 1 we list the other parity game equivalences that we study in more detail in the subsequent sections

Strong direct simulation equivalence (\(\equiv _{sd}\)), see Sect. 5.1;

Direct simulation equivalence (\(\equiv _{ d }\)) [23,24,25], see Sect. 5.1;

Delayed simulation equivalence (\(\equiv _{ de }\)) [23], see Sect. 5.2;

Delayed simulation equivalence, \(\textit{even}\,\)biased (\(\equiv _{ de }^{e}\)) [23], see Sect. 5.2.1;

Delayed simulation equivalence, \(\textit{odd\,}\)biased (\(\equiv _{ de }^{o}\)) [23], see Sect. 5.2.1;
In the lattice, an arrow from one equivalence to the other indicates that the first equivalence is finer than the latter. The number on an arrow refers to the theorem in this paper that claims this strictly finerthan relation between the equivalences.
The original definitions of the equivalences listed above vary in nature. Strong, governed, stuttering, and governed stuttering bisimilarity are defined coinductively, whereas direct simulation and all variations of delayed simulation are defined as simulation games. Furthermore, the direct and delayed simulation games define a preorder, whereas the others define an equivalence relation; the preorders are lifted to equivalence relations in the standard way.
Direct and delayed simulation equivalence
We introduce the direct simulation preorder and the induced direct simulation equivalence in Sect. 5.1. In Sect. 5.2, we recall the delayed simulation preorder, the induced delayed simulation equivalence and two biased versions of the delayed simulation preorder and equivalence. Throughout these sections, we assume an arbitrary parity game \(\mathcal {G}\!\!=\!\! (V,\rightarrow ,\varOmega ,\mathcal {P})\).
Direct simulation and direct simulation equivalence
Direct simulation is one of the most basic preorders studied for parity games. It is difficult to trace the exact origins of the definition, but it was suggested (though not formally defined) in [23] and appeared earlier in the setting of alternating Büchi automata [24]. We here follow the gamebased definition as given in [25].
The game is played on pairs of vertices (v, w), where duplicatortries to prove that v is simulated by w, and Spoiler tries to disprove this fact (hence the names of the players). The game proceeds in rounds, such that either duplicatoror Spoiler plays first according to the rules in Table 1. The player who is to play first, and the vertex on which this player has to play is determined by \(\mathcal {P}(v)\) and \(\mathcal {P}(w)\), i.e. the players that own the parity game vertices that constitute the current configuration. For example, in a configuration (v, w) with , Spoiler gets to play first on w by moving to some \(w'\) such that \(w \rightarrow w'\) in the parity game, and duplicatorhas to respond from v by playing to some \(v'\) such that \(v \rightarrow v'\) in the parity game.
The intuition here is as follows. Starting in a pair of vertices (v, w), Spoiler produces, step by step, an \(\omega \)word of priorities which it runs as two ‘simultaneous plays’ from v and w. Spoiler gets to control which successor is chosen at \(\textit{even}\,\)vertices reached by the play that started in v, whereas for the play that started in w, duplicatorcontrols the choice of successor for \(\textit{even}\,\)vertices. This is reversed at \(\textit{odd\,}\)vertices, in which case the player hands over control over the play to her opponent who then gets to choose the successor.
Since the game is total, play continues indefinitely, and duplicatorwins if she can always match Spoiler ’s move with the same priority, showing that the states are equivalent. If duplicatorcannot always match the priority, this means Spoiler found a way to distinguish two vertices. The game is formally defined as follows.
Definition 5
(Direct simulation game) The direct simulation game is played on configurations drawn from \(V \times V\), and it is played in rounds. A round of the game proceeds as follows:

1.
The players move from (v, w) to \((v', w')\) such that \(v \rightarrow v'\) and \(w \rightarrow w'\), according to the rules in Table 1;

2.
Play continues in the next round from the newly reached position \((v', w')\).
An infinite play \((v_0, w_0), (v_1, w_1), \ldots \) is won by duplicatorif \(\varOmega (v_j) = \varOmega (w_j)\) for all j, i.e., duplicatorwas able to mimic every move from Spoiler with a move to a vertex with equal priority. In all other cases Spoiler wins the play.
We say that v is directly simulated by w, denoted \(v \sqsubseteq _{ d }w\) whenever duplicatorhas a winning strategy from (v, w) in the direct simulation game.
Example 1
In the parity game in Fig. 2, \(v_0 \sqsubseteq _{ d }v_1\). Observe that from \((v_0, v_1)\), duplicatorcan choose both successors in the direct simulation game. We do not have \(v_1 \sqsubseteq _{ d }v_0\), since from configuration \((v_1, v_0)\), Spoiler ’s move \(v_1 \rightarrow v_3\) cannot be matched from \(v_0\). Note that additionally we have \(v_0 \sqsubseteq _{ d }v_2\) and \(v_2 \sqsubseteq _{ d }v_0\), and, of course, \(v_i \sqsubseteq _{ d }v_i\) for all i.
Direct simulation is a preorder: reflexivity is easily seen to hold (duplicatorcan follow a copycat strategy), but transitivity is more involved. In the setting of alternating Büchi automata, direct simulation was shown to be transitive using strategy composition, see [22, 24]. Following essentially the same technique one can show transitivity of direct simulation for parity games. We use the direct simulation preorder to obtain direct simulation equivalence in the standard way.
Definition 6
(Direct simulation equivalence [23, 25]) Vertices v and w are direct simulation equivalent, denoted \(v \equiv _{ d }w\), iff \(v \sqsubseteq _{ d }w\) and \(w \sqsubseteq _{ d }v\).
The alternative coinductive definition of direct simulation which we present next, allows for a more straightforward proof of transitivity. Our definition below was taken from [26], where it is also referred to as governed simulation. Intuitively, if , then for every successor \(v \rightarrow v'\), player even must be able to force to some \(w'\) from w in one step, such that \(v' \mathrel {R^{}}w'\). The case where more or less directly reflects the last two cases in Table 1: if duplicatorowns w, she merely needs to find \(v \rightarrow v'\) and \(w \rightarrow w'\) such that \(v' \mathrel {R^{}}w'\), otherwise, for every such \(w'\) she needs to find a \(v \rightarrow v'\) with \(v' \mathrel {R^{}}w'\).
Definition 7
(Direct simulation relation [26]) A relation \({\mathrel {R^{}}} \subseteq V \times V\) is a direct simulation if and only if \(v \mathrel {R^{}}w\) implies

\(\varOmega (v) = \varOmega (w)\);

if , then for each \(v' \in v^\bullet \), ;

if , then .
We write \(v \le _{d}w\), if and only if there is a direct simulation relation \(\mathrel {R^{}}\) such that \(v \mathrel {R^{}}w\). Likewise, we write \(v \rightleftarrows _{ d }w\) iff \(v \le _{d}w\) and \(w \le _{d}v\).
The theorem below states that the gamebased and coinductive definitions of direct simulation coincide.
Theorem 2
For all \(v, w \in V\), we have

\(v \le _{d}w\) if and only if \(v \sqsubseteq _{ d }w\), and

\(v \rightleftarrows _{ d }w\) if and only if \(v \equiv _{ d }w\).
Proof
For the first part of the statement, both implications follow straightforwardly from the definitions, using a case distinction on the players of v and w. The second part then follows from the definitions of \(\equiv _{ d }\) and \(\rightleftarrows _{ d }\). \(\square \)
Proposition 1
The relations \(\le _{d}\) and \(\rightleftarrows _{ d }\) are a preorder and an equivalence relation, respectively. Moreover, \(\le _{d}\) itself is a direct simulation relation.
Proof
For transitivity, one can check that for direct simulation relations R and S, the relation \(R \circ S\), defined as \(v \mathrel {(R \circ S)} w\) iff there is some u such that \(v \mathrel {R^{}}u\) and \(u \mathrel {S}w\), is again a direct simulation relation. \(\square \)
Strong direct simulation. If we impose an additional constraint on direct simulation, viz. we do not allow to relate vertices owned by different players, we obtain the stronger notion strong direct simulation. Clearly, this notion again is a preorder. We write \(v \le _{sd}w\) iff there is some strong direct simulation relation that relates v and w, and we write \(v \equiv _{sd}w\) iff \(v \le _{sd}w\) and \(w \le _{sd}v\). Note that in the parity game in Fig. 2, we still have \(v_0 \equiv _{sd}v_2\), but \(v_0 \not \le _{sd}v_1\).
Delayed simulation
Direct simulation equivalence is limited in its capability to relate vertices. The reason for this is that in each step of the simulation game, duplicatoris required to match with a move to a vertex with exactly the same priority. Following Etessami et al. [19], in [23], a more liberal notion of simulation called delayed simulation is considered. In this notion matching may be delayed. The idea is that in the winning condition of a play of a parity game, only the priorities that occur infinitely often are of importance. Therefore, intuitively it is allowed to delay matching a given (dominating) priority for a finite number of rounds.
The delayed simulation game is, like the direct simulation game, played on an arena consisting of configurations that contain a pair of vertices, and turns are taken according to the same rules, see Table 1. These configurations are now extended with a third parameter which is used to keep track of the dominating priority that still needs to be matched by duplicator. We say that this is the obligation that still needs to be met by duplicator.
An obligation in a delayed simulation game is either a natural number (corresponding to a priority in the game) or the symbol \(\checkmark \); the latter is used to indicate the absence of obligations, signifying that duplicatorhas matched the dominating priority played by Spoiler . We denote the set of obligations by K. Given two priorities and an existing obligation, a new obligation is obtained using the function \(\gamma ^{}:\mathbb {N}\times \mathbb {N}\times K \rightarrow K\). Let \(n,m \in \mathbb {N}\), and \(k \in K \setminus \{ \checkmark \}\), then \(\gamma ^{}\) is defined as follows:
For vertices \(v,w \in V\) and obligations \(k \in K\), we typically write \(\gamma ^{}(v,w,k)\) to denote \(\gamma ^{}(\varOmega (v), \varOmega (w), k)\).
The intuition behind this update is as follows. Either, with the priorities of the states reached after taking a step, the pending obligation is fulfilled, and the configuration reached by taking a step does itself not give rise to a new obligation, in which case the result is \(\checkmark \). Otherwise the new obligation is the minimum of the priorities passed to the function, or the current obligation. This represents the dominating priority that still needs to be met by duplicator. Note that there are two ways to fulfil the pending obligation. Either the first argument renders the pending obligation superfluous since it is smaller and odd (corresponding to a vertex with a dominating odd priority), or the second argument is such that it matches the pending obligation (corresponding to a vertex with a dominating even priority).
Definition 8
(Delayed simulation game [23]) A delayed simulation game is a game played by players Spoiler and duplicatoron an arena consisting of positions drawn from \(V \times V\) and obligations taken from K. The game is played in rounds. Assuming (v, w) is the current position, and k the current obligation, a round of the game proceeds as follows:

1.
Spoiler and duplicatorpropose moves \(v \rightarrow v'\) and \(w \rightarrow w'\) according to the rules in Table 1.

2.
The game continues from \((v', w')\) with obligation \(\gamma ^{}(v', w', k)\).
An infinite play \((v_0, w_0, k_0), (v_1, w_1, k_1), \ldots \) is won by duplicatoriff \(k_j = \checkmark \) for infinitely many j. This means that duplicatorwas always able to eventually fulfil all pending obligations. In all other cases Spoiler wins the game.
We say that v is delayed simulated by w, denoted \(v \sqsubseteq _{ de }w\) just whenever duplicatorhas a winning strategy from (v, w) with obligation \(\gamma ^{}(v, w, \checkmark )\) in the delayed simulation game.
Example 2
In the parity game in Fig. 3, \(v_i \sqsubseteq _{ de }v_j\) for all i, j. Observe that, with respect to direct simulation, vertices cannot be related to each other.
Delayed simulation is, like direct simulation, a preorder. The proof thereof is substantially more involved than the proof that direct simulation is a preorder, requiring an analysis of 24 cases, some of which are rather intricate. For details, we refer to [22]; we here only repeat this result.
Proposition 2
The relation \(\sqsubseteq _{ de }\) is a preorder.
We obtain delayed simulation equivalence in the standard way.
Definition 9
(Delayed simulation equivalence [23]) Vertices v and w are delayed simulation equivalent, denoted \(v \equiv _{ de }w\), iff \(v \sqsubseteq _{ de }w\) and \(w \sqsubseteq _{ de }v\).
Next, we give an alternative, coinductive definition for delayed simulation. Since the moves in the game for delayed simulation and direct simulation match, one may expect that such a characterisation can be obtained by a moreorless straightforward enhancement of the direct simulation relation. This is partly true: indeed, the moves of the game are captured in a way similar to how this is done for direct simulation. However, the winning condition of delayed simulation requires that infinitely often all obligations are met. This requires ‘nonlocal’ reasoning that must somehow be captured through a coinductive argument. Meeting an obligation is typically a progress property, requiring an inductive argument rather than a coinductive argument.
To combine both aspects in a single, coinductive definition, we draw inspiration from Namjoshi’s notion of wellfounded bisimulation [45]. Wellfounded bisimulation is a relation which is equivalent to stuttering equivalence, but which permits local reasoning by introducing a wellfoundedness criterion. We use a similar wellfoundedness requirement in our coinductive definition, ensuring progress is made towards fulfilling obligations. This moreover requires, as can be expected, that our coinductive relation ranges not only over pairs of vertices but also over obligations. For a relation \({\mathrel {R^{}}} \subseteq V \times K \times V\), we write \(v \mathrel {R^{k}} w\) if v and w are related under pending obligation k. The wellfoundedness restriction thus enables us to express that \(v\mathrel {R^{k}} w\) holds if we can build a coinductive argument that ultimately depends on pairs of vertices \(v',w'\) related under obligation \(\checkmark \); viz. \(v'\mathrel {R^{\checkmark }} w'\).
Definition 10
(Wellfounded delayed simulation) A relation \({\mathrel {R^{}}} \subseteq V \times K\times V\) is a wellfounded delayed simulation iff there is a wellfounded order \(<\cdot \) on \(V \times V \times K\) such that for all \(v,w \in V\) and \(k_{}\in K\) for which \(v \mathrel {R^{k_{}}} w\) holds, also:

implies for all \(v' \in v^\bullet \), ;

implies .
Vertex v is wellfounded delayed simulated by w, denoted \(v \le _{de}w\), iff there exists a wellfounded delayed simulation \(\mathrel {R^{}}\) such that \(v \mathrel {R^{\gamma ^{}(v,w,\checkmark )}} w\).
In the remainder of this section we show that this definition is equivalent to the gamebased definition. We first show that vertices that are related by wellfounded delayed simulation are also related by delayed simulation. Essentially we show that for a pair of related vertices \(v \mathrel {R^{\gamma (v,w,k)}} w\) (where \(\mathrel {R^{}}\) is a wellfounded delayed simulation), a winning strategy for duplicatorfrom configuration \((v,w, \gamma (v,w,k))\) in the delayed simulation game can be inferred. The proof contains two steps: first, we show that the moves in the game are made between configurations that correspond to related vertices, and second we show that when a move is made according to the induced strategy, eventually a configuration with obligation \(\checkmark \) is reached, relying on the wellfoundedness of the ordering \(<\cdot \).
Lemma 7
For \(v,w \in V\), \(v \le _{de}w\) implies \(v \sqsubseteq _{ de }w\).
Proof
We prove the stronger statement that if there is a wellfounded delayed simulation \(\mathrel {R^{}}\) such that \(v \mathrel {R^{\gamma ^{}(v, w, k_{})}} w\), for \(k \in K\), then duplicatorhas a strategy to win the delayed simulation game from \((v, w, \gamma ^{}(v, w, k_{}))\). The result then follows immediately from the observation that \(v \mathrel {R^{\gamma ^{}(v, w, \checkmark )}} w\), and hence duplicatorwins the game from \((v, w, \gamma ^{}(v, w, \checkmark ))\).
We first show that duplicatorhas a strategy to move between positions (v, w) with obligation \(k_{}\) for which \(v \mathrel {R^{k_{}}} w\) to positions \((v',w')\) and obligation \(k_{}'\) for which \(v' \mathrel {R^{k_{}'}} w'\). Assume that \(v \mathrel {R^{k}} w\). We distinguish four cases based on the owner of v and w.

. In the delayed simulation game, this corresponds to the vertex \((v, w, k_{})\), in which Spoiler is to move first. Spoiler first plays an arbitrary move \(v \rightarrow v'\). By definition of the wellfounded delayed simulation, there is \(w \rightarrow w'\) such that \(v' \mathrel {R^{\gamma ^{}(v', w', k_{})}} w'\); duplicatormatches with this \(w'\).

. In the delayed simulation game, from position \((v, w, k_{})\), Spoiler is to make both moves, so there is no duplicatorstrategy to be defined. Observe that wellfounded delayed simulation guarantees that for all \(v \rightarrow v'\) and \(w \rightarrow w'\), \(v' \mathrel {R^{\gamma ^{}(v', w', k_{})}} w'\).

. duplicatorplays twice in the delayed simulation game. According to the wellfounded delayed simulation, there exist \(v \rightarrow v'\) and \(w \rightarrow w'\) such that \(v' \mathrel {R^{\gamma ^{}(v', w', k_{})}} w'\). duplicatorplays such moves.

. In the delayed simulation game, Spoiler is to move first, say \(w \rightarrow w'\). From the wellfounded delayed simulation, we find that for all such moves, there exists some \(v \rightarrow v'\) such that \(v' \mathrel {R^{\gamma ^{}(v', w', k_{})}} w'\). duplicatorplays to this \(w'\).
It remains to be shown that for all configurations \((v, w, k_{})\) such that \(v \mathrel {R^{k_{}}} w\), this duplicatorstrategy is, indeed, winning for duplicator. Observe that it suffices to show that, if \(k_{}\ne \checkmark \), eventually a configuration \((v', w', \checkmark )\) is reached. This follows, since in every round in the game above moves are made from \((v, w, k_{})\) to \((v', w', k_{}')\) such that \((v', w', k_{}') <\cdot (v, w, k_{})\). Since \(<\cdot \) is a wellfounded order, this can only be repeated finitely many times, and eventually all obligations are met. \(\square \)
Before we show the converse, we first show that a winning strategy for player duplicatorin the delayed simulation game induces a wellfounded order on those configurations won by duplicator. Essentially this is based on the observation that, since duplicator’s strategy is winning from a given configuration, all partial plays starting in that configuration that follow duplicator’s strategy reach a configuration with obligation \(\checkmark \) in a finite number of steps.
Lemma 8
The winning strategy for duplicatorin the delayed simulation game induces a wellfounded order on \(V \times V \times K\) for those \((v,w,k_{})\) for which duplicatorwins position (v, w) with obligation \(k_{}\).
Proof
Observe that the delayed simulation game has a Büchi winning condition. Hence those configurations in the game that duplicatorcan win, can be won using a memoryless strategy. For the remainder of the proof, fix such a winning memoryless strategy.
For each position (v, w) and obligation \(k_{}\) that is won by duplicator, we extract a finite tree from the solitaire game that is induced by duplicator’s strategy by taking the (infinite) unfolding of the game starting in \((v, w, k_{})\), and pruning each branch at the first node with obligation \(\checkmark \). Since the strategy is duplicatorwinning, this tree is finite. Furthermore, if \((v, w, k_{})\) appears in the tree of a different configuration, the subtree rooted in \((v, w, k_{})\) in that particular subtree is identical to the tree of \((v, w, k_{})\).
These trees determine a wellfounded order \(<\cdot \) as follows: \((v', w', \ell ) <\cdot (v, w, k_{})\) iff the height of the tree rooted in \((v', w', \ell )\) is less than that of the tree rooted in \((v, w, k_{})\). \(\square \)
The following corollary immediately follows from the existence of the wellfounded order.
Corollary 1
In the delayed simulation game, for every position (v, w) with obligation \(k_{}\) from which duplicatorhas a winning strategy, if the game proceeds according to this strategy to some position \((v', w')\) with obligation \(\ell = \gamma ^{}(v',w', k_{})\), we have \(k_{}= \checkmark \) or \((v', w', \ell ) <\cdot (v, w, k_{})\).
Finally, we show that any delayed simulation is also a wellfounded delayed simulation.
Lemma 9
For \(v, w \in V\), \(v \sqsubseteq _{ de }w\) implies \(v \le _{de}w\).
Proof
The relation \({\mathrel {R^{}}} \subseteq V \times K\times V\) defined as \(v \mathrel {R^{k_{}}} w\) if duplicatorwins the delayed simulation game from (v, w) with obligation \(k_{}\) is a wellfounded delayed simulation. This result follows from two observations. First, a wellfounded order on configurations won by duplicatorexists, according to Lemma 8. Second, that the transfer condition is satisfied follows from the duplicatorwinning strategy in the delayed simulation game, using a straightforward case distinction the players of related vertices. Furthermore, in every step, the wellfounded order on configurations won by duplicatordecreases in every step according to Corollary 1. \(\square \)
The following theorem, stating that delayed simulation and wellfounded delayed simulation coincide, now follows directly.
Theorem 3
For all \(v,w \in V\) we have \(v \sqsubseteq _{ de }w\) if and only if \(v \le _{de}w\).
Proof
Follows immediately from Lemmata 7 and 9.
Biased delayed simulations.
As observed in [23], quotienting is problematic for delayed simulation: no sensible definition of quotienting appears to exist such that it guarantees that the quotient is again delayed simulation equivalent to the original game. Fritz and Wilke ‘mend’ this by introducing two variations (so called biased delayed simulations) on delayed simulation which do permit some form of quotienting although these are not unique. We briefly describe these variations below.
Evenbiased delayed simulation. The evenbiased delayed simulation game, and its coinductive variant, are identical to their delayed simulation and wellfounded delayed simulation counterparts. The only difference lies in the update function on obligations. Given two priorities and an existing obligation, a new obligation is obtained using the update function \(\gamma ^{e} :\mathbb {N}\times \mathbb {N}\times K\rightarrow K\), where:
We again abbreviate \(\gamma ^{e}(\varOmega (v),\varOmega (w),k)\) by \(\gamma ^{e}(v,w,k)\).
Using the new update function in the delayed simulation game ensures that a pending obligation is only changed back to \(\checkmark \) by a small even priority; a small odd priority does not change the obligation. We say that v is evenbiased delayed simulated by w, denoted \(v \sqsubseteq _{ de }^{e}w\) iff duplicatorhas a winning strategy from (v, w) with obligation \(\gamma ^{e}(v, w, \checkmark )\) in the evenbiased delayed simulation game.
Likewise, we obtain wellfounded, evenbiased delayed simulation by replacing all occurrences of \(\gamma ^{}\) by \(\gamma ^{e}\) in Definition 10. Vertex v is wellfounded, evenbiased delayed simulated by w, denoted \(v \le _{de}^{e}w\), iff there exists a wellfounded, evenbiased delayed simulation preorder \(\mathrel {R^{}}\) such that \(v \mathrel {R^{\gamma ^{e}(v, w, \checkmark )}} w\).
Oddbiased delayed simulation. The oddbiased delayed simulation is defined in a similar way as the evenbiased delayed simulation. Instead of small even priorities leading to an update of a pending obligation, small odd priorities lead to a change in the obligation. Given two priorities and an existing obligation, a new obligation is obtained using the update function \(\gamma ^{o} :\varOmega \times \varOmega \times K\rightarrow K\), where:
The gamebased and coinductive definitions are analogous to the evenbiased version.
Governed bisimulation and governed stuttering bisimulation
In this section we consider essentially two notions of bisimulation for parity games, and some derived notions. First, in Sect. 6.1, we introduce governed bisimulation, which was studied under various guises in e.g. [25, 39, 42]. Governed bisimulation is, as we demonstrate in that section, closely related to direct simulation. Next, in Sect. 6.2, governed stuttering bisimulation [11, 14, 39] is introduced. The adjective ‘governed’ reflects that there is always one player that governs, or forces a local choice (and response to such a choice) in a play.
Governed bisimulation
Our definition of governed bisimulation, as presented below, is based on the one from [42] where it is defined in the closely related setting of Boolean equation systems; because of its capabilities to relate conjunctive and disjunctive equations, it was dubbed idempotence identifying bisimulation. It was rephrased for parity games in [39] and there named governed bisimulation.
Definition 11
(Governed bisimulation) A symmetric relation \({\mathrel {R^{}}}\subseteq V\times V\) is a governed bisimulation iff \(v \mathrel {R^{}}w\) implies

\(\varOmega (v) = \varOmega (w)\);

if \(\mathcal {P}(v) \ne \mathcal {P}(w)\), then \(v' \mathrel {R^{}}w'\) for all \(v' \in v^\bullet \) and \(w' \in w^\bullet \);

for all \(v' \in v^\bullet \) there is some \(w' \in w^\bullet \) such that \(v' \mathrel {R^{}}w'\).
Vertices v and w are said to be governed bisimilar, denoted , if and only if there is a governed bisimulation \(\mathrel {R^{}}\) such that \(v \mathrel {R^{}}w\).
Example 3
In the parity game in Fig. 4, we have for all i, , and furthermore, and . Observe that we have \(v_0 \equiv _{ d }v_1\), where \(v_0 \le _{d}v_1\) is witnessed by relation \(R_1 = \{ (v_i, v_i) \mid 0 \le i \le 5 \} \cup \{ (v_0, v_1) \}\), and \(v_1 \le _{d}v_0\) is witnessed by \(R_2 = \{ (v_i, v_i) \mid 0 \le i \le 5 \} \cup (v_1, v_0), (v_4, v_2), (v_4, v_3) \}\). We do, however, not have , since the latter would require \(v_4\) to be related to \(v_2\), but from \(v_4\) the step \(v_2 \rightarrow v_5\) cannot be mimicked.
Governed bisimulation is such that vertices owned by different players can only be related whenever all their successors are related. It turns out that this is exactly what is obtained when imposing a symmetry requirement on direct simulation. As a result, we have the following proposition.
Proposition 3
We have iff there is a symmetric direct simulation relation \(\mathrel {R^{}}\) such that \(v \mathrel {R^{}}w\).
As a consequence, we immediately find that governed bisimilarity is an equivalence relation.
Theorem 4
is an equivalence relation on parity games.
Proof
Follows from combining Proposition 3 and Proposition 1. \(\square \)
Additionally, the direct simulation game, extended with the possibility for Spoiler to switch to a symmetric position in the game play, gives the direct bisimulation game, as defined by Etessami et al. [19] for Büchi games.
Definition 12
(Direct bisimulation game) The direct bisimulation game is played on configurations drawn from \(V \times V\), and it is played in rounds. A round of the game starting in (v, w) proceeds as follows:

1.
Spoiler chooses \((u_0, u_1) \in \{ (v,w), (w,v) \}\);

2.
The players move from \((u_0, u_1)\) according to the rules in Table 1

3.
Play continues in the next round from the newly reached position.
An infinite play \((v_0, w_0), (v_1, w_1), \ldots \) is won by duplicatorif \(\varOmega (v_j) = \varOmega (w_j)\) for all j, i.e., duplicatorwas able to mimic every move from Spoiler with a move to a vertex with equal priority. In all other cases Spoiler wins the play.
We write \(v \equiv _{ g }w\) whenever duplicatorhas a winning strategy from (v, w) in the direct bisimulation game.
The following theorem now formally establishes the correspondence between governed bisimilarity and direct bisimulation.
Theorem 5
For all \(v, w \in V\), we have if and only if \(v \equiv _{ g }w\).
Proof
Straightforward from the definitions, using a case distinction on the players of v and w. \(\square \)
Strong Bisimulation. If we again impose the additional constraint on governed bisimulation that we do not allow to relate vertices owned by different players, we obtain a notion called strong bisimulation [39]. The derived notion of strong bisimilarity, denoted and defined as iff there is some strong bisimulation relation that relates v and w, is an equivalence relation.
Governed stuttering bisimulation
The (bi)simulation games discussed so far all have in common that the gameplay proceeds in ‘lockstep’: duplicatormust match every move proposed by Spoiler with a proper countermove. In a sense, this ignores the fact that the parity condition is not sensitive to finite repetitions of priorities but only cares about infinite repetitions. The insensitivity of the parity condition to finite repetitions is reminiscent to the notion of stuttering in process theory. Indeed, as we demonstrate in what follows, governed bisimulation can be weakened such that it becomes insensitive to finite stuttering, but remains sensitive to infinite stuttering. The resulting relation is called governed stuttering bisimulation.
Essentially, governed stuttering bisimulation is obtained by porting stuttering equivalence for Kripke structures [7] to the setting of parity games. Intuitively, governed stuttering bisimulation requires that a move from a vertex v to \(v'\) is matched by a finite (and potentially empty) sequence of moves from w, through vertices that remain related to v, to arrive at some \(w'\) that is related to \(v'\). In addition, every divergent play from a vertex v (i.e. a play that remains confined to a single equivalence class) should be matched with a divergent play from a related vertex w.
Details, however, are subtle. In stuttering equivalence it suffices to have the ability to move or diverge and match such moves or divergences with some move or a divergence. In contrast, in the parity game setting, we have to consider the abilities of a player, knowing that a single player does not control the moves from all vertices. Only moves and divergences that can be forced by a player count, and matching of such moves and divergences must be done through moves or divergences that the same player can force. Figure 5 illustrates some of these concepts. While in the depicted parity game there is an infinite play that passes through the two leftmost vertices with priority 0, neither \(\textit{even}\,\)nor \(\textit{odd\,}\)can force such an infinite play. As a result, we may ignore such infinite plays, and in this sense, the abilities (for both players) from those two vertices are no different from the abilities both players have from the two rightmost vertices with priority 0.
The definition of governed stuttering bisimulation presented below is based on [11, 14]. For our definition, we strongly rely on our notation to denote that a player is able to ‘force play’. The idea is that, if from one vertex v another equivalence class \(\mathcal {C}\) can be reached in a single step, then from a related vertex, the player owning v can force the play to end up in the same equivalence class \(\mathcal {C}\) (irrespective of the opponent’s moves). Furthermore, if a player can force the game to stay in the equivalence class of v indefinitely, than can also do so from the related vertex.
Definition 13
(Governed stuttering bisimulation) Let \({\mathrel {R^{}}} \subseteq V \times V\) be an equivalence relation. Then \(\mathrel {R^{}}\) is a governed stuttering bisimulation if and only if \(v \mathrel {R^{}}w\) implies

(a)
\(\varOmega (v) = \varOmega (w)\);

(b)
\(v \rightarrow \mathcal {C}\) implies .

(c)
\(v \mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) implies \(w\mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) for .
Vertices v and w are governed stuttering bisimilar, denoted , iff a governed stuttering bisimulation \(\mathrel {R^{}}\) exists such that \(v \mathrel {R^{}}w\).
Example 4
The parity game in Fig. 5 nicely illustrates the key properties of governed stuttering bisimulation: for \(v_j\) in \(\{ v_0, v_1, v_3, v_4 \}\) we have neither nor . Furthermore, for all these vertices, both players can force the game to reach vertex \(v_2\). Therefore, all vertices with the same priorities are related by . Also note that the vertices with priority 0 are not related by, e.g., governed bisimulation since the latter is sensitive to counting, and \(v_0\) and \(v_1\) can reach multiple equivalence classes.
Proving that is a governed stuttering bisimulation relation is technically involved. In particular, all standard proof techniques for doing so break down or become too complex to manage. Instead of a large monolithic proof of the result, we proceed in small steps by gradually rephrasing the above definition to one that is ultimately more easily seen to be an equivalence. Our first step in this direction is to remove the asymmetry in clause (b) of the definition of governed stuttering bisimulation. Before we do so, we state a useful lemma that allows us to strengthen the conclusion of Lemma 5. Essentially, the lemma states the following. Suppose that there is a vertex v in its equivalence class such that all its outgoing edges are sure to leave the equivalence class into some set . Then for all vertices u related to v, any transition leaving the equivalence class is also sure to end up in .
Lemma 10
Let R be a governed stuttering bisimulation. Let \(\mathcal {U}\subseteq {V}_{/R} \setminus \{[v]_{R}\}\). If , then \(u^\bullet \setminus [v]_{R} \subseteq \bigcup \mathcal {U}\) for all \(u \in [v]_{R}\).
Proof
Let v be such that \(v \rightarrow \mathcal {U}\) for some \(\mathcal {U}\subseteq {V}_{/R} \setminus \{[v]_{R}\}\). Suppose \(u \rightarrow \mathcal {C}\) for some \(\mathcal {C}\notin \mathcal {U}\cup \{[v]_{R}\}\). Since , by Definition 13, we have . But and \(\mathcal {C}\notin \mathcal {U}\) so . Contradiction. \(\square \)
Now consider the following alternative for governed stuttering bisimulation. Instead of requiring that, if v leaves to another equivalence class \(\mathcal {C}\) directly, than the player owning v has a strategy to do so from any related vertex, we make the second requirement symmetric. Informally, if a player \( i _{}\) has a strategy to force the play to some equivalence class \(\mathcal {C}\), then she also has a strategy to achieve the same from any related vertex.
Proposition 4
Let \({\mathrel {R^{}}} \subseteq V \times V\) and \(v,w\in V\). Then \(\mathrel {R^{}}\) is a governed stuttering bisimulation iff \(\mathrel {R^{}}\) is an equivalence relation and \(v \mathrel {R^{}}w\) implies:

(a)
\(\Omega (v) = \Omega (w)\);

(b)
iff for all ;

(c)
\(v \mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) iff \(w\mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) for all .
Proof
The proof for the implication from right to left follows immediately. We focus on the implication from left to right. Assume that \(\mathrel {R^{}}\) is a governed stuttering bisimulation. We prove the second condition only; the other two conditions follow immediately from Definition 13 and symmetry of R. Let \( i _{}\) be an arbitrary player and assume that for given \(v \in V\) and \(\mathcal {C}\in {V}_{/\mathrel {R^{}}} \setminus \{[v]_{\mathrel {R^{}}}\}\). Let \(S = \{u \in [v]_{R} ~~ u^\bullet \cap \mathcal {C}\ne \emptyset \}\). We distinguish two cases.

Case \(V_{ i _{}} \cap S \ne \emptyset \). Let \(u \in S \cap V_{ i _{}}\). Since \(u \rightarrow \mathcal {C}\), follows from Definition 13.

Case \(S \subseteq V_{\lnot i _{}}\). By Lemma 5, there is a \(u \in S\) for which \(u^\bullet \subseteq \mathcal {C}\) and by Lemma 10 (for \(\mathcal {U}= \{\mathcal {C}\}\)), \(u^\bullet \setminus [v]_{R} \subseteq \mathcal {C}\) for all \(u \in [v]_{R}\). Furthermore, by Lemma 3, implies . Then, by Definition 13, and by Lemma 3, . But since \(u^\bullet \subseteq \mathcal {C}\cup [v]_{R}\) for all u, the desired follows from Lemma 6. \(\square \)
While the above alternative characterisation of governed stuttering bisimulation is now fully symmetric, using the restriction on the class \(\mathcal {C}\) that is considered in clause (b) we were still not able to give an insightful proof that is an equivalence relation using this definition. We therefore further generalise this clause such that it is phrased in terms of sets of classes. So, if from a vertex v, player \( i _{}\) can force the play to a set of classes \(\mathcal {U}\), then she can do so from any related vertex.
A perhaps surprising sideresult of this generalisation is that the divergence requirement of clause (c) becomes superfluous. This is due to the duality between diverging in a set by one player, and forcing to the complement of that set by the other as shown in Lemma 3. Note that this generalisation is not trivial, as is in general neither equivalent to saying that and , nor or .
Proposition 5
Let \({\mathrel {R^{}}} \subseteq V \times V\) and \(v,w\in V\). Then \(\mathrel {R^{}}\) is a governed stuttering bisimulation iff \(\mathrel {R^{}}\) is an equivalence relation and \(v \mathrel {R^{}}w\) implies:

(a)
\(\varOmega (v) = \varOmega (w)\);

(b)
iff for all .
Proof
We show that the second condition is equivalent to the conjunction of the last two conditions in Proposition 4. We split the proof into an ifpart and an onlyifpart.
 \(\Leftarrow \) :

The second condition from Proposition 4 is equivalent to the second condition above if we let \(\mathcal {U}\) range only over singleton sets (if , take \(\mathcal {U}= \{\mathcal {C}\}\)). The third condition is equivalent to the second condition above, where \(\mathcal {U}= {V}_{/\mathrel {R^{}}} \setminus \{[v]_{\mathrel {R^{}}}\}\). This can be seen by appealing to Lemma 3.
 \(\Rightarrow \) :

Let \(\mathrel {R^{}}\) be a governed stuttering bisimulation and let \(v,w \in V\) such that \(v \mathrel {R^{}}w\). Assume that for some \(\mathcal {U}\subseteq {V}_{/\mathrel {R^{}}} \setminus \{[v]_{\mathrel {R^{}}}\}\). Let \(S = \{u \in [v]_{\mathrel {R^{}}} ~~ u \rightarrow \mathcal {U}\}\). Using similar arguments as in the proof of Proposition 4, distinguishing cases on emptiness on \(S \cap V_{ i _{}}\), we obtain that .
In the previous proposition, we lifted the notion of forcing play via the current equivalence class towards a target class, to the notion of forcing a play via the current equivalence class towards a set of target classes. Towards the proving transitivity of governed stuttering bisimulation, we introduce a final generalisation in the proposition below; rather than forcing play towards a set of target classes via the current equivalence class, we now allow the play to be forced to that set via a set of equivalence classes. For this definition, we identify a set of equivalence classes \(\mathcal {U}\) with the set of states to allow us to write instead of .
Proposition 6
Let \({\mathrel {R^{}}} \subseteq V \times V\) and \(v,w\in V\). Then \(\mathrel {R^{}}\) is a governed stuttering bisimulation iff \(\mathrel {R^{}}\) is an equivalence relation and \(v \mathrel {R^{}}w\) implies:

(a)
\(\varOmega (v) = \varOmega (w)\);

(b)
iff for all such that \([v]_{\mathrel {R^{}}} \in \mathcal {U}\) and \([v]_{\mathrel {R^{}}} \notin \mathcal {T}\).
Proof
We show that the second condition is equivalent to the second condition in Proposition 5. We split the proof into an ifcase and an onlyifpart.
 \(\Leftarrow \) :

The second condition from Proposition 5 is equivalent to the second condition above if we fix \(\mathcal {U}= \{[v]_{\mathrel {R^{}}}\}\).
 \(\Rightarrow \) :

Let \(\mathrel {R^{}}\) be a governed stuttering bisimulation and let \( i _{}, v, w, \mathcal {U}\) and \(\mathcal {T}\) be as described. Assume that ; under this assumption we will prove that . The proof for the implication in the other direction is completely symmetric. Let \(\sigma \in \mathbb {S}_{ i _{}}\) be such that and consider the set of paths originating in v that are allowed by \(\sigma \). All these paths must have a prefix \(v \dots v', u\) such that but . Call these prefixes the \(\sigma \)prefixes of v. We proceed by induction on the length of the longest such prefix. If the longest prefix has length 2, then all prefixes have length 2, implying that \(v \mathop {\,\!_{ i _{}}\smash \rightarrow } \mathcal {T}\). In particular, and by Proposition 5 also , which proves . As the induction hypothesis, assume that if \(u \mathrel {R^{}}u'\), and the longest \(\sigma \)prefix of u is shorter than the longest \(\sigma \)prefix of v, then . Note that every \(\sigma \)prefix \(p_{}\) of v must have a first position n such that \({p_{}}[n] \notin [v]_{\mathrel {R^{}}}\). Collect all these \({p_{}}[n]\) in a set U, and notice that for all \(u \in U\), also . Furthermore, . By Proposition 5, . Now consider an arbitrary . Because there is some \(u \in U\) such that \(u \mathrel {R^{}}u'\), its longest \(\sigma \)prefix is shorter than the longest \(\sigma \)prefix of v, and because for such u, we can use the induction hypothesis to derive that . The above in particular implies two facts: , and for all . Using these, we can now apply Lemma 4 to conclude . \(\square \)
With this last characterisation, it is now straightforward to prove that governed stuttering bisimilarity is an equivalence relation. We do so by showing that the transitive closure of the union of two governed stuttering bisimulations R and S is again a governed stuttering bisimulation. The generalisation from classes to sets of classes allows us to view equivalence classes in \((R \cup S)^*\), i.e., the transitive closure of \(R \cup S\), as the union of sets of equivalence classes of R (or S), giving us an easy way to compare the effect of the second requirement of Proposition 6 on \((R \cup S)^*\) with its effect on R and S.
Theorem 6
is an equivalence relation.
Proof
We show that \((R \cup S)^*\) is a governed stuttering bisimulation if R and S are, by showing that \((R \cup S)^*\) satisfies the conditions of Proposition 6 if R and S do. If \(v,w \in V\) are related under \((R \cup S)^*\), then there exists a sequence of vertices \(u_0, \dots , u_n\) such that \(v \mathrel {R^{}}u_0 \mathrel {S}\dots \mathrel {R^{}}u_n \mathrel {S}w\) (the strict alternation between the two relations can always be achieved because R and S are reflexive). By transitivity of \(=\) we then have \(\varOmega (v) = \varOmega (w)\), so the first property is satisfied.
For the second property, assume that for some and some \(\mathcal {U},\mathcal {T}\subseteq {V}_{/(R \cup S)^*}\) such that \([v]_{(R \cup S)^*} \in \mathcal {U}\) and \([v]_{(R \cup S)^*} \notin \mathcal {T}\). We need to prove that . Note that R and S both refine \((R \cup S)^*\), so we can find sets \(\mathcal {U}_R \subseteq {V}_{/\mathrel {R^{}}}\) and \(\mathcal {U}_S \subseteq {V}_{/\mathrel {S}}\) such that . Because , also , and by Proposition 6 then , which is equivalent to . By a simple inductive argument we now arrive at , which is equivalent to . \(\square \)
As a sideresult of the proof of Theorem 6, we find that the union of all governed stuttering bisimulations is again a governed stuttering bisimulation, which coincides with governed stuttering bisimilarity.
In order to better understand the differences between governed stuttering bisimulation and, e.g. delayed simulation equivalence, we next provide a gamebased characterisation of the relation. While in this new game, Spoiler and duplicatorstill move according to the same rules as in the delayed simulation game, duplicatornow has more freedom to choose a new configuration: she can now also choose to ‘rollback’ one of the proposed moves. This allows her to postpone matching a move. Of course, such moves may not be postponed indefinitely, so some additional mechanism is needed to keep track of duplicator’s progress so as to prevent duplicatorfrom becoming too powerful. For this, we follow [16], and we use a system of challenges and rewards: a \(\dagger \)challenge indicates duplicatordecided to match a move by Spoiler by not moving; a \(\checkmark \)reward indicates duplicatormatched a move by Spoiler by making a countermove, and a challenge (k, u) taken from \(\{0,1\} \times V\) indicates that duplicatoris in the process of matching a move to vertex u. We let C denote the set of challenges \((\{0,1\} \times V) \cup \{\dagger ,\checkmark \}\).
Definition 14
(Governed Stuttering bisimulation game) The governed stuttering bisimulation game is played on an arena of configurations drawn from \((V \times V) \times C\), and it is played in rounds. A round of the game starting in a configuration ((v, w), c) proceeds as follows:

1.
Spoiler chooses to play from \((u_0,u_1) \in \{(v,w), (w, v)\}\);

2.
the players move from \((u_0,u_1)\) to \((t_0,t_1)\) according to the rules in Table 1;

3.
duplicatorselects a new configuration drawn from the following set:
$$\begin{aligned} \begin{array}{lll} \{ &{} ((t_0,t_1),\checkmark ), &{} \\ &{} ((u_0,t_1),\gamma ^{}(c,(0,t_0),v,u_0)), &{} \\ &{} ((t_0,u_1),\gamma ^{}(c,(1,t_1),w,u_1)) &{} \} \end{array} \end{aligned}$$where update \(\gamma ^{}\) is defined as follows:
An infinite play \(((v_0,w_0),c_0), ((v_1,w_1),c_1), \dots \) is won by duplicatoriff \(\varOmega (v_j) = \varOmega (w_j)\) for all j and \(c_k = \checkmark \) for infinitely many k. duplicatorwins the governed stuttering bisimulation game for a position (v, w) iff she has a strategy that wins all plays starting in configuration \(((v,w),\checkmark )\).
We write \(v \equiv _{ g,st }w\) whenever duplicatorwins the governed stuttering bisimulation game for position (v, w).
Observe that in the governed stuttering game, duplicatorearns, as explained before, a \(\checkmark \) reward whenever she continues playing in the position determined at the end of step 2. However, she also earns a \(\checkmark \) whenever Spoiler decides to drop a pending challenge or, in step 1 of a round, switch positions. The example below illustrates some of the intricacies in the game play.
Example 5
Consider the parity game depicted in Fig. 5. In this parity game, all vertices with priority 0 are related by . The game illustrates why duplicatorgains a \(\checkmark \) reward whenever Spoiler does not respect a pending challenge. This can be seen as follows: consider the game starting in \(((v_1, v_3), \checkmark )\) and suppose Spoiler decides to play \(v_1 \rightarrow v_2\). The only suitable response by duplicatoris to play \(v_3 \rightarrow v_4\). New configurations \(((v_2, v_4), \checkmark )\) and \(((v_2, v_3, \checkmark ))\) are not an option for duplicatorsince he immediately loses due to the different priorities of \(v_2\) and \(v_4\) or \(v_3\) respectively. The new configuration chosen by duplicatorwill hence be \(((v_1, v_3), (0, v_2))\), challenging Spoiler to play \(v_1 \rightarrow v_2\) again in the next round. From this configuration, if Spoiler indeed plays \(v_1 \rightarrow v_2\), duplicatorcan match with \(v_4 \rightarrow v_2\), and play stays in \(((v_2, v_2), \checkmark )\) indefinitely, leading to a win from duplicator. Now, let us consider what happens if Spoiler plays \(v_1 \rightarrow v_0\) instead. Spoiler did not respect the challenge, and duplicatormatches with \(v_4 \rightarrow v_3\), and we end up in \(((v_1, v_3), \checkmark )\) again. If duplicatorwould not have earned a \(\checkmark \) reward in this case, play would have ended up in \(((v_1, v_3), (0, v_0))\) instead, and, if in the next round Spoiler again ignores the challenge, play can alternate indefinitely between \(((v_1, v_3), (0, v_0))\) and \(((v_1, v_4), (0, v_2))\), which would result in a win for Spoiler . This is undesirable since we already observed that \(v_1, v_3\) and \(v_4\) are governed stuttering bisimilar.
For the remainder of this section we turn our attention to relating the above game to governed stuttering bisimulation. Our next result states that whenever vertices v, w are governed stuttering bisimilar, duplicatorwins all plays starting in configuration \(((v,w),\checkmark )\). We sketch the main ideas behind the proof; details can be found in the “Appendix”.
Proposition 7
For all \(v,w \in V\) if then \(v \equiv _{ g,st }w\).
Proof
The proof proceeds by showing that duplicatorhas a strategy that ensures (1) that plays allowed by this strategy move along configurations of the form \(((u_0,u_1),c)\) for which and (2) duplicatornever gets stuck playing according to this strategy and 3) there is a strictly decreasing measure between two consecutive non\(\checkmark \) configurations on any play allowed by this strategy. Together, this implies that duplicatorhas a winning strategy for configurations \(((v,w),\checkmark )\). \(\square \)
We next establish that vertices related through the governed stuttering bisimulation game are related by governed stuttering bisimulation. A straightforward proof thereof is hampered by the fact that any purported governed stuttering bisimulation relation is, by definition, required to be an equivalence relation. However, proving that the governed stuttering bisimulation game induces an equivalence relation is rather difficult. The strategy employed to prove the stated result is to use contraposition; this requires showing that for any given pair of nongoverned stuttering bisimilar vertices we can construct a strategy that is winning for Spoiler . Note that we can do so because the governed stuttering bisimulation game has a Büchi winning condition, which implies the game is determined. This strategy is based on a fixpoint characterisation of governed stuttering bisimilarity, given below.
Definition 15
Let \({\mathrel {R^{}}} \subseteq V \times V\) be an equivalence relation on V. The predicate transformer \(\mathcal {F} : V \times V \rightarrow V \times V\) is defined as follows:
The predicate transformer \(\mathcal {F}\) has the following properties, both of which can be proven straightforwardly.
Lemma 11
\(\mathcal {F}(R)\) is an equivalence relation for any equivalence relation R on V.
Lemma 12
\(\mathcal {F}\) is a monotone operator on the complete lattice of equivalence relations on V.
Using these properties, the following corollary now follows immediately.
Corollary 2
We have , where \(\nu \) is the greatest fixed point.
Proof
Follows from the fact that for \(R = \nu \mathcal {F}\) and \(\nu \mathcal {F} = \mathcal {F}(\nu \mathcal {F})\) the definition of \(\mathcal {F}\) reduces to the definition of governed stuttering bisimulation. \(\square \)
We finally state our completeness result. Again, we only outline the main steps of the proof; details can be found in the “Appendix”.
Proposition 8
For all \(v,w \in V\) if \(v \equiv _{ g,st }w\) then .
Proof
We essentially prove the contrapositive of the statement, i.e. for all \(v,w \in V\), if , then also \(v \not \equiv _{ g,st }w\). Let . By Corollary 2, then also \((v,w) \notin \nu \mathcal {F}\). By the KnasterTarskiKleene fixpoint approximation theorem, we thus have \((v,w) \notin \bigcap \limits _{k \ge 1} \mathcal {F}^k(V \times V)\). Using induction, one can prove, for \(R^k = \bigcap _{l \le k} R^l\), that for all \(k \ge 1\):
For the inductive case, one can construct a strategy for Spoiler that guarantees he never gets stuck and for which every play allowed by the strategy either (1) visits some configuration \(((t_0,t_1),c')\) for which the induction hypothesis applies, or (2) is such that there are only a finite number of \(\checkmark \) rewards along the play. \(\square \)
Propositions 7 and 8 lead to the following theorem.
Theorem 7
For all \(v,w \in V\) we have iff \(v \equiv _{ g,st }w\).
Stuttering Bisimulation. When we impose the additional constraint on governed stuttering bisimulation that we do not allow to relate vertices owned by different players, we obtain a notion called stuttering bisimulation [13]. The derived notion of stuttering bisimilarity, denoted and defined as iff there is some stuttering bisimulation relation that relates v and w, is an equivalence relation.
Quotienting
Simulation and bisimulation equivalences are often used to reduce the size of graphs by factoring out vertices that are equivalent, i.e. by computing quotient structures. This can be particularly interesting if computationally expensive algorithms must be run on the graph: whenever the analysis such algorithms perform on the graphs are insensitive to (bi)simulation equivalence, they can be run on the smaller quotient structures instead. In our setting, the same reasoning applies: typically, parity game solving is expensive and it may therefore pay off to first compute a quotient structure and only then solve the resulting quotient structure.
In this section, we show that most of the (bi)simulation relations we studied in the previous two sections have unique quotient structures. A fundamental property of quotienting is that the resulting quotient structure of a game should again be equivalent to the original game. This requires that we lift our equivalences to relations between two different game graphs. We do so in the standard way.
Definition 16
Let \(\mathcal {G}_j = (V_j,\rightarrow _j,\varOmega _j,\mathcal {P}_j)\), for \(j = 1, 2\), be arbitrary parity games. We say that \(\mathcal {G}_1 \sim \mathcal {G}_2\), for an equivalence relation \(\sim \) defined on the vertices of a parity game, whenever in the disjoint union of \(\mathcal {G}_1\) and \(\mathcal {G}_2\), for all \(v_1 \in V_1\) there is some \(v_2 \in V_2\) such that \(v_1 \sim v_2\) and for all \(\bar{v}_2 \in V_2\) there is some \(\bar{v}_1 \in V_1\) such that \(\bar{v}_1 \sim \bar{v}_2\).
Simulation equivalence quotients
Quotienting for delayed simulation equivalence is, as observed in [22, 23], problematic, and only the biased versions admit some form of quotienting. However, the quotients for biased delayed simulation equivalences are not unique, see also Lemma 3.5 in [22]. We here only consider quotienting for direct simulation equivalence; for the quotients of delayed simulation equivalence we refer to the aforementioned works.
The equivalence classes of direct simulation equivalence determine the set of vertices of the quotient structure. Defining the transition relation of the quotient structure is a bit more subtle. As observed in [9], a unique quotient structure of simulation equivalence for Kripke structures exists, but requires that vertices have no transitions to a pair of vertices, one of which is sometimes referred to as a ‘little brother’ of the other one (a vertex that is simulated by, but not equivalent to the other vertex). That is, transitions to nonmaximal vertices are omitted.
While in the setting of Kripke structures, only transitions to maximal successor vertices must be retained, depending on the owner of the source vertex, in our setting we need to consider maximal or minimal successor vertices.
Definition 17
Let \(V' \subseteq V\) be an arbitrary nonempty set of vertices. An element v is:

minimal among \(V'\) iff for all \(u \in V'\) for which \(u \le _{d}v\), also \(v \le _{d}u\);

maximal among \(V'\) iff for all \(u \in V'\) for which \(v \le _{d}u\), also \(u \le _{d}v\).
For a given vertex v, a successor \(v' \in v^\bullet \) is in the set \(\min _{\le _{d}}(v)\) iff \(v'\) is minimal among \(v^\bullet \); likewise, \(v' \in v^\bullet \) is in the set \(\max _{\le _{d}}(v)\) iff \(v'\) is maximal among \(v^\bullet \).
Since \(\le _{d}\) is a preorder, \(\min _{\le _{d}}(v)\) and \(\max _{\le _{d}}(v)\) are nonempty sets.
An additional complication in defining a unique quotient structure is that a single equivalence class may contain vertices owned by \(\textit{even}\,\)and vertices owned by \(\textit{odd\,}\). It turns out that the owner of such equivalence classes can be chosen arbitrarily: we prove that such classes have a unique successor equivalence class. For equivalence classes with exactly one successor, we can assign a unique owner; we choose to assign such classes to player \(\textit{even}\,\).
Definition 18
(Direct simulation equivalence quotient) The direct simulation equivalence quotient of \((V,\rightarrow ,\varOmega ,\mathcal {P})\) is the structure \(({V}_{/\equiv _{ d }}, \rightarrow ', \varOmega ', \mathcal {P}')\), where, for \(\mathcal {C},\mathcal {C}' \in {V}_{/\equiv _{ d }}\):

\(\varOmega '(\mathcal {C}) = \min \{\varOmega (v) ~~ v \in \mathcal {C}\}\),

\(\mathcal {C}\rightarrow ' \mathcal {C}'\) iff
Observe that it is not obvious that \(\rightarrow '\) is a total edge relation. The lemma below allows us to establish that this is the case. It establishes that for an equivalence class consisting entirely of vertices owned by \(\textit{odd\,}\)(respectively, \(\textit{even}\,\)), the set of minimal (respectively, maximal) successors of all vertices in this class are the same. The vertices in equivalence classes that consist of both \(\textit{even}\,\)and \(\textit{odd\,}\)vertices the set of minimal successors of \(\textit{odd\,}\)vertices is the same as the set of maximal successors of the \(\textit{even}\,\)vertices.
Lemma 13
Let \(\mathcal {C}\in {V}_{/\equiv _{ d }}\). Then:

If then \([\min _{\le _{d}}(v)]_{\equiv _{ d }} = [\min _{\le _{d}}(w)]_{\equiv _{ d }}\) for all \(v,w \in \mathcal {C}\),

If then \([\max _{\le _{d}}(v)]_{\equiv _{ d }} = [\max _{\le _{d}}(w)]_{\equiv _{ d }}\) for all \(v,w \in \mathcal {C}\),

If and then for all and we have \([\max _{\le _{d}}(v)]_{\equiv _{ d }} = [\min _{\le _{d}}(w)]_{\equiv _{ d }}\).
Proof
We prove the first and the third statement; the proof for the second statement is analogous to that of the first.

Suppose . Pick \(v,w \in \mathcal {C}\). Let \(v' \in \min _{\le _{d}}(v)\). Since \(v' \in v^\bullet \) and \(w \le _{d}v\), we have \(w' \le _{d}v'\) for some \(w' \in w^\bullet \). This implies that there is some \(w'' \in \min _{\le _{d}}(w)\) such that \(w'' \le _{d}v'\); for, if \(w' \notin \min _{\le _{d}}(w)\), then there must be some \(w'' \in \min _{\le _{d}}(w)\) such that \(w'' \le _{d}w'\). But then also \(w'' \le _{d}v'\).
We next show that also \(v' \le _{d}w''\). Since \(v \le _{d}w\) and \(w'' \in w^\bullet \) we have \(v'' \le _{d}w''\) for some \(v'' \in v^\bullet \). Since \(w'' \le _{d}v'\) and \(v'' \le _{d}w''\), we have \(v'' \le _{d}v'\). But since \(v' \in \min _{\le _{d}}(v)\), this implies \(v' \equiv _{ d }v''\). But from \(v' \le _{d}v''\) and \(v'' \le _{d}w''\) we obtain \(v' \le _{d}w''\). Hence, \(v' \equiv _{ d }w''\) for some \(w'' \in \min _{\le _{d}}(w)\).

Suppose \(\mathcal {P}(v) \ne \mathcal {P}(w)\) for some \(v,w \in \mathcal {C}\). Pick \(v,w \in \mathcal {C}\) such that and . Since \(w \le _{d}v\), there must be \(w' \in w^\bullet \) and \(v' \in v^\bullet \) such that \(w' \le _{d}v'\). Fix such \(v'\) and \(w'\). Since \(v \le _{d}w\) we find that for all \(v'' \in v^\bullet \) and \(w'' \in w^\bullet \) we have \(v'' \le _{d}w''\). In particular, \(v' \le _{d}w'\). So \(v' \equiv _{ d }w'\).
Next, since for all \(w'' \in w^\bullet \) we have \(v' \le _{d}w''\) and \(v' \equiv _{ d }w'\), we also have \(w' \le _{d}w''\) for all \(w'' \in w^\bullet \). But this implies \(w' \in \min _{\le _{d}}(w)\), and, in particular, \([\min _{\le _{d}}(w)]_{\equiv _{ d }} = 1\). Likewise, we deduce \(v' \in \max _{\le _{d}}(v)\) and \([\max _{\le _{d}}(v)]_{\equiv _{ d }} = 1\). We thus find \([\max _{\le _{d}}(v)]_{\equiv _{ d }} = \{ [v']_{\equiv _{ d }} \} = \{ [w']_{\equiv _{ d }} \} = [\min _{\le _{d}}(w)]_{\equiv _{ d }}\). \(\square \)
As a consequence of the above lemma, we obtain the following two results, essentially confirming that the direct simulation equivalence quotient is welldefined:
Corollary 3
Let \(({V}_{/\equiv _{ d }},\rightarrow ',\varOmega ', \mathcal {P}')\) be a direct simulation equivalence quotient of some parity game \((V,\rightarrow ,\varOmega ,\mathcal {P})\). Then for all \(\mathcal {C},\mathcal {C}' \in {V}_{/\equiv _{ d }}\):

if and for some \(v \in \mathcal {C}\), \(v' \in \mathcal {C}'\) also \(v' \in \min _{\le _{d}}(v)\), then \(\mathcal {C}\rightarrow ' \mathcal {C}'\).

if and for some , \(v' \in \mathcal {C}'\) also \(v' \in \max _{\le _{d}}(v)\), then \(\mathcal {C}\rightarrow ' \mathcal {C}'\).
Corollary 4
The direct simulation reduced quotient structure associated to a parity game \((V, \rightarrow \varOmega , \mathcal {P})\) is again a parity game.
We next establish that the direct simulation quotient of a parity game is equivalent to the original parity game.
Proposition 9
Let \(\mathcal {G}= (V, \rightarrow , \varOmega , \mathcal {P})\) be a parity game and \(\mathcal {G}_q = ({V}_{/\equiv _{ d }}, \rightarrow ', \varOmega ', \mathcal {P}')\) its direct simulation quotient. Then \(\mathcal {G}_q \equiv _{ d }\mathcal {G}\).
Proof
\(\mathcal {G}_q \le _{d}\mathcal {G}\), follows from the observation that relation \(H \subseteq {V}_{/\equiv _{ d }} \times V\), defined as , is a direct simulation relation. Similarly, \(\mathcal {G}\le _{d}\mathcal {G}_q\), follows from the fact that relation \(H \subseteq V \times {V}_{/\equiv _{ d }}\), given by , is a direct simulation relation. \(\square \)
From the above result it follows that quotienting is ‘safe’ in the sense that one can solve the quotient game and still extract the solution of the original parity game. It is not hard to see that the size of a quotient is at most as large as the original game. From a computational point of view it thus makes sense to solve the quotient game instead of the original game. We finally establish that the quotient is unique. Combined with the fact that the quotient game is at most as large as the original game we find that each parity game has a unique, smallest quotient. This essentially confirms that by quotienting we achieve a maximal reduction.
Theorem 8
Let \(\mathcal {G},\mathcal {G}'\) be two parity games and let \(\mathcal {G}_q\) and \(\mathcal {G}_q'\) be their direct simulation equivalence quotients, respectively. Then \(\mathcal {G}\equiv _{ d }\mathcal {G}'\) iff the two structures \(\mathcal {G}_q = ({V}_{/\equiv _{ d }}, \rightarrow _q, \varOmega _q, \mathcal {P}_q)\) and \(\mathcal {G}_q' = ({V'}_{/\equiv _{ d }}, \rightarrow '_q, \varOmega '_q, \mathcal {P}'_q)\) are isomorphic.
Proof
The proof that isomorphism of \(\mathcal {G}_q\) and \(\mathcal {G}_q'\) implies \(\mathcal {G}\equiv _{ d }\mathcal {G}'\) follows essentially from Proposition 9 and that isomorphic structures are also direct simulation equivalent.
The proof that \(\mathcal {G}\equiv _{ d }\mathcal {G}'\) implies that \(\mathcal {G}_q\) and \(\mathcal {G}_q'\) are isomorphic structures follows the following steps. Assume that \(\mathcal {G}\equiv _{ d }\mathcal {G}'\). Let \(f \subseteq {V'}_{/\equiv _{ d }} \times {V}_{/\equiv _{ d }}\) be defined as \((\mathcal {C}', \mathcal {C}) \in f\) iff \(\mathcal {C}' \equiv _{ d }\mathcal {C}\). Note that for all \((\mathcal {C}',\mathcal {C}) \in f\) we have \(\varOmega '_q(\mathcal {C}') = \varOmega _q(\mathcal {C})\).
We first show that f is a total bijective function from \({V'}_{/\equiv _{ d }}\) to \({V}_{/\equiv _{ d }}\). For injectivity and functionality of f we reason as follows. Suppose f is not functional. Then there is some \(v' \in V'\) and two \(v,\bar{v} \in V\) such that \([v]_{\equiv _{ d }} \ne [\bar{v}]_{\equiv _{ d }}\), \(([v']_{\equiv _{ d }},[v]_{\equiv _{ d }}) \in f\) and \(([v']_{\equiv _{ d }},[\bar{v}]_{\equiv _{ d }}) \in f\). Then by definition, \(v' \equiv _{ d }v\) and \(v' \equiv _{ d }\bar{v}\). But then also \(v \equiv _{ d }\bar{v}\), contradicting that \([v]_{\equiv _{ d }} \ne [\bar{v}]_{\equiv _{ d }}\). So f is functional. The proof that \(f^{1}\) is a function from \({V}_{/\equiv _{ d }}\) to \({V'}_{/\equiv _{ d }}\) is similar. We may therefore conclude that f is an injective function.
For surjectivity of f, we observe that by definition of \(\mathcal {G}\equiv _{ d }\mathcal {G}'\), for each \(v \in V\) there is some \(v' \in V'\) such that \(v \equiv _{ d }v'\). Hence, for each \([v]_{\equiv _{ d }} \in {V}_{/\equiv _{ d }}\) there is some \([v]_{\equiv _{ d }} \in {V'}_{/\equiv _{ d }}\) such that \(([v']_{\equiv _{ d }}, [v]_{\equiv _{ d }}) \in f\). Similarly, we can show that \(f^{1}\) is surjective and therefore f is total bijection.
We next prove that \(\mathcal {P}_q'(\mathcal {C}) = \mathcal {P}_q(f(\mathcal {C}))\). Towards the contrary, assume that whereas for some \(\mathcal {C}\). Then and for all \(v \in \mathcal {C}\) we have \(\min _{\le _{d}}(v) > 1\), and there is some \(w \in f(\mathcal {C})\) satisfying either , or \(\min _{\le _{d}}(w) = 1\). Let \(w \in f(\mathcal {C})\) be such and pick an arbitrary \(v \in \mathcal {C}\). We distinguish two cases.

Case . Since \(w \equiv _{ d }f(\mathcal {C}) \equiv _{ d }\mathcal {C}\equiv _{ d }v\) we have \(w \le _{d}v\) in particular. Pick an arbitrary \(w' \in w^\bullet \). Then, since , we have \(w' \le _{d}v'\) for all \(v' \in v^\bullet \); more specifically, we have \(w' \le _{d}v_1'\) and \(w' \le _{d}v_2'\) for \(v_1',v_2' \in \min _{\le _{d}}(v)\) such that \(v_1' \not \equiv _{ d }v_2'\). Since \(v_1',v_2'\) are minimal elements, we thus also have \(v_1' \le _{d}w'\) and \(v_2' \le _{d}w'\) and hence \(v_1' \equiv _{ d }w'\) and \(v_2' \equiv _{ d }w'\). But from this we obtain \(v_1' \equiv _{ d }v_2'\). Contradiction.

Case \(\min _{\le _{d}}(w) = 1\). Without loss of generality we may assume that . Since \(w \equiv _{ d }f(\mathcal {C}) \equiv _{ d }\mathcal {C}\equiv _{ d }v\) we also have \(w \le _{d}v\). Let \(v_1',v_2' \in \min _{\le _{d}}(v)\) be such that \(v_1' \not \equiv _{ d }v_2'\). Then there must be some \(w_1', w_2' \in w^\bullet \) such that \(w_1' \le _{d}v_1'\) and \(w_2' \le _{d}v_2'\). Let \(w_1',w_2'\) be such; without loss of generality, we may assume that \(w_1'\) and \(w_2'\) are minimal. Since \(v_1',v_2'\) are minimal, we find that \(v_1' \le _{d}w_1'\) and \(v_2' \le _{d}w_2'\) and hence \(v_1' \equiv _{ d }w_1'\) and \(v_2' \equiv _{ d }w_2'\). But because \(v_1' \not \equiv _{ d }v_2'\) we have \(w_1' \not \equiv _{ d }w_2'\). Since \(w_1'\) and \(w_2'\) are minimal we have \([\min _{\le _{d}}(w)]_{\equiv _{ d }} \ge 2\). Contradiction.
Hence, \(\mathcal {P}'_q(\mathcal {C}) = \mathcal {P}_q(f(\mathcal {C}))\).
Finally, we prove that \(\mathcal {C}\rightarrow '_q \mathcal {C}'\) iff \(f(\mathcal {C}) \rightarrow _q f(\mathcal {C}')\). Suppose \(\mathcal {C}\rightarrow '_q \mathcal {C}'\) but not \(f(\mathcal {C}) \rightarrow _q f(\mathcal {C}')\). Assume . The case where is similar. Since \(\mathcal {C}\equiv _{ d }f(\mathcal {C})\), there must be some \(\mathcal {D}\) such that \(f(\mathcal {C}) \rightarrow _q \mathcal {D}\) and \(\mathcal {C}' \le _{d}\mathcal {D}\). But then also \(\mathcal {C}\rightarrow _q' \mathcal {C}''\) and \(\mathcal {D} \le _{d}\mathcal {C}''\) for some \(\mathcal {C}''\). Then \(\mathcal {C}' \le _{d}\mathcal {C}''\). Distinguish two cases:

Case \(\mathcal {C}' = \mathcal {C}''\). Then \(f(\mathcal {C}') = f(\mathcal {C}'') = \mathcal {D}\), contradicting our assumption that \(f(\mathcal {C}) \not \rightarrow _q f(\mathcal {C}')\).

Case \(\mathcal {C}' \ne \mathcal {C}''\). Then we have \(\mathcal {C}\rightarrow _q' \mathcal {C}'\) and \(\mathcal {C}\rightarrow _q' \mathcal {C}''\) and \(\mathcal {C}' \le _{d}\mathcal {C}''\). But this means that vertices in \(\mathcal {C}'\) are not maximal. Hence, \(\mathcal {G}_q'\) does not have a transition \(\mathcal {C}\rightarrow _q' \mathcal {C}'\). \(\square \)
Corollary 5
The direct simulation equivalence quotient of \(\mathcal {G}\) is a unique (upto isomorphism) smallest parity game direct simulation equivalent to \(\mathcal {G}\).
Governed bisimulation and governed stuttering bisimulation quotients
We first define the governed bisimulation quotient. It is essentially a simplification of the direct simulation equivalence quotient.
Definition 19
(Governed bisimulation quotient) The governed bisimulation quotient of \((V,\rightarrow ,\varOmega ,\mathcal {P})\) is the structure , where, for :

\(\varOmega '(\mathcal {C}) = \min \{\varOmega (v) ~~ v \in \mathcal {C}\}\),

\(\mathcal {C}\rightarrow ' \mathcal {C}'\) if and only if
We next state some elementary results concerning the governed bisimulation quotient. Since the proofs of these results follow the corresponding ones we presented in the previous section, we omit their details.
Proposition 10
Let \(\mathcal {G}= (V, \rightarrow , \varOmega ,\mathcal {P})\) be a parity game and be its governed bisimulation quotient. Then .
Proof
Follows from the fact that the relation \(R = \{(v,\mathcal {C}), (\mathcal {C},v)~~ v \in \mathcal {C}\}\), is a governed bisimulation relation. \(\square \)
Theorem 9
Let \(\mathcal {G},\mathcal {G}'\) be two parity games and let \(\mathcal {G}_q\) and \(\mathcal {G}_q'\) be their direct simulation equivalence quotients, respectively. Then iff the two structures and are isomorphic.
Corollary 6
The governed bisimulation quotient of \(\mathcal {G}\) is a unique (upto isomorphism) smallest parity game that is governed bisimilar to \(\mathcal {G}\).
As a consequence of the above results, we find that each parity game has a unique and minimal governed bisimilar parity game.
We next define the governed stuttering bisimulation quotient. It requires some subtlety to properly deal with divergences and ensure that a unique player is assigned to an equivalence class. More specifically, we cannot simply assign the player of an arbitrary vertex in an equivalence class to this equivalence class; instead, the player assigned to an equivalence class must be the one that has the ability to force play from that equivalence class.
Definition 20
(Governed stuttering bisimulation quotient)
The governed stuttering bisimulation quotient of \((V,\rightarrow ,\varOmega ,\mathcal {P})\) is the structure , where, for :

\(\varOmega '(\mathcal {C}) = \min \{\varOmega (v) ~~ v \in \mathcal {C}\}\),

\(\mathcal {C}\rightarrow ' \mathcal {C}'\) if and only if
The results below essentially mirror those results we presented in the previous section for the direct simulation equivalence quotient. Details of the proofs are once again omitted since they follow the exact same line of reasoning as those from the previous section.
Proposition 11
Let \(\mathcal {G}= (V, \rightarrow , \varOmega ,\mathcal {P})\) be a parity game and be its governed stuttering bisimulation quotient. Then .
Proof
Follows from the fact that relation , defined as \(R = \{(v,\mathcal {C}), (\mathcal {C},v), (v,w), (\mathcal {C},\mathcal {C})~~ v,w \in \mathcal {C}\}\), is a governed stuttering bisimulation. \(\square \)
Theorem 10
Let \(\mathcal {G},\mathcal {G}'\) be two parity games and let \(\mathcal {G}_q\) and \(\mathcal {G}_q'\) be their direct simulation equivalence quotients, respectively. Then iff the two structures and are isomorphic.
Corollary 7
The governed stuttering bisimulation quotient of \(\mathcal {G}\) is a unique (upto isomorphism) smallest parity game that is governed stuttering bisimilar to \(\mathcal {G}\).
A comparison of discriminating power
In this section, we compare the discriminative power of each of the equivalences discussed in the preceding sections, essentially justifying the lattice we illustrated in Sect. 4. This permits us to assess the reductive power of each of the studied equivalences that admit (unique) quotienting. For each of the equivalences we show which other equivalences it strictly refines. Incomparability results are described separately.
We first focus on proving the righthand side of the lattice we presented in Sect. 4. That is, we first compare isomorphism, strong bisimilarity and governed bisimilarity and then focus on the various simulation equivalences.
Theorem 11
Isomorphism is strictly finer than strong bisimilarity.
Proof
Clearly, every pair of isomorphic parity games is a pair of strong bisimilar parity games. Strictness follows from a standard example:
Clearly, both vertices in the left parity game are strongly bisimilar to the vertex in the right parity game, and vice versa. However, these vertices are not isomorphic. \(\square \)
The following theorem relates strong bisimilarity to stuttering equivalence, governed bisimulation and strong direct simulation equivalence, and except for the comparison to governed bisimulation it is essentially the counterpart of the classical theorems in the setting of Kripke structures.
Theorem 12
Strong bisimilarity is strictly finer than strong direct simulation equivalence, stuttering bisimulation equivalence and governed bisimulation equivalence.
Proof
We sketch each of the refinements:

Every strong bisimulation relation is a direct simulation relation. Since such a relation is symmetric, every pair of parity games related via strong bisimilarity is also related via strong direct simulation equivalence. Strictness follows from the parity game in Fig. 4, in which \(v_0\) and \(v_1\) are strong direct simulation equivalent but not strongly bisimilar.

Every strong bisimulation relation is a stuttering bisimulation relation, this follows directly from the definitions. Strictness follows from the parity game in Fig. 5, in which \(v_3\) and \(v_4\) are stuttering bisimilar, but not strongly bisimilar.

Every strong bisimulation relation is a governed bisimulation relation, this follows directly from the definitions. Strictness follows from the parity game in Fig. 6, which is minimal modulo strong bisimilarity, but vertices \(v_2\) and \(v_3\) are governed bisimilar. \(\square \)
We next state, without proof, a result that essentially follows by definition.
Theorem 13
Strong direct simulation equivalence strictly refines direct simulation equivalence.
The following refinement results follow a line of reasoning similar to the ones seen before.
Theorem 14
Governed bisimulation equivalence strictly refines direct simulation equivalence and governed stuttering bisimulation equivalence.
Proof
We again sketch both refinements.

Refinement follows directly from the observation that a governed bisimulation is a symmetric direct simulation. Strictness follows from examples similar to those discriminating strong bisimilarity and simulation equivalence.

It follows from the definitions that every governed bisimulation is also a governed stuttering bisimulation. The strictness of the refinement follows from the example in Fig. 5 in which all vertices with priority 0 are governed stuttering bisimilar, but none are governed bisimilar. \(\square \)
Theorem 15
Governed bisimulation equivalence and strong direct simulation equivalence are incomparable.
Proof
This follows from the parity game in Fig. 4 in which vertices \(v_0\) and \(v_6\) are governed bisimilar, but not strong direct simulation equivalent. Furthermore, \(v_0\) and \(v_1\) are strong direct simulation equivalent, but not governed bisimilar. \(\square \)
Regarding direct simulation and (biased) delayed simulations we have the following result due to Fritz and Wilke [23].
Theorem 16
[23] Direct simulation equivalence is strictly finer than even and oddbiased delayed simulation equivalence. Even and oddbiased delayed simulation are incomparable, and both are strictly finer than delayed simulation equivalence. Delayed simulation equivalence in turn is strictly finer than winner equivalence.
This completes the results underlying the righthand side of the lattice we presented in Sect. 4.
We next focus on the lefthand side of the lattice.
Theorem 17
Stuttering bisimilarity is incomparable to governed bisimilarity, strong direct simulation, direct simulation and all delayed simulation variations.
Proof
In the parity game in Fig. 5, \(v_3\) and \(v_4\) are stuttering bisimilar but they cannot be related under governed bisimilarity, (strong) direct simulation equivalence, nor any of the delayed simulation equivalences. For the other direction, consider vertices \(v_0\), \(v_1\) and \(v_6\) from the parity game in Fig. 4. None of these vertices are stuttering bisimilar, whereas \(v_0\) and \(v_6\) are governed bisimilar, \(v_0\) and \(v_1\) are strong direct simulation equivalent, and all three are direct simulation equivalent, and therefore also delayed simulation equivalent. \(\square \)
Theorem 18
Governed stuttering bisimilarity is incomparable to direct simulation, strong direct simulation, and all delayed simulation variations.
Proof
Follows from the same examples as used in the proof of Theorem 17. \(\square \)
Theorem 19
Stuttering bisimilarity strictly refines governed stuttering bisimilarity.
Proof
It follows from the definitions that every stuttering bisimulation is also a governed stuttering bisimulation. Strictness follows from the parity game in Fig. 5 in which \(v_0\) and \(v_1\) are governed stuttering bisimilar, but not stuttering bisimilar. \(\square \)
To complete the lattice, we next show that governed stuttering bisimilarity is strictly finer than winner equivalence. In order to prove this result, we must first lift the concept of governed stuttering bisimilarity to paths. Being able to reason about paths then allows us to show that every memoryless strategy of player \(\textit{odd\,}\)(respectively, \(\textit{even}\,\)) in the original game induces a strategy of player \(\textit{odd\,}\)(respectively, \(\textit{even}\,\)) in the quotient game such that all plays in the quotient game, allowed by this strategy, have a ‘matching’ play allowed by the strategy in the original game. If we then consider a strategy that is winning for a player in the original game, it follows that this strategy is also winning for this player in the quotient game, and vice versa.
We first lift governed bisimilarity from vertices to paths. Paths of length 1 are equivalent if the vertices they consist of are equivalent. If paths p and q are equivalent, then iff v is equivalent to the last vertex in q, and iff . An infinite path p is equivalent to a path q if for all finite prefixes of p there is an equivalent prefix of q and vice versa.
Lemma 14
Let \((V, {\rightarrow }, \mathcal {P}, \varOmega )\) be a parity game, and let be its quotient. Let \(v \in V\), and such that \(v \in \mathcal {C}\). For all players \( i _{}\), and all \(\sigma \in \mathbb {S}_{ i _{}}\) there is some \(\psi \in \mathbb {S}_{ i _{}}^*\) such that for all there is a such that .
Proof
Let \(\sigma \in \mathbb {S}_{ i _{}}\) be an arbitrary strategy for player \( i _{}\). Before we construct a strategy \(\psi \in \mathbb {S}_{ i _{}}^*\) for player \( i _{}\) in the quotient game, we introduce some auxiliary functions which help defining this strategy. Define an arbitrary complete ordering \(<\cdot \) on vertices, and define the following for finite paths q starting in \(\mathcal {C}\), where \(\min _{<\cdot } \emptyset \) is defined to be \(\bot \):
We next show that it is possible to define a strategy \(\psi \in \mathbb {S}_{ i _{}}^*\) for finite plays such that if for some \(p \in \varPi _{\sigma }^{}(v)\), then:
Let be such that for and assume \(\mathcal {P}'(\mathcal {C}') = i _{}\). In case \(\mathsf {div}(q)\) and \(\mathcal {C}' \rightarrow ' \mathcal {C}'\), then obviously \(\psi (q)\) can be defined as \(\mathcal {C}'\). We proceed to show that if \(\lnot \mathsf {div}(q)\) or \(\mathcal {C}' \not \rightarrow ' \mathcal {C}'\), then 1) \(\mathsf {next}(q) \ne \bot \), and 2) we can set . We show the first by distinguishing two cases:
 Case \(\lnot \mathsf {div}(q)\).:

It follows straightforwardly that \(\mathsf {next}(q) \ne \bot \).
 Case \(\mathcal {C}' \not \rightarrow ' \mathcal {C}'\).:

Because \(\mathcal {C}'\) is a vertex is a quotient graph, . Consider the path for which , and assume that p is of the form . Since , also and hence . Then by Lemma 3, . Let \(\sigma ' \in \mathbb {S}_{\lnot i _{}}\) be such that and consider the unique path such that , . Then is such that , \(pr \mathop {\,\!_{\sigma }\smash \rightarrow } v'\) and . Hence, \(\mathsf {next}(q) \ne \bot \).
Next, we show that we can set . Since \(\mathsf {next}(q) \ne \bot \), there must be some such that \(v'' = \mathsf {next}(q)\), and . Also, , since and . As \(\mathcal {C}'\) is a vertex in a quotient graph, this implies . Hence, we can set .
Now that we have shown that it is always possible to define a strategy adhering to the restrictions above, let \(\psi \) be such a strategy. We show using induction on n that for all n
For \(n=0\), this is trivial, because . For \(n = m + 1\), assume as the induction hypothesis that . Let \(q \in \varPi _{\psi }^{n}(\mathcal {C})\) and let and \(\bar{q} \in \varPi _{\psi }^{m}(\mathcal {C})\) such that and . Distinguish cases on the player who owns \(\mathcal {C}'\).
 Case \(\mathcal {P}'(\mathcal {C}') \ne i _{}.\) :

Then \(\mathcal {C}'' = \psi (\bar{q})\). The induction hypothesis yields some such that , therefore \(\mathcal {C}'' = \mathcal {C}'\) if \(\mathsf {div}(\bar{q})\) and \(\mathcal {C}' \rightarrow ' \mathcal {C}'\), and otherwise . If \(\mathcal {C}' = \mathcal {C}''\), then \(\mathsf {div}(\bar{q})\), so there must be some such that and therefore also some such that . If , there must be some such that and and . By definition, for such p.
 Case \(\mathcal {P}'(\mathcal {C}') \ne i _{}.\) :

From the induction hypothesis, obtain a such that .
Without loss of generality we may assume that \(\bar{p}\) is finite. Note that .
We distinguish two cases.

Case \(\mathcal {C}' = \mathcal {C}''\). Then we have .

Case \(\mathcal {C}' \ne \mathcal {C}''\). Let \(v'\) be the last vertex in \(\bar{p}\). Because , also . So let \(\sigma ' \in \mathbb {S}_{\lnot i _{}}\) be such that . Now consider an infinite path \(\bar{p} p\) such that \(\sigma \Vdash \bar{p} p\) and \(\sigma ' \Vdash \bar{p} p\). For some index \(k \ge 0\), it must be the case that and for all \(l < k\). So .
Finally, we prove that for all there is a such that . Let \(q \in \varPi _{\psi }^{\omega }(\mathcal {C})\). Then by the above, we find that there is some \(p \in \varPi _{\sigma }^{}(v)\). Suppose p is finite and \(p = \bar{p} v'\) for some vertex \(v'\). Since q is a path through the quotient graph, q must be of the form \(\bar{q} \mathcal {C}^\omega \) for some \('\).

Case \(\mathcal {P}'(\mathcal {C}) = i _{}\). Then \(\psi (\bar{q} \mathcal {C}) = \mathcal {C}\), and thus \(\mathsf {div}(\bar{q} \mathcal {C})\) by definition of \(\psi \). But then there must be some \(p' \in \varPi _{\sigma }^{\omega }(v)\) such that .

Case \(\mathcal {P}'(\mathcal {C}) \ne i _{}\). Since \(\mathcal {C}\rightarrow ' \mathcal {C}\) we have and since , also . Let \(\sigma ' \in \mathbb {S}_{\lnot i _{}}\) be such that . Then there is an infinite path \(p' \in \varPi _{\sigma '}^{\omega }(v')\) such that \(\sigma \Vdash p'\) and . But then \(\sigma \Vdash \bar{p}\!p'\) and . \(\square \)
Finally this allows us to prove the result that governed stuttering bisimilar vertices are won by the same player.
Theorem 20
Governed stuttering bisimilarity strictly refines winner equivalence.
Proof
Let \(\mathcal {G}= (V, {\rightarrow }, \mathcal {P}, \varOmega )\) be a parity game, and let \(v, w \in V\) such that . Let be the governed stuttering bisimulation quotient of \(\mathcal {G}\), and let be such that . By transitivity of , also . Now suppose that player \( i _{}\) has a winning strategy \(\sigma \) from v. Then by Lemma 14, \( i _{}\) has a strategy \(\psi \) from \(\mathcal {C}\) such that for every play \(q \in \varPi _{\psi }^{}(\mathcal {C})\) there is a play \(p \in \varPi _{\sigma }^{}(v)\) such that . Because the priorities occurring infinitely often on such p and q are the same, \(\psi \) is also winning for \( i _{}\). If \(\lnot i _{}\) had a winning strategy \(\sigma '\) from w, then we could repeat this argument to construct a winning strategy for \(\lnot i _{}\) from \(\mathcal {C}\), but this would be contrary to the fact that parity games are determined. Therefore, w must also be won by player \( i _{}\). \(\square \)
Together with all results in this section (essentially comparing the (bi)simulations discussed in this paper) that precede this final result, we find that each of the equivalences described in this paper strictly refines winner equivalence, as also reflected by the lattice we illustrated in Sect. 4. Therefore, any pair of vertices, related by any of the equivalences described, is won by the same player.
Conclusion
Preorders and equivalences for parity games have been studied on a number of occasions, see [11, 13, 14, 22, 23, 34, 39]. A major motivation for some of these is that they provide the prospect of simplifying games prior to solving them. In this paper, we reconsidered several of the parity game relations previously defined by us, viz. (governed) bisimulation and (governed) stuttering bisimulation. More specifically, we gave detailed proofs showing that our relations are equivalences, they have unique parity game quotients and they approximate the winning regions of parity games. Furthermore, we showed that our coinductively defined equivalence relations admit gamebased definitions; the latter facilitated the comparison of our equivalences to the gamebased definitions of relations for parity games found in the literature. For the latter relations, we additionally gave coinductive definitions. Finally, we showed that, unlike e.g. delayed simulation or any of its biased versions, our equivalence relations give rise to unique quotients.
There are several natural continuations of this research. First, the experiments that were conducted in [14, 39] showed that parity games that could not be solved become solvable by preprocessing the games using an \(\mathcal {O}(m n)\) stuttering bisimulation minimisation algorithm or an \(\mathcal {O}(m n^2)\) governed stuttering bisimulation minimisation algorithm. While the average reduction achieved by governed stuttering bisimulation of parity games encoding typical practical decision problems exceeded 80%, see [39], the overall gain in speed otherwise was not significant. It would be worthwhile to establish whether this is still true when using the \(\mathcal {O}(m \log {n})\) stuttering equivalence minimisation algorithm of [31]. Moreover, it would be interesting to see whether the \(\mathcal {O}(m n^2)\) time complexity of governed stuttering bisimulation can be reduced using ideas from [31]. Similarly, we believe that our coinductive rephrasing of delayed simulation will help to devise a more efficient algorithm for computing it, using a partition refinement approach.
Finally, an interesting line of investigation is to see whether the incomparable notions of governed stuttering bisimulation and delayed simulation equivalence can be married. Given that we have established gamebased and coinductive definitions for both relations, defining such a relation now seems within reach. The resulting relation would be closer to winning equivalence and perhaps even shed light on ways to efficiently solve parity games in general.
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Appendix: Detailed proofs of propositions 7 and 8
Appendix: Detailed proofs of propositions 7 and 8
Before we address Propositions 7 and 8, we first state the definition of the variant function we will use in the proof of Proposition 7 and we state three lemmata that characterise properties of this variant function.
Definition 21
(Governed stuttering bisimulation game measure) We define a measure with respect to for a configuration \(((u_0,u_1),c)\) in the governed stuttering bisimulation game as follows:
where \(\textit{expel}(u_0,u_1)\) denotes the number of steps before \(\mathcal {P}(u_0)\)’s opponent is forced from and \(\textit{exit}(u_0,u_1,u_2)\) denotes the number of steps it takes for \(\mathcal {P}(u_0)\) to force play to from \(u_1\). Formally, we have:
where for \(U, T \subseteq V\) and \(v \in U\):
Measures are ordered lexicographically, i.e. \((m_0, m_1) < (n_0, n_1)\) iff \(m_0< n_0 \mathrel {\vee }(m_0 = n_0 \mathrel {\wedge }m_1 < n_1)\).
We first prove some basic properties for the function \( m \).
Lemma 15
For \(u,v \in V\), .
Proof
First, observe that (apart from \(c = \dagger \)), the conditions in the second and third clause of the definition of \( m \) are complementary. Furthermore observe that, for all \(u_0, u_1\) such that , we have \(\textit{expel}(u_0, u_1) > 0\) since . The result then immediately follows. \(\square \)
Lemma 16
Let \(U,T \subseteq V\), such that \(U \cap T = \emptyset \) and let \(u \in U\). For all players \( i _{}\), if then \( dist _{ i _{},U}(u,T) > \min \{ dist _{ i _{},U}(v,T) \mid u \rightarrow v \mathrel {\wedge }v \in U \cup T\}\).
Proof
Assume and let \(n = dist _{ i _{},U}(u,T)\). Hence \(u \in {_{U}}\! Attr ^{n}_{ i _{}}(T)\) and \(u \not \in {_{U}}\! Attr ^{n1}_{ i _{}}(T)\). Observe that \(n > 0\) since \(u \in U\) and \(U \cap T = \emptyset \). We proceed by a case distinction on \(\mathcal {P}(u)\).

\(\mathcal {P}(u) = i _{}\). Since n is such that \(u \notin {_{U}}\! Attr ^{n1}_{ i _{}}(T)\), we have . Let v be such, and observe that \( dist _{ i _{},U}(v,T) \le n  1 < n\). The result then follows immediately.

\(\mathcal {P}(u) \ne i _{}\). As n is such that \(u \notin {_{U}}\! Attr ^{n1}_{ i _{}}(T)\), we have . Hence . Again the result follows immediately. \(\square \)
We also prove the following stronger result in case u is owned by the opponent.
Lemma 17
Let \(U,T \subseteq V\), such that \(U \cap T = \emptyset \) and \(u \in U \cap V_{\lnot i _{}}\). Then implies \( dist _{ i _{},U}(u,T) > \max \{ dist _{ i _{},U}(v,T) \mid u \rightarrow v \mathrel {\wedge }v \in U \cup T \}\).
Proof
Let \(u \in U\cap V_{\lnot i _{}}\) such that . Suppose \(n = dist _{ i _{},U}(u,T)\). Then \(u \in {_{U}}\! Attr ^{n}_{ i _{}}(T)\) and \(u \not \in {_{U}}\! Attr ^{n1}_{ i _{}}(T)\). Since \(u \notin V_{ i _{}}\), , hence ; furthermore, such v are in \(U \cup T\), and the result follows immediately. \(\square \)
Proposition 7
For all \(v,w \in V\) if then .
Proof
We prove for all governed stuttering bisimilar vertices that there is a duplicatorwinning strategy in the governed stuttering bisimulation game from configuration \(((v,w),\checkmark )\).
We show this by constructing a duplicatorstrategy that moves between governed stuttering bisimilar vertices, and that makes sure that from every configuration ((v, w), c), within a finite number of steps another configuration \(((v',w'),\checkmark )\) is reached. As a consequence, the duplicatorstrategy is such that it passes through configurations with reward \(\checkmark \) infinitely often, hence the duplicatorstrategy is winning.
Formally, we preserve the invariant \(\Phi \) which is the conjunction of the following for configurations \(((u_0,u_1),c)\):

,

,

,

.
In addition, we prove that from every configuration \(((u_0, u_1),c)\), within a finite number of steps a configuration \(((u_0',u_1'),\checkmark )\) is reached by showing that \( m \) is a variant function. That is, if, in a round, we move from configuration \(((u_0,u_1),c)\) to configuration \(((u_0',u_1'),c')\) with \(c \ne \checkmark \) and \(c' \ne \checkmark \), then \( m (u_0,u_1,c) > m (u_0',u_1',c')\).
From these two observations, the result immediately follows. Note that initially we are in a configuration \(((v,w),\checkmark )\); hence \(\Phi \) is satisfied trivially. Suppose the game has reached a configuration \(((u_0, u_1),c)\) satisfying \(\Phi \). In step 1 of the round, Spoiler chooses to play from \((t_0,t_1)\), taken from \((u_0, u_1)\) or \((u_1,u_0)\). We remark that if Spoiler decides to play from \((u_1,u_0)\), then, regardless of step 2, any pending challenge or \(\dagger \) will be replaced by a \(\checkmark \) at the end of step 3. For this case, we therefore do not need to argue that \( m \) decreases.
We distinguish cases based on which player can force a divergence in the coinductive definition and consider duplicator’s options in step 2 and 3 of the round, and prove that duplicatorcan always arrive in a new configuration \(((t_0',t_1'),c')\) that satisfies \(\Phi \) and for which, if \(c' \ne \checkmark \) and \(c \ne \checkmark \), \( m (t_0,t_1,c) > m (t_0',t_1',c')\).

and . This case is trivial, as in that case exactly one (reachable) equivalence class exists.

. Since also . We distinguish cases based on the owners of the vertices.

. Spoiler plays \(t_0 \rightarrow w_0\).

Case there is some \(w_1 \in t_1^\bullet \) such that . Then duplicatorplays to such a \(w_1\). The new configuration is \(((w_0,w_1),\checkmark )\).

Case there is no \(w_1 \in t_1^\bullet \) such that . Then duplicatorplays to a \(w_1 \in t_1^\bullet \) for which with minimal \( m (t_0,w_1,(0,w_0))\); the existence of a follows from .
New configuration: if \(c \in \{\checkmark ,\dagger ,(0,w_0)\}\) and \(u_0 = t_0\) then the new configuration is \(((t_0,w_1),(0,w_0))\), and else it is \(((t_0,w_1),\checkmark )\).
Progress: we demonstrate \( m (t_0,w_1,(0,w_0)) < m (t_0,t_1,c)\) for \(c \in \{\dagger ,(0,w_0)\}\). If \(c = \dagger \) this follows from Lemma 15. In case \(c = (0,w_0)\), this follows from Lemmata 16 and 17.


. Spoiler plays \(t_1 \rightarrow w_1\). Since , all \(w_1 \in t_1^\bullet \) satisfy . The same holds for all \(w_0 \in t_0^\bullet \). duplicatorcan thus play arbitrary \(t_0 \rightarrow w_0\).
New configuration: \(((w_0, w_1),\checkmark )\)

, . Spoiler plays \(t_0 \rightarrow w_0\) and \(t_1 \rightarrow w_1\). Since , also . We distinguish two further cases.

Case .
New configuration: \(((w_0,w_1),\checkmark )\).

Case .
New configuration: if \(c \in \{\checkmark ,\dagger ,(0,w_0)\}\) and \(u_0 = t_0\) then the new configuration is \(((t_0, w_1),(0,w_0))\); else the new configuration is \(((t_0,w_1),\checkmark )\). Observe that .
Progress: we demonstrate \( m (t_0,w_1,(0,w_0)) < m (t_0,t_1,c)\) for \(c \in \{\dagger ,(0,w_0)\}\). If \(c = \dagger \) this follows from Lemma 15. In case \(c = (0,w_0)\), this follows from Lemmata 16 and 17.


, . duplicatorplays \(t_1 \rightarrow w_1\) such that and \(t_0 \rightarrow w_0\). Such a \(w_1\) exists because .
New configuration: \(((w_0, w_1), \checkmark )\).


and . So, as before, . This case is dual to the previous one.

and . We consider the owners of \(t_0\) and \(t_1\).

. Spoiler plays \(t_0 \rightarrow w_0\). We distinguish two cases.

Case there is some \(w_1 \in t_1^\bullet \) for which . duplicatorplays \(t_1 \rightarrow w_1\) such that .
New configuration: \(((w_0,w_1),\checkmark )\).

Case there is no \(w_1 \in t_1^\bullet \) for which .

Case . Then for all \(w_1 \in t_1^\bullet \), .
New configuration: \(((w_0,t_1),\dagger )\) if \(c \in \{\checkmark ,\dagger \}\) and \(u_0 = t_0\); else \(((w_0,t_1),\checkmark )\).
Progress: \( m (t_0,w_1,\dagger ) < m (t_0,t_1,\dagger )\) follows from Lemma 17.

Case . In this case duplicatorplays \(t_1 \rightarrow w_1\) such that and \( m (t_0,w_0,(0,w_0))\) is minimal.
New configuration: if \(c \in \{\checkmark ,\dagger ,(0,w_0)\}\) and \(u_0 = t_0\) then the new configuration is \(((t_0,w_1),(0,w_0))\) and else \(((t_0,w_1),\checkmark )\).
Progress: we must show \( m (t_0,w_1,(0,w_0)) < m (t_0,t_1,\dagger )\) for \(c \in \{\dagger ,(0,w_0)\}\). In case \(c = \dagger \) this follows from Lemma 15. In case \(c = (0,w_0)\), this follows from Lemmata 16 and 17.



. Spoiler plays \(t_1 \rightarrow w_1\).

Case there is some \(w_0 \in t_0^\bullet \) for which . duplicatorplays \(t_0 \rightarrow w_0\) such that .
New configuration: \(((w_0,w_1),\checkmark )\).

Case there is no \(w_0 \in t_0^\bullet \) for which .

Case . Then for all \(w_0 \in t_0^\bullet \), . duplicatorplays some arbitrary \(t_0 \rightarrow w_0\).
New configuration: \(((t_0,w_1),\dagger )\) if \(c \in \{\checkmark ,\dagger \}\) and \(u_1 = t_1\); else \(((t_0,w_1),\checkmark )\).
Progress: \( m (w_0,t_1,\dagger ) < m (t_0,t_1,\dagger )\) follows from Lemma 17.

Case . In this case duplicatorplays \(t_0 \rightarrow w_0\) such that and \( m (w_0,t_0,(1,w_1))\) is minimal.
New configuration: if \(c \in \{\checkmark ,\dagger ,(1,w_1)\}\) and \(u_1 = t_1\) the new configuration is \(((w_0,t_1),(1,w_1))\) and else it is \(((w_0,t_1),\checkmark )\).
Progress: we must show \( m (w_0,t_1,(1,w_1)) < m (t_0,t_1,c)\) for \(c \in \{\dagger ,(1,w_1)\}\). In case \(c = \dagger \) this follows from Lemma 15. In case \(c = (1,w_1)\), this follows from Lemmata 16 and 17.



. Spoiler plays \(t_0 \rightarrow w_0\) and \(t_1 \rightarrow w_1\). In case then either or . We distinguish three cases:

Case .
New configuration: \(((w_0, w_1),\checkmark )\).

Case and .
New configuration: \(((w_0, t_1), (1,w_1))\) if \(c \in \{\checkmark ,\dagger ,(1,w_1)\}\) and \(u_1 = t_1\); else \(((w_0,t_1),\checkmark )\).
Progress: we must show \( m (w_0,t_1,(1,w_1)) < m (t_0,t_1,c)\) for \(c \in \{\dagger ,(1,w_1)\}\). In case \(c = \dagger \) this follows from Lemma 15. In case \(c = (1,w_1)\) this follows from Lemmata 16 and 17.

Case and .
New configuration: \(((t_0,w_1),(0,w_0))\) if \(c \in \{\checkmark ,\dagger ,(0,w_0)\}\) and \(u_0 = t_0\); else \(((t_0,w_1),\checkmark )\).
Progress: we must show \( m (t_0,w_1,(0,w_0)) < m (t_0,t_1,c)\) for \(c \in \{\dagger ,(0,w_0)\}\). In case \(c = \dagger \) this follows from Lemma 15. In case \(c = (0,w_0)\) this follows from Lemmata 16 and 17.


.

Case there are \(w_0 \in t_0^\bullet \) and \(w_1 \in t_1^\bullet \) such that . Then duplicatorplays to such \(w_0\) and \(w_1\).
New configuration: \(((w_0,w_1),\checkmark )\).

Case there are no \(w_0 \in t_0^\bullet \) and \(w_1 \in t_1^\bullet \) such that .

case there is some \(w_0 \in t_0^\bullet \) such that . Then duplicatorplays to \(w_0\) that is such while minimising \( m (w_0,t_1,\dagger )\).
New configuration: \(((w_0,t_1),\dagger )\) if \(u_1 = t_1\); else it is \(((w_0,t_1),\checkmark )\).
Progress: we first show that \(u_1 = t_1\) implies \(c \in \{\dagger ,\checkmark \}\). Towards a contradiction, assume \(c = (0,t)\) for some t. By our invariant, this implies . Since \((u_0,u_1) \in \{(t_0,t_1),(t_1,t_0)\}\) and \(u_1 = t_1\) we have \(u_0 = t_0\). But then both and . Contradiction. Towards another contradiction, assume \(c = (1,t)\) for some t. By our invariant, this implies . This contradicts \(u_1 = t_1\) since . It therefore suffices to show \( m (w_0,t_1,\dagger ) < m (t_0,t_1,\dagger )\). This follows from the fact that we minimised \( m (w_0,t_1,\dagger )\) and Lemmata 16 and 17.

case there is some \(w_1 \in t_1^\bullet \) such that . Then duplicatorplays to \(w_1\) that is such while minimising \( m (t_0,w_1,\dagger )\).
New configuration: \(((t_0,w_1),\dagger )\) if \(u_0 = t_0\); else it is \(((t_0,w_1),\checkmark )\).
Progress: using arguments, similar to those in the previous case, it follows that \(c \in \{\dagger ,\checkmark \}\). It therefore suffices to show \( m (t_0,w_1,\dagger ) < m (t_0,t_1,\dagger )\). This follows from the fact that we minimised \( m (t_0,w_1,\dagger )\) and Lemmata 16 and 17. \(\square \)



We next focus on proving that every pair of vertices related through the governed stuttering bisimulation game are in fact governed stuttering bisimilar.
Proposition 8
For all \(v,w \in V\) if then .
Proof
We prove the contrapositive of the statement, i.e. for all \(v,w \in V\), if , then also \(v \not \equiv _{ g,st }w\). Let . By Corollary 2, then also \((v,w) \notin \nu \mathcal {F}\). By the KnasterTarskiKleene fixpoint approximation theorem, we thus have \((v,w) \notin \bigcap \limits _{k \ge 1} \mathcal {F}^k(V \times V)\). Let \(R^k\) denote the relation \(\mathcal {F}^k(V \times V)\); i.e., \(R^k\) is the relation obtained by applying the operator \(\mathcal {F}\) ktimes. Note that because of monotonicity, \(R^k = \bigcap _{l \le k} R^l\). We next prove, using induction, that for all \(k \ge 1\):

Base case \(k = 1\). Observe that \(R^1 = \{ (v,w) \in V \times V \mid \varOmega (v) = \varOmega (w) \}\). Spoiler wins the governed stuttering bisimulation game for all configurations \(((u_0,u_1),c)\) satisfying \((u_0,u_1) \notin R^1\): all plays starting in such a configuration trivially violate duplicator’s winning condition.

Inductive step. Assume that the statement holds for some \(k \ge 1\). Pick an arbitrary position \((u_0,u_1)\) for which \((u_0,u_1) \notin R^{k+1}\) and let c be an arbitrary challenge/reward. We must show that Spoiler wins the governed stuttering bisimulation game for these. Recall that we have \(R^{k+1} \subseteq R^k\).
If \((u_0,u_1) \notin R^{k}\), then by the induction hypothesis, Spoiler wins the governed stuttering bisimulation game from the configuration \(((u_0,u_1), c)\). Observe that by definition of \(\mathcal {F}\), we have for all \((v,w) \in R^k \setminus R^{k+1}\) that there are and \(\mathcal {U},\mathcal {T} \subseteq V_{/R^{k}}\) for which
Let \( i _{}, \mathcal {U}, \mathcal {T}\) be such that (\(\star \)). We focus on the case ; the case that is fully dual. Assume that and not ; the case in which not but is symmetric. Note that we can assume that \(\mathcal {T} \cap \mathcal {U} = \emptyset \), as iff for any \(\mathcal {U},\mathcal {T}\). We may therefore also simplify \(\mathcal {U} \setminus \mathcal {T}\) to \(\mathcal {U}\).
Let \(\sigma \in \mathbb {S}_{ i _{}}\) be the (memoryless) strategy underlying . Using \(\sigma \), we construct a winning strategy for Spoiler for configuration ((v, w), c). We first show that Spoiler can invariantly move between configurations \(((t_0,t_1),c)\) that satisfy the following property \(\Phi \):
Let \(((t_0,t_1),c)\) be a configuration for which \(\Phi \) holds. For all such configurations Spoiler ’s move in step 1 of a round is to play from \((t_0,t_1)\); i.e. Spoiler does not switch positions. We distinguish three main cases, showing that duplicatorhas no other option than to choose a new configuration that satisfies \(\Phi \).

1.
Case \(c \in \{ \dagger ,\checkmark \}\). We furthermore distinguish cases based on the players of \(t_0\) and \(t_1\).

Case . Since , Spoiler proposes to move from \(t_0\) to \(\sigma (t_0)\). duplicatorproposes \(u_1 \in t_1^\bullet \). Observe that not . duplicatorthen can propose to continue in: \(((\sigma (t_0),u_1),\checkmark )\), \(((t_0,u_1), (0,\sigma (t_0)))\), or \(((\sigma (t_0),t_1), \dagger )\). Clearly, all new configurations satisfy \(\Phi \).

Case . Spoiler proposes to move from \(t_1\) to \(u_1\) such that not . Such \(u_1\) exists. duplicatorproposes \(u_0 \in t_0^\bullet \). Observe that \(t_0 \mathop {\,\!_{\sigma }\smash \rightarrow } u_0\). duplicatorproposes to continue in: \(((u_0,u_1),\checkmark )\), \(((t_0,u_1),\dagger )\), or \(((u_0,t_1),(1,u_1))\). All new configurations satisfy \(\Phi \).

Case , . Since Spoiler proposes to move from \(t_0\) to \(\sigma (t_0)\) and from \(t_1\) to \(u_1\) such that not . Note that such \(u_1\) exists. duplicatorproposes to continue in: \(( (\sigma (t_0),u_1),\checkmark )\), \(( (t_0,u_1), (0,\sigma (t_0)))\), or \(( (\sigma (t_0),t_1),(1,u_1))\). Again, all new configurations satisfy \(\Phi \).

Case , . duplicatorproposes to move from \(t_1\) to \(u_1\) and from \(t_0\) to \(u_0\). Since , we have \(t_0 \mathop {\,\!_{\sigma }\smash \rightarrow } u_0\) and because of \(\Phi \), we have not . duplicatorthen proposes to continue in: \(((u_0,u_1), \checkmark )\), \(((t_0,u_1), \dagger )\), or \(((u_0,t_1), \dagger )\). All new configurations satisfy \(\Phi \).


2.
Case \(c = (0,t)\). Because of \(\Phi \), we have and . Then Spoiler plays from configuration \((t_0,t_1)\). We furthermore distinguish cases based on the owner of \(t_1\).

Case . Spoiler proposes to move from \(t_0\) to t. duplicatorproposes \(u_1 \in t_1^\bullet \). Observe that not . duplicatorproposes to continue in on of the following configurations: \(((t,u_1),\checkmark )\), \(((t_0,u_1),(0,t))\), or \(((t,t_1),\dagger )\)

Case . Spoiler proposes to move from \(t_0\) to t and from \(t_1\) to \(u_1\) such that not . Such \(u_1\) exists. duplicatorproposes to continue in: \(((t,u_1),\checkmark )\), \(((t_0,u_1),(0,t))\), or \(((t,t_1), \checkmark )\)
In both cases, the new rounds satisfy \(\Phi \).


3.
Case \(c = (1,t)\). Because of \(\Phi \), we have not and . Then Spoiler plays from configuration \((t_0,t_1)\). We furthermore distinguish cases based on the owner of \(t_0\).

Case . Spoiler proposes to move from \(t_0\) to \(\sigma (t_0)\) and from \(t_1\) to t. duplicatorproposes to continue in: \(((\sigma (t_0),t),\checkmark )\), \(((t_0,t),\checkmark )\), or \(((\sigma (t_0),t_1),(1,t))\).

Case . Spoiler proposes to move from \(t_1\) to t. duplicatorproposes \(u_0 \in t_0^\bullet \). Observe that \(t_0 \mathop {\,\!_{\sigma }\smash \rightarrow } u_0\). duplicatorproposes to continue in: \(((u_0,t),\checkmark )\), \(((t_0,t),\dagger )\), or \(((u_0,t_1),(1,t))\).

We next observe that for any \((t_0,t_1)\) for which \(t_0, t_1\) meet the premiss of \(\Phi \), but not the conclusion, Spoiler can, in a single round, move to a configuration that either does not meet \(\Phi \)’s premiss or to one that meets \(\Phi \)’s conclusion. More specifically, suppose that \([t_0]_{R^k} = [t_1]_{R^k}\) but one of the following holds:

1.
but not ;

2.
\(c = (0,t)\) implies or not \(t_0 \mathop {\,\!_{\sigma }\smash \rightarrow } t\);

3.
\(c = (1,t)\) implies , \(t \notin t_1^\bullet \), or .
Whenever we are in case 1, Spoiler switches positions in step 1 of a round and follows the strategy outlined above. Whenever we are in case 2 or 3, Spoiler drops challenge c in step 1 and plays as if \(c \in \{\dagger ,\checkmark \}\). In all three cases, duplicatoris rewarded a \(\checkmark \) as the new challenge at the end of the round and \(\Phi \) holds trivially.
Summarising, we find that for configurations \(((t_0,t_1),c)\) for which both (\(\star \)) and \(\Phi \) hold, Spoiler can move to another configuration that either meets \(\Phi \) or is such that \(\Phi \)’s premiss is violated. For configurations \(((t_0,t_1),c)\) for which (\(\star \)) but not \(\Phi \) holds, Spoiler can move in a single round to a configuration for which she can henceforth maintain \(\Phi \) as an invariant or for which \(\Phi \)’s premiss is violated.
We finally argue that when Spoiler plays according to the above strategy, she wins all plays. Observe that we only need to show this for all infinite plays that pass only through positions (u, t) for which \((u,t) \in R^k\); for those plays that at some point pass along a position \((u,t) \notin R^k\), our induction hypothesis yields a winning strategy for Spoiler .
Let \((v_0,w_0)\ (v_1,w_1)\ (v_2,w_2) \dots \) be an infinite sequence of positions on an infinite play \(\pi \) that is allowed by Spoiler ’s strategy, such that for all l, \([v_l]_{R^k} = [w_l]_{R^k}\). Towards a contradiction, assume that duplicatorwins \(\pi \). Observe that \([v_l]_{R^k} = [w_l]_{R^k}\) implies that \(\varOmega (v_l) = \varOmega (w_l)\) for all \((v_l,w_l) \in R^k \setminus R^{k+1}\); therefore we can only arrive at a contradiction by showing that duplicatorearns a finite number of \(\checkmark \) rewards along \(\pi \). By invariant \(\Phi \), for all positions \((v_l,w_l)\), for \(l \ge 1\), we have . Let \(\delta (v_l,w_l)\) denote the length of the longest path from \(v_l\) to reach \(\mathcal {T}\) when playing according to \(\sigma \). Note that \(\delta \) is finite and decreases along the positions in \(\pi \), but never reaches 0, as all vertices remain in \(\mathcal {U}\). This means that for some m, we have \(\delta (v_m,w_m) = \delta (v_n,w_n)\) for all \(n \ge m\). Fix this m. Moreover, there must be some u such that:
But this means that, once Spoiler ’s strategy reaches the position containing u, all remaining \(\checkmark \)’s earned by duplicatormust be due to Spoiler switching positions or discarding a challenge in step 1 of each new round. As we explained, Spoiler switches positions and/or drops a challenge only in the first round when starting in a configuration that does not satisfy \(\Phi \); she never does so afterwards. Therefore, duplicatorearns no \(\checkmark \) rewards when \(\pi \) reaches a configuration containing a position with u. But then duplicatorearns only a finite number of \(\checkmark \) rewards along \(\pi \), contradicting the assumption that duplicatorwins \(\pi \). Therefore, Spoiler has a strategy to win from any configuration \(((u_0,u_1),c)\) for which \((u_0,u_1) \notin R^k\). \(\square \)
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Cranen, S., Keiren, J.J.A. & Willemse, T.A.C. Parity game reductions. Acta Informatica 55, 401–444 (2018). https://doi.org/10.1007/s002360170301x
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DOI: https://doi.org/10.1007/s002360170301x