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Acta Informatica

, Volume 55, Issue 5, pp 401–444 | Cite as

Parity game reductions

  • Sjoerd Cranen
  • Jeroen J. A. KeirenEmail author
  • Tim A. C. Willemse
Open Access
Original Article

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 game-based 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.

1 Introduction

We study preorders and equivalences defined on parity games. Such games are turn-based 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 second-order 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 speed-up 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 Groote-Vaandrager 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 speed-ups 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 idempotence-identifying 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 side-note, 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 odd-biased versions. A complicating factor is the fact that these relations have only game-based 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 game-based definitions. Likewise, we give game-based 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 odd-biased) 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 non-trivial, 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 first-order logic that is definable in modal logic [4], and the fragment of monadic second-order logic that is captured by the modal \(\mu \)-calculus. Bisimulation and simulation-like relations, called consistent correlations [54] and consistent consequence [25] for PBESs, a fixpoint-logic based framework which is closely related to parity games, were imperative to prove the soundness of the syntax-based 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 well-founded bisimulation, a notion equivalent to stuttering bisimulation which solely relies on local reasoning by introducing a well-founded order into the relation. Still, at its basis, well-founded 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 game-based definitions for weak behavioural equivalences. Yin et al. [55] describe branching bisimulation games for normed process algebra. A game-based characterisation of divergence-blind 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 game-based characterisation of branching bisimulation (with explicit divergence) that does not need such a preprocessing step. Our game-based characterisation of governed stuttering bisimulation is based on the same ideas, but requires extensions to the two-player 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.

2 Parity games

A parity game is a two-player graph game, played by two players \(\textit{even}\,\)and \(\textit{odd\,}\)  (denoted Open image in new window and Open image in new window ) 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,

  • Open image in new window 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., Open image in new window and Open image in new window . The function \(\mathcal {P}\) induces a partitioning of V into a set of vertices Open image in new window owned by player \(\textit{even}\,\)and a set of vertices Open image in new window owned by player \(\textit{odd\,}\); we use \(\mathcal {P}\) and Open image in new window 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 thanm 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 Open image in new window , 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 Open image in new window wins, otherwise player Open image in new window 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 \(j < n-1\) it is the case that if \(\sigma \) is defined for Open image in new window , then Open image in new window . The definition of allows is extended to infinite paths in the obvious manner. We generalise the definition of Open image in new window to paths allowed by a strategy \(\sigma \); formally, we define:The definition for infinite paths is generalised in the same way and denoted Open image in new window . By Open image in new window we denote Open image in new window , 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 well-known 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 Open image in new window such that winning strategies Open image in new window from Open image in new window and Open image in new window from Open image in new window exist. Furthermore, for Open image in new window , 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 Open image in new window of a parity game.

3 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 single-step relation \(\mathop {\,\!_{\sigma }\smash \rightarrow } \subseteq \rightarrow \) that contains only those edges allowed by \(\sigma \):
$$\begin{aligned} \mathop {\,\!_{\sigma }\smash \rightarrow } \mathop {=}\limits ^{\varDelta }\{ (v, u) \in \rightarrow \mid \text { if } \sigma (v) \text { is defined then } u = \sigma (v) \} \end{aligned}$$
In other words, \(\mathop {\,\!_{\sigma }\smash \rightarrow }\) contains those edges (vu) 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 Open image in new window , 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 Open image in new window 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 Open image in new window , 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, Open image in new window 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 Open image in new window .

Proof

Observe that the implication from left to right holds by definition. So assume that for some \(\sigma \in \mathbb {S}_{ i _{}}^*\), we have Open image in new window . Note that Open image in new window iff Open image in new window . 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 Open image in new window and the priorities of all other vertices to 1 iff Open image in new window .

Since Open image in new window , 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 Open image in new window in the resulting graph, but then also Open image in new window in the original graph, and hence also the required Open image in new window . \(\square \)
The complement of these relations is denoted by a slashed version of the corresponding arrow, e.g., Open image in new window can be written Open image in new window . 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, thenAll 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 Open image in new window 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 Open image in new window 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 Open image in new window iff \(v \in {_{U}}\! Attr ^{}_{ i _{}}(T)\).

Proof

We first introduce some additional notation. Let Open image in new window denote that there is a \(\sigma \in \mathbb {S}_{ i _{}}\) such that for all Open image in new window , there is some \(m \le n\) such that \({p_{}}[m] \in T\), and \({p_{}}[j] \in U\) for all \(j < m\). Note that Open image in new window iff there is some n such that Open image in new window . Using induction on n, it follows that Open image in new window 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.

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.

Proof

By Lemma 2, Open image in new window 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 Open image in new window we find \(v \in {_{U}}\! Attr ^{}_{ i _{}}(T)\). Combined, we find \(v \in {_{U}}\! Attr ^{}_{ i _{}}(T')\) which, by Lemma 2, yields the desired Open image in new window . \(\square \)

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 Open image in new window implies \(V_{ i _{}} \cap S \ne \emptyset \) or for some \(u \in S\), \(u^\bullet \subseteq T\).

Proof

Assume Open image in new window 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 Open image in new window . 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 \(v \in U\), Open image in new window implies Open image in new window 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 Open image in new window , 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 \({p_{}}[j-1]^\bullet \subseteq U \cup T'\), also \({p_{}}[j] \in T'\) and therefore Open image in new window . Thus Open image in new window . \(\square \)

4 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 game-based characterisations, we first introduce relations and (bi)simulation games.

4.1 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 \(U \subseteq V\), Open image in new window , and Open image in new window .

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 \(\{ [v]_{\mathrel {R^{}}} \mid v \in V \}\).

4.2 (Bi)simulation games

Some of the relations we consider in this paper are defined using two-player games. The games we consider are, essentially, Ehrenfeucht-Fraïssé games. For an extensive review of their applications in computer science, the interested reader is referred to [52].

Like parity games, the two-player games we use to characterise relations are instances of two-player, infinite-duration 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 two-player 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 in-depth treatment of the underlying theory.

4.3 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
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 finer-than relation between the equivalences.
Fig. 1

Lattice of equivalences for parity games. The numbers on the edges refer to the legend shown to the right, which in turn refers to the theorems that witness the existence of the edge

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.

5 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 \(\mathcal {G}\!\!=\!\! (V,\rightarrow ,\varOmega ,\mathcal {P})\).

5.1 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 game-based definition as given in [25].

The game is played on pairs of vertices (vw), 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 (vw) with Open image in new window , 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 (vw), 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. 1.

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

     
  2. 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 (vw) in the direct simulation game.
Table 1

Allowed moves in a (bi)simulation game

Note that when playing on v the allowed moves are \(v \rightarrow v'\) where \(v \rightarrow v'\) is an edge in the parity game, and likewise for w

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.

Fig. 2

Parity game with \(v_0 \sqsubseteq _{ d }v_1\), \(v_0 \sqsubseteq _{ d }v_2\), \(v_2 \sqsubseteq _{ d }v_0\), and for all \(v_i\), \(v_i \sqsubseteq _{ d }v_i\)

Direct simulation is a preorder: reflexivity is easily seen to hold (duplicatorcan follow a copy-cat 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 Open image in new window , 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 Open image in new window 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
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 game-based 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 \(v_0 \not \le _{sd}v_1\).

5.2 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 (vw) is the current position, and k the current obligation, a round of the game proceeds as follows:
  1. 1.

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

     
  2. 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 (vw) 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 ij. Observe that, with respect to direct simulation, vertices cannot be related to each other.

Fig. 3

Parity game in which all vertices are delayed simulation equivalent

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 more-or-less 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 ‘non-local’ 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 well-founded bisimulation [45]. Well-founded bisimulation is a relation which is equivalent to stuttering equivalence, but which permits local reasoning by introducing a well-foundedness criterion. We use a similar well-foundedness 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 well-foundedness 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

(Well-founded delayed simulation) A relation \({\mathrel {R^{}}} \subseteq V \times K\times V\) is a well-founded delayed simulation iff there is a well-founded 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:Vertex v is well-founded delayed simulated by w, denoted \(v \le _{de}w\), iff there exists a well-founded 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 game-based definition. We first show that vertices that are related by well-founded 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 well-founded 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 well-foundedness 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 well-founded 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 (vw) 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.
  • Open image in new window . 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 well-founded delayed simulation, there is \(w \rightarrow w'\) such that \(v' \mathrel {R^{\gamma ^{}(v', w', k_{})}} w'\); duplicatormatches with this \(w'\).

  • Open image in new window . 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 well-founded delayed simulation guarantees that for all \(v \rightarrow v'\) and \(w \rightarrow w'\), \(v' \mathrel {R^{\gamma ^{}(v', w', k_{})}} w'\).

  • Open image in new window . duplicatorplays twice in the delayed simulation game. According to the well-founded delayed simulation, there exist \(v \rightarrow v'\) and \(w \rightarrow w'\) such that \(v' \mathrel {R^{\gamma ^{}(v', w', k_{})}} w'\). duplicatorplays such moves.

  • Open image in new window . In the delayed simulation game, Spoiler is to move first, say \(w \rightarrow w'\). From the well-founded 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 duplicator-strategy 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 well-founded 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 well-founded 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 well-founded order on \(V \times V \times K\) for those \((v,w,k_{})\) for which duplicatorwins position (vw) 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 (vw) 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 duplicator-winning, 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 well-founded 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 well-founded order.

Corollary 1

In the delayed simulation game, for every position (vw) 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 well-founded 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 (vw) with obligation \(k_{}\) is a well-founded delayed simulation. This result follows from two observations. First, a well-founded order on configurations won by duplicatorexists, according to Lemma 8. Second, that the transfer condition is satisfied follows from the duplicator-winning strategy in the delayed simulation game, using a straightforward case distinction the players of related vertices. Furthermore, in every step, the well-founded order on configurations won by duplicatordecreases in every step according to Corollary 1. \(\square \)

The following theorem, stating that delayed simulation and well-founded 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.

5.2.1 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.

Even-biased delayed simulation. The even-biased delayed simulation game, and its coinductive variant, are identical to their delayed simulation and well-founded 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:
$$\begin{aligned} \gamma ^{e}(n,m,k_{})&= {\left\{ \begin{array}{ll} k_{}&{} \text {if } m \preccurlyeq n, n \text { odd}, n \le k_{}, \\ &{} \text { and } (m \text { odd or } k_{}< m) \\ \gamma ^{}(n,m,k_{}) &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
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 even-biased delayed simulated by w, denoted \(v \sqsubseteq _{ de }^{e}w\) iff duplicatorhas a winning strategy from (vw) with obligation \(\gamma ^{e}(v, w, \checkmark )\) in the even-biased delayed simulation game.

Likewise, we obtain well-founded, even-biased delayed simulation by replacing all occurrences of \(\gamma ^{}\) by \(\gamma ^{e}\) in Definition 10. Vertex v is well-founded, even-biased delayed simulated by w, denoted \(v \le _{de}^{e}w\), iff there exists a well-founded, even-biased delayed simulation preorder \(\mathrel {R^{}}\) such that \(v \mathrel {R^{\gamma ^{e}(v, w, \checkmark )}} w\).

Odd-biased delayed simulation. The odd-biased delayed simulation is defined in a similar way as the even-biased 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:
$$\begin{aligned} \gamma ^{o}(n,m,k_{})&= {\left\{ \begin{array}{ll} k_{}&{} \text {if } m \preccurlyeq n, m \text { even}, m \le k_{}, \\ &{} \text { and } (n \text { even or } k_{}< n) \\ \gamma ^{}(n,m,k_{}) &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
The game-based and coinductive definitions are analogous to the even-biased version.

6 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.

6.1 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 Open image in new window , 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, Open image in new window , and furthermore, Open image in new window and Open image in new window . 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 Open image in new window , 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.
Fig. 4

Parity game in which \(v_2\) and \(v_3\) are governed bisimilar. Vertices \(v_0\), \(v_1\) and \(v_6\) are direct simulation equivalent. Vertices \(v_0\) and \(v_6\) are governed bisimilar but not strong direct simulation equivalent. Vertices \(v_0\) and \(v_1\) are strong direct simulation equivalent, but not governed bisimilar

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 Open image in new window 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

Open image in new window 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 (vw) proceeds as follows:
  1. 1.

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

     
  2. 2.

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

     
  3. 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 (vw) 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 Open image in new window 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 Open image in new window and defined as Open image in new window iff there is some strong bisimulation relation that relates v and w, is an equivalence relation.

6.2 Governed stuttering bisimulation

The (bi)simulation games discussed so far all have in common that the game-play proceeds in ‘lock-step’: 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 left-most 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 right-most vertices with priority 0.
Fig. 5

Equal priorities are related by Open image in new window Open image in new window Neither player can force play to visit only 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
  1. (a)

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

     
  2. (b)

    \(v \rightarrow \mathcal {C}\) implies Open image in new window .

     
  3. (c)

    \(v \mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) implies \(w\mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) for Open image in new window .

     
Vertices v and w are governed stuttering bisimilar, denoted Open image in new window , 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 Open image in new window nor Open image in new window . 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 Open image in new window . 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 Open image in new window 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 Open image in new window . Then for all vertices u related to v, any transition leaving the equivalence class is also sure to end up in Open image in new window .

Lemma 10

Let R be a governed stuttering bisimulation. Let \(\mathcal {U}\subseteq {V}_{/R} \setminus \{[v]_{R}\}\). If Open image in new window , 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 Open image in new window , by Definition 13, we have Open image in new window . But Open image in new window and \(\mathcal {C}\notin \mathcal {U}\) so Open image in new window . 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:
  1. (a)

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

     
  2. (b)
     
  3. (c)

    \(v \mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) iff \(w\mathop {\,\!_{ i _{}}\smash \mapsto ^{}_{\mathrel {R^{}}}}\) for all Open image in new window .

     

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 Open image in new window 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.

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 Open image in new window 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 side-result 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 Open image in new window is in general neither equivalent to saying that Open image in new window and Open image in new window , nor Open image in new window or Open image in new window .

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:
  1. (a)

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

     
  2. (b)
     

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 if-part and an only-if-part.
\(\Leftarrow \)

The second condition from Proposition 4 is equivalent to the second condition above if we let \(\mathcal {U}\) range only over singleton sets (if Open image in new window , 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 Open image in new window 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 Open image in new window .

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 Open image in new window to allow us to write Open image in new window instead of Open image in new window .

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:
  1. (a)

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

     
  2. (b)

    Open image in new window iff Open image in new window for all Open image in new window 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 if-case and an only-if-part.
\(\Leftarrow \)

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 Open image in new window ; under this assumption we will prove that Open image in new window . The proof for the implication in the other direction is completely symmetric. Let \(\sigma \in \mathbb {S}_{ i _{}}\) be such that Open image in new window 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 Open image in new window but Open image in new window . 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, Open image in new window and by Proposition 5 also Open image in new window , which proves Open image in new window . As the induction hypothesis, assume that if \(u \mathrel {R^{}}u'\), Open image in new window and the longest \(\sigma \)-prefix of u is shorter than the longest \(\sigma \)-prefix of v, then Open image in new window . 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 Open image in new window . Furthermore, Open image in new window . By Proposition 5, Open image in new window . Now consider an arbitrary Open image in new window . 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 Open image in new window for such u, we can use the induction hypothesis to derive that Open image in new window . The above in particular implies two facts: Open image in new window , and Open image in new window for all Open image in new window . Using these, we can now apply Lemma 4 to conclude Open image in new window . \(\square \)

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

Open image in new window 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 Open image in new window for some Open image in new window 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 Open image in new window . 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 Open image in new window . Because Open image in new window , also Open image in new window , and by Proposition 6 then Open image in new window , which is equivalent to Open image in new window . By a simple inductive argument we now arrive at Open image in new window , which is equivalent to Open image in new window . \(\square \)

As a side-result 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 game-based 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 ‘roll-back’ 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 (ku) 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 ((vw), c) proceeds as follows:
  1. 1.

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

     
  2. 2.

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

     
  3. 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 (vw) 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 (vw).

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 Open image in new window . 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 vw 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 Open image in new window 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 Open image in new window 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 non-governed 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 Open image in new window , 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 Open image in new window .

Proof

We essentially prove the contrapositive of the statement, i.e. for all \(v,w \in V\), if Open image in new window , then also \(v \not \equiv _{ g,st }w\). Let Open image in new window . By Corollary 2, then also \((v,w) \notin \nu \mathcal {F}\). By the Knaster-Tarski-Kleene 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 Open image in new window 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 Open image in new window and defined as Open image in new window iff there is some stuttering bisimulation relation that relates v and w, is an equivalence relation.

7 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 \(\bar{v}_1 \sim \bar{v}_2\).

7.1 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 non-maximal 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 non-empty 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 non-empty 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 \(\mathcal {C},\mathcal {C}' \in {V}_{/\equiv _{ d }}\):

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:

Proof

We prove the first and the third statement; the proof for the second statement is analogous to that of the first.
  • Suppose Open image in new window . 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 Open image in new window and Open image in new window . 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 well-defined:

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 \(\mathcal {C},\mathcal {C}' \in {V}_{/\equiv _{ d }}\):

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 Open image in new window , 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 Open image in new window , 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 Open image in new window whereas Open image in new window for some \(\mathcal {C}\). Then Open image in new window 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 Open image in new window , 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 Open image in new window . 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 Open image in new window , 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 Open image in new window . 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 Open image in new window . The case where Open image in new window 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 (up-to isomorphism) smallest parity game direct simulation equivalent to \(\mathcal {G}\).

7.2 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 Open image in new window , where, for Open image in new window :

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 Open image in new window be its governed bisimulation quotient. Then Open image in new window .

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 Open image in new window iff the two structures Open image in new window and Open image in new window are isomorphic.

Corollary 6

The governed bisimulation quotient of \(\mathcal {G}\) is a unique (up-to 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 Open image in new window , where, for Open image in new window :

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 Open image in new window be its governed stuttering bisimulation quotient. Then Open image in new window .

Proof

Follows from the fact that relation Open image in new window , 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 Open image in new window iff the two structures Open image in new window and Open image in new window are isomorphic.

Corollary 7

The governed stuttering bisimulation quotient of \(\mathcal {G}\) is a unique (up-to isomorphism) smallest parity game that is governed stuttering bisimilar to \(\mathcal {G}\).

8 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 right-hand 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. \(\square \)

Fig. 6

Parity game which is minimal with respect to strong bisimilarity. Vertices \(v_2\) and \(v_3\) are governed bisimilar

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 odd-biased delayed simulation equivalence. Even- and odd-biased 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 right-hand side of the lattice we presented in Sect. 4.

We next focus on the left-hand 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 Open image in new window iff v is equivalent to the last vertex in q, and Open image in new window iff Open image in new window . 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 Open image in new window be its quotient. Let \(v \in V\), and Open image in new window 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 Open image in new window there is a Open image in new window such that Open image in new window .

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 Open image in new window such that if Open image in new window for some \(p \in \varPi _{\sigma }^{}(v)\), then:Let Open image in new window be such that Open image in new window for Open image in new window 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 Open image in new window . 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, Open image in new window . Consider the path Open image in new window for which Open image in new window , and assume that p is of the form Open image in new window . Since Open image in new window , also Open image in new window and hence Open image in new window . Then by Lemma 3, Open image in new window . Let \(\sigma ' \in \mathbb {S}_{\lnot i _{}}\) be such that Open image in new window and consider the unique path Open image in new window such that Open image in new window , Open image in new window . Then Open image in new window is such that Open image in new window , \(pr \mathop {\,\!_{\sigma }\smash \rightarrow } v'\) and Open image in new window . Hence, \(\mathsf {next}(q) \ne \bot \).

Next, we show that we can set Open image in new window . Since \(\mathsf {next}(q) \ne \bot \), there must be some Open image in new window such that \(v'' = \mathsf {next}(q)\), Open image in new window and Open image in new window . Also, Open image in new window , since Open image in new window   and Open image in new window . As \(\mathcal {C}'\) is a vertex in a quotient graph, this implies Open image in new window . Hence, we can set Open image in new window .
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 nFor \(n=0\), this is trivial, because Open image in new window . For \(n = m + 1\), assume as the induction hypothesis that Open image in new window . Let \(q \in \varPi _{\psi }^{n}(\mathcal {C})\) and let Open image in new window and \(\bar{q} \in \varPi _{\psi }^{m}(\mathcal {C})\) such that Open image in new window and Open image in new window . 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 Open image in new window such that Open image in new window , therefore \(\mathcal {C}'' = \mathcal {C}'\) if \(\mathsf {div}(\bar{q})\) and \(\mathcal {C}' \rightarrow ' \mathcal {C}'\), and otherwise Open image in new window . If \(\mathcal {C}' = \mathcal {C}''\), then \(\mathsf {div}(\bar{q})\), so there must be some Open image in new window such that Open image in new window and therefore also some Open image in new window such that Open image in new window . If Open image in new window , there must be some Open image in new window such that Open image in new window and Open image in new window and Open image in new window . By definition, Open image in new window for such p.

Case \(\mathcal {P}'(\mathcal {C}') \ne i _{}.\)

From the induction hypothesis, obtain a Open image in new window such that Open image in new window .

Without loss of generality we may assume that \(\bar{p}\) is finite. Note that Open image in new window .

We distinguish two cases.
Finally, we prove that for all Open image in new window there is a Open image in new window such that Open image in new window . 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 \('\).

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 Open image in new window . Let Open image in new window be the governed stuttering bisimulation quotient of \(\mathcal {G}\), and let Open image in new window be such that Open image in new window . By transitivity of Open image in new window , also Open image in new window . 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 Open image in new window . 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.

9 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 game-based definitions; the latter facilitated the comparison of our equivalences to the game-based 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 game-based 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.

Notes

Acknowledgements

The authors would like to thank the reviewers for their comments and suggestions, which helped to improve this paper.

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Authors and Affiliations

  1. 1.Department of Computer Science and MathematicsEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Faculty of Management, Science and TechnologyOpen University of the NetherlandsHeerlenThe Netherlands
  3. 3.Institute for Computing and Information SciencesRadboud UniversityNijmegenThe Netherlands

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