Witnesses for Fixpoint Games on Lattices

König, Barbara; Messing, Karla

doi:10.4230/LIPIcs.ICALP.2026.182

Witnesses for Fixpoint Games on Lattices

Barbara König

Universität Duisburg-Essen, Germany Karla Messing

Universität Duisburg-Essen, Germany

Abstract

We construct witnesses that can be used to derive strategies in fixpoint games and provide proof that the least fixpoint of a function is either above or not below some given bound. We rely on a lattice-theoretical approach, including a Galois connection that connects a lattice representing the “logic universe”, where the witness lives, with another lattice representing the “behaviour universe”, over which the function is defined. In fact we consider two types of games – primal and dual games – and in both cases show how to derive winning strategies in the game from witnesses and construct witnesses from strategies. The two games differ wrt. their rules and the choice of basis of the lattice.

The theory can be instantiated to well-known examples: in particular we compare with the construction of distinguishing formulas in standard bisimilarity and behavioural metrics for probabilistic systems. As a new case study we consider witnesses for certifying lower bounds for the termination probability for Markov chains.

Keywords and phrases:

fixpoint games, lattice theory, witnesses, bisimilarity, Galois connections, Hennessy-Milner theorem, distinguishing formulas, behavioural metrics

Category:

Track B: Automata, Logic, Semantics, and Theory of Programming

Copyright and License:

2012 ACM Subject Classification:

Theory of computation

\rightarrow

Concurrency ; Theory of computation

\rightarrow

Logic and verification

Related Version:

Full Version: https://arxiv.org/abs/2603.11908 [26]

Funding:

The authors were supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 434050016 (SpeQt).

DOI:

10.4230/LIPIcs.ICALP.2026.182

Event:

53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Editors:

Sayan Bhattacharya, Danupon Nanongkai, Michael Benedikt, and Gabriele Puppis

Series and Publisher:

Leibniz International Proceedings in Informatics, Schloss Dagstuhl – Leibniz-Zentrum für Informatik

1 Introduction

In concurrency theory, there are many concepts that arise as least or greatest fixpoints of functions, for instance termination probability in Markov chains [1, 21], values for Markov decision processes [5] and simple stochastic games [11], bisimilarity [31, 35] or behavioural metrics [14, 9, 13, 16, 39, 2]. Limiting the discussion to least fixpoints (which can always be done without loss of generality via dualization), such a fixpoint characterization can also provide upper bounds via pre-fixpoints. However here we are instead interested in lower bounds, respectively in showing that upper bounds do not hold. In the case of bisimilarity, the first task is equivalent to certifying that two states are bisimilar, while the second means certifying that they are not. The latter notion is also known as apartness and has recently garnered increased attention [19, 37].

One way to witness non-bisimilarity is via so-called distinguishing formulas [10]. The Hennessy-Milner theorem [22] guarantees that in a finitely-branching transition system two states are bisimilar if and only if they satisfy the same formulas of the Hennessy-Milner logic, a form of modal logic [35]. Hence, if two states are non-bisimilar, there must be a formula distinguishing them. A similar notion of distinguishing formulas has been studied in the quantitative setting of behavioural metrics [29], where such a formula certifies that the distance of two states is above some bound (or a sequence of formulas is constructed that certifies the distance in the limit). Distinguishing formulas can in particular be used for purposes of explainability, e.g. explaining why two states are non-bisimilar as opposed to just stating this fact. Although their existence follows directly from the Hennessy-Milner theorem, their construction in general does not and has to be described separately.

Our aim is to lift the notion of distinguishing formulas – here called witnesses for greater generality – to a lattice-theoretic setting, where we assume a Galois connection connecting the “logic universe” with the “behaviour universe”. (In the case of bisimilarity logical formulas live in the logic universe, while bisimulation relations live in the behaviour universe.)

Apart from (least) fixpoints and logics, there is a third equivalent characterization that relies on games, that also provides us with the concept of (winning) strategies. In particular, we consider fixpoint games over continuous lattices [3] and we will show how to translate witnesses into game strategies and vice versa. In fact, there are two games for least fixpoints, a primal game and a dual game, where the dual game is derived by taking the game for the greatest fixpoint and dualizing the order. In both games the existential player $\exists$ makes the first move, followed by the universal player $\forall$ . In the primal game the witness provides the strategy for the existential player, while in the dual game we obtain the strategy of the universal player, hence both cases have to be treated separately. Both versions also have different requirements: in the primal case we assume continuity of the behaviour function and the lattice under consideration must have a basis consisting of irreducibles. In the dual case the function has to be co-proper.

Our results and constructions assume a continuous (resp. co-continuous) lattice and use the so-called way-below and way-above relations as well as notions from Scott topology [20]. In particular we focus on constructing finitary formulas corresponding to finitary strategies. As applications we will consider bisimilarity as well as behavioural metrics for probabilistic systems. As a new case study we will show how to provide witnesses for certifying lower bounds for the termination probability for Markov chains.

The theory provides a framework and guideline for witnesses and distinguishing formulas, in particular we provide generic algorithms for deriving strategies from witnesses and witness construction. We believe that viewing this construction in a more general framework is valuable, in particular since its usefulness seems to extend beyond behavioural equivalences and metrics.

Setup.

Assume a monotone function $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ (also called behaviour function) on a complete lattice $(\mathbb{B},\sqsubseteq)$ and we are interested in its least fixpoint $\mu\operatorname{\mathsf{beh}}$ . Then – due to the Knaster-Tarski theorem [36] – in order to show that a given lattice element $b\in\mathbb{B}$ is above $\mu\operatorname{\mathsf{beh}}$ ( $\mu\operatorname{\mathsf{beh}}\sqsubseteq b$ ), it is sufficient to find $p\in\mathbb{B}$ that satisfies $\operatorname{\mathsf{beh}}(p)\sqsubseteq p\sqsubseteq b$ (where the first inequality states that $p$ is a pre-fixpoint), from which we can immediately deduce $\mu\operatorname{\mathsf{beh}}\sqsubseteq b$ .

Our interest however lies in certifying the negated statement via a witness, that is $\mu\operatorname{\mathsf{beh}}\not\sqsubseteq b$ (or alternatively $b\ll\mu\operatorname{\mathsf{beh}}$ where $\ll$ is the way-below order). For the application examples this means to show that two states are not bisimilar or that the behavioural distance of two states is strictly larger than some bound. In this case it helps to consider a fixpoint game [3] where the aim of one player ( $\exists$ , defender) is to prove that the inequality $\mu\operatorname{\mathsf{beh}}\sqsubseteq b$ holds, while it is the aim of the opponent ( $\forall$ , attacker) to show that it does not.

In particular, we want to represent the strategy of the attacker. One way to do this is to look for a witness (also known as distinguishing formula [10, 29]), from which such a strategy can be derived. In lattice-theoretic terms, such witnesses live in another lattice – denoted by $\mathbb{L}$ – that is related to $\mathbb{B}$ via a Galois connection $\alpha\colon\mathbb{L}\to\mathbb{B}$ , $\gamma\colon\mathbb{B}\to\mathbb{L}$ . We assume the existence of a logic function $\operatorname{\mathsf{log}}\colon\mathbb{L}\to\mathbb{L}$ such that the left adjoint of the Galois connection maps its least fixpoint to the least fixpoint of the behaviour function ( $\alpha(\mu\operatorname{\mathsf{log}})=\mu\operatorname{\mathsf{beh}}$ ). Now the aim is to look for $\ell\sqsubseteq\mu\operatorname{\mathsf{log}}$ that induces a strategy for the fixpoint game for $\operatorname{\mathsf{beh}}$ on $\mathbb{B}$ .

In fact, it turns out that there are two games one can consider on $\mathbb{B}$ , a primal game and a dual game. Depending on which game is chosen, this influences the notion of witnesses and the translation between witnesses and strategies.

A full version of the paper, which includes all proofs and some additional information can be found at [26].

2 Preliminaries

We recall some basic definitions about lattices, Galois connections, the Hennessy-Milner framework based on Galois connections as introduced in [7], as well as Scott topology [20].

Partial Orders and Lattices.

A complete lattice $(\mathbb{L},\sqsubseteq)$ consists of a set $\mathbb{L}$ and a partial order $\sqsubseteq$ defined on $\mathbb{L}$ such that every subset $Y\subseteq\mathbb{L}$ has a least upper bound $\bigsqcup Y$ and a greatest lower bound $\bigsqcap Y$ . The bottom and top elements of $\mathbb{L}$ are denoted by $\bot$ , $\top$ respectively

By Knaster-Tarski [36] every monotone function $f\colon\mathbb{L}\to\mathbb{L}$ has a least fixpoint $\mu f$ and a greatest fixpoint $\nu f$ . The least fixpoint can be obtained by Kleene iteration over the ordinals, i.e., $\mu f=\bigsqcup_{i\in\mathsf{Ord}}f^{i}(\bot)$ , dually for the greatest fixpoint $\nu f$ .

We recall some notions on (co-)continuous lattices from [20].

Let $(\mathbb{L},\sqsubseteq)$ be a lattice. A subset $D\subseteq\mathbb{L}$ is directed if $D\neq\emptyset$ and every finite subset of $D$ has an upper bound in $D$ . For $\ell,\ell^{\prime}\in\mathbb{L}$ we say that $\ell$ is way-below $\ell^{\prime}$ ( $\ell\ll\ell^{\prime}$ ) iff for all directed subsets $D\subseteq\mathbb{L}$ , the relation $\ell^{\prime}\sqsubseteq\bigsqcup D$ implies the existence of $d\in D$ with $\ell\sqsubseteq d$ . The lattice $\mathbb{L}$ is called continuous if for all $\ell\in\mathbb{L}$ , $\ell=\bigsqcup\{\ell^{\prime}\in\mathbb{L}\mid\ell^{\prime}\ll\ell\}$ . (Note that on a continuous lattice, whenever $\ell\ll\ell^{\prime}$ and $\ell^{\prime}\sqsubseteq\bigsqcup D$ for a directed set $D$ , there even exists $d\in D$ such that $\ell\ll d$ [20].)

We also use the dual notions and call a set $F$ filtered if $F\not=\emptyset$ and every finite subset of $F$ has a lower bound in $F$ ; and say that $\ell$ is way-above $\ell^{\prime}$ ( $\ell^{\prime}\operatorname{\triangleleft\!\triangleleft}\ell$ ) iff for all filtered subsets $F\subseteq\mathbb{L}$ , the relation $\ell^{\prime}\sqsupseteq\bigsqcap F$ implies the existence of $f\in F$ such that $\ell\sqsupseteq f$ .¹¹1Note that in this terminology, “way-above” is not simply the inverse of “way below”. The lattice $\mathbb{L}$ is called co-continuous if for all $\ell\in\mathbb{L}$ , $\ell=\bigsqcap\{\ell^{\prime}\in\mathbb{L}\mid\ell^{\prime}\operatorname{% \triangleright\!\triangleright}\ell\}$ .

For $\ell\in\mathbb{L}$ , we write $\mathop{\uparrow}\ell=\{\ell^{\prime}\in\mathbb{L}\mid\ell\sqsubseteq\ell^{% \prime}\}$ , $\mathop{\downarrow}\ell=\{\ell^{\prime}\in\mathbb{L}\mid\ell\sqsupseteq\ell^{% \prime}\}$ , $\mathop{\mathrel{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}}\ell=\{% \ell^{\prime}\in\mathbb{L}\mid\ell\ll\ell^{\prime}\}$ and $\mathop{\mathrel{\rotatebox[origin={c}]{90.0}{$\operatorname{\triangleleft\!% \triangleleft}$}}}\ell=\{\ell^{\prime}\in\mathbb{L}\mid\ell\operatorname{% \triangleright\!\triangleright}\ell^{\prime}\}$ . The set of ordinals will be denoted by $\mathsf{Ord}$ .

Example 1.

Let $X$ be a set. For the powerset lattice $(\mathcal{P}(X),\subseteq)$ , a set $S\subseteq X$ is way-below $S^{\prime}\subseteq X$ ( $S\ll S^{\prime}$ ) if $S\subseteq S^{\prime}$ and $S$ finite. Since every set arises as the union of finite sets, powerset lattices are continuous. The sets $S,S^{\prime}$ are in the way-above relation ( $S\operatorname{\triangleleft\!\triangleleft}S^{\prime}$ ) iff $S\subseteq S^{\prime}$ and $S^{\prime}$ is co-finite (complement wrt. the superset $X$ of a finite set). Powerset lattices are co-continuous, since every set is the intersection of co-finite sets.

For the lattice $([0,1],\leq)$ it holds that two elements $r,s\in[0,1]$ are in the way-below relation ( $r\ll s$ ) if both are $0$ or $r<s$ . Every real number is the supremum of strictly smaller numbers, hence $[0,1]$ is a continuous lattice as well. It holds that $r\operatorname{\triangleleft\!\triangleleft}s$ iff both are $1$ or $r<s$ .²²2Note that $1\leq\bigsqcap F$ can only hold if $F=\{1\}$ due to the requirement that every filtered set is non-empty. The lattice $[0,1]$ is co-continuous, since every number is the infimum of numbers that are strictly larger.

We need the notion of basis, i.e., a subset of the lattice that allows generating each element (either via join or meet). A join basis of a lattice $\mathbb{L}$ is a subset $J_{\mathbb{L}}\subseteq\mathbb{L}$ such that for each element $\ell\in\mathbb{L}$ , $\ell=\bigsqcup\{b\in J_{\mathbb{L}}\mid b\sqsubseteq\ell\}$ . A meet basis of $\mathbb{L}$ is a subset $M_{\mathbb{L}}\subseteq\mathbb{L}$ such that for each element $\ell\in\mathbb{L}$ , $\ell=\bigsqcap\{m\in M_{\mathbb{L}}\mid m\sqsupseteq\ell\}$ .

When $\mathbb{L}$ is a continuous lattice and $J_{\mathbb{L}}$ is a join basis of $\mathbb{L}$ , then it holds – for all $\ell\in\mathbb{L}$ – that $\ell=\bigsqcup_{b\ll\ell,b\in J_{\mathbb{L}}}b$ . The dual holds for a meet basis and the way-above relation.

Example 2.

For a powerset lattice $(\mathcal{P}(X),\subseteq)$ , one choice of join basis is to consider all singletons. A possible meet basis is the set of all co-singletons, i.e., all sets of the form $\overline{\{x\}}=X\backslash\{x\}$ for $x\in X$ .

For the lattice $([0,1],\leq)$ a possible join basis consists of $(0,1]\cap\mathbb{Q}$ .

We also need the following notion of irreducibility. We call an element $b\in\mathbb{L}$ of a lattice $\mathbb{L}$ way-below irreducible (or simply irreducible) if whenever $b\ll\bigsqcup F$ for $F\subseteq\mathbb{L}$ finite, then $b\ll f$ for some $f\in F$ (where $F$ is not necessarily directed).

Note that since $\bot\ll\bot=\bigsqcup\emptyset$ , $\bot$ can not be an irreducible.

Example 3.

For the powerset lattice $(\mathcal{P}(X),\subseteq)$ , singletons are way-below irreducible. For the lattice $([0,1],\leq)$ , all elements – apart from $0$ – are way-below irreducible.

Galois Connections and Adjoint Logic.

We summarize the Galois connection approach from [7] that can be used to give an abstract account of the Hennessy-Milner theorem. Intuitively, the Galois connection relates a “logical universe” and a “behavioural universe”.

Let $\mathbb{L}$ , $\mathbb{B}$ be two lattices. A Galois connection from $\mathbb{L}$ to $\mathbb{B}$ is a pair $\alpha\dashv\gamma$ of monotone functions $\alpha\colon\mathbb{L}\to\mathbb{B}$ , $\gamma\colon\mathbb{B}\to\mathbb{L}$ such that for all $\ell\in\mathbb{L}$ : $\ell\sqsubseteq\gamma(\alpha(\ell))$ and for all $d\in\mathbb{B}$ : $\alpha(\gamma(d))\sqsubseteq d$ . It is well-known that a left adjoint $\alpha$ preserves all suprema while a right adjoint $\gamma$ preserves all infima.

We also fix two monotone functions $\operatorname{\mathsf{log}}\colon\mathbb{L}\to\mathbb{L}$ and $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ and by Knaster-Tarski [36] they both have least fixpoints (denoted by $\mu\operatorname{\mathsf{log}}$ , $\mu\operatorname{\mathsf{beh}}$ ).

From [7] it follows that $\alpha$ preserves least fixpoints of $\operatorname{\mathsf{log}}$ , $\operatorname{\mathsf{beh}}$ ( $\alpha(\mu\operatorname{\mathsf{log}})=\mu\operatorname{\mathsf{beh}}$ ) whenever $\alpha\circ\operatorname{\mathsf{log}}=\operatorname{\mathsf{beh}}\circ% \operatorname{\alpha}$ . This condition also implies that $\alpha$ preserves all stages of the
Kleene iteration over the ordinals, i.e., $\alpha(\operatorname{\mathsf{log}}^{i}(\bot))=\operatorname{\mathsf{beh}}^{i}(\bot)$ for all $i\in\mathsf{Ord}$ .

Example 4 (Bisimilarity & Hennessy-Milner Logic).

As a running example we consider standard bisimilarity [31]. Here we follow [7] (using arbitrary relations instead of only equivalence relations) and work – for simplicity – with unlabelled transition systems, i.e., pairs $(X,\to)$ consisting of a state space $X$ and a transition relation $\to\ \subseteq\,X\times X$ . We write $\mathit{succ}(x)=\{x^{\prime}\in X\mid x\to x^{\prime}\}$ and assume finite branching, i.e., $\mathit{succ}(x)$ is finite for all $x\in X$ .

We define lattices $\mathbb{L}=(\mathcal{P}(\mathcal{P}(X)),\subseteq)$ , $\mathbb{B}=(\mathit{Rel}(X),\supseteq)$ , where $\mathit{Rel}(X)$ is the set of all relations $R\subseteq X\times X$ . The Galois connection is:

	$\displaystyle\alpha(\mathcal{S})$	$\displaystyle=$	$\displaystyle\{(x_{1},x_{2})\in X\times X\mid\forall S\in\mathcal{S}\colon(x_{% 1}\in S\iff x_{2}\in S)\}$
	$\displaystyle\gamma(R)$	$\displaystyle=$	$\displaystyle\{S\subseteq X\mid\forall(x_{1},x_{2})\in R\colon(x_{1}\in S\iff x% _{2}\in S)\}.$

Intuitively, $\alpha$ generates an equivalence on $X$ from a set of subsets of $X$ and $\gamma$ maps a relation to all subsets of $X$ that are closed under this relation.

As logic function we consider $\operatorname{\mathsf{log}}\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(% \mathcal{P}(X))$ with $\operatorname{\mathsf{log}}(\mathcal{S})=\Diamond[\operatorname{\mathsf{cl}}^{% \cup,\lnot}_{f}(\mathcal{S})]$ , where $f[A]=\{f(a)\mid a\in A\}$ for a function $f\colon X\to Y$ and $A\subseteq X$ , $\operatorname{\mathsf{cl}}^{\cup,\lnot}_{f}$ closes $\mathcal{S}$ under finite intersections and complement and $\Diamond(S)=\{x\in X\mid\exists x^{\prime}\in S:\ x\to x^{\prime}\}$ .

The corresponding behaviour function $\operatorname{\mathsf{beh}}\colon\mathit{Rel}(X)\to\mathit{Rel}(X)$ is the standard (monotone) bisimilarity map: given $R\in\mathit{Rel}(X)$ it holds that $x_{1}\mathrel{\operatorname{\mathsf{beh}}(R)}x_{2}$ iff

\forall y_{1}\in\mathit{succ}(x_{1})\,\exists{y_{2}\in\mathit{succ}(x_{2})}% \colon y_{1}\mathrel{R}y_{2}\land\forall y_{2}\in\mathit{succ}(x_{2})\,\exists% {y_{1}\in\mathit{succ}(x_{1})}\colon y_{1}\mathrel{R}y_{2}.

From [7] it follows that $\alpha\circ\operatorname{\mathsf{log}}=\operatorname{\mathsf{beh}}\circ% \operatorname{\alpha}$ . In particular this means that $(x_{1},x_{2})\in\alpha(\mu\operatorname{\mathsf{log}})$ iff $x_{1},x_{2}$ are bisimilar (denoted by $x_{1}\sim x_{2}$ ), which is exactly the statement of the Hennessy-Milner theorem [22].

In the rest of the paper, we assume that $\mathbb{L}$ and $\mathbb{B}$ are both complete lattices, with monotone endofunctions $\operatorname{\mathsf{log}}$ respectively $\operatorname{\mathsf{beh}}$ , and a Galois connection $\alpha\dashv\gamma$ as above, such that $\alpha\circ\operatorname{\mathsf{log}}=\operatorname{\mathsf{beh}}\circ% \operatorname{\alpha}$ . Intuitively, this means that the functions $\operatorname{\mathsf{log}}$ and $\operatorname{\mathsf{beh}}$ satisfy the natural requirement of being “in step” with each other.

Scott Topology.

We need some concepts from Scott topology, in particular open sets, as well as continuous and proper maps (cf. [20]).

A subset $O\subseteq\mathbb{L}$ of a complete lattice $(\mathbb{L},\sqsubseteq)$ is called (Scott) open iff $O=\mathop{\uparrow}O$ , and $\bigsqcup D\in O$ implies $D\cap O\not=\emptyset$ for all directed sets $D\subseteq\mathbb{L}$ . The collection of all Scott open subsets of $\mathbb{L}$ is called the Scott topology of $\mathbb{L}$ and denoted by $\sigma(\mathbb{L})$ . A set $C\subseteq\mathbb{L}$ is called compact if whenever $C$ can be covered by open sets ( $C\subseteq\bigcup_{i\in I}O_{i}$ , $O_{i}$ open), there exists a finite subcover ( $C\subseteq\bigcup_{i\in J}O_{i}$ where $J\subseteq I$ and $J$ finite).

In a continuous lattice the sets of the form $\mathop{\mathrel{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}}\ell$ ( $\ell\in\mathbb{L}$ ) provide a basis for the Scott open sets [20]. That is, each Scott open set is the union of such sets.

Furthermore each set of the form $\mathop{\uparrow}\ell$ (for $\ell\in\mathbb{L}$ ) and their finite unions can easily be seen to be compact.

Example 5.

In a powerset lattice $\mathcal{P}(X)$ , the open sets are the families of finite character, i.e., $O\subseteq\mathcal{P}(X)$ such that $S\in O$ iff $F\in O$ for some finite subset $F$ of $S$ [20].

A function $f:\mathbb{S}\rightarrow\mathbb{T}$ between lattices is called (Scott-)continuous if and only if it is continuous with respect to the Scott topologies (i.e. $f^{-1}(U)\in\sigma(\mathbb{S})$ for all $U\in\sigma(\mathbb{T})$ ). Equivalently, $f$ is continuous if it preserves suprema of directed sets (i.e. $f(\bigsqcup D)=\bigsqcup f(D)$ for all directed subsets $D$ of $\mathbb{S}$ ). Note that a continuous function reaches its smallest fixpoint in $\omega$ steps.

A map $f:\mathbb{S}\rightarrow\mathbb{T}$ is proper if the inverse image of a compact set is also compact. It is called proper wrt. $B\subseteq\mathbb{T}$ if the inverse image of each $\mathop{\uparrow}\ell$ (for $\ell\in B$ ) is compact.

All these notions have duals that can be obtained by flipping the order and will be denoted by co-open, co-compact, co-continuous and co-proper.

Example 6 (Continuity and Properness).

Consider the bisimilarity map $\operatorname{\mathsf{beh}}$ introduced earlier (cf. Example 4). Since the transition system is assumed to be finitely branching, this map is co-continuous wrt. $\subseteq$ (see also [31]). It is also proper wrt. $\subseteq$ and a basis consisting of singletons $\{(x_{1},x_{2})\}\subseteq X\times X$ . Wrt. $\supseteq$ the map is hence continuous and co-proper wrt. a basis of co-singletons.

3 Witnesses

We will now introduce witnesses, analogous to distinguishing formulas in Hennessy-Milner logic that witness the non-bisimilarity or distance between two states (see [35]). A witness guarantees a lower bound for the least fixpoint respectively shows that a given element is not an upper bound.

We will later explain how witnesses can be constructed and how they are related to the winning strategies of the attacker in a game. We distinguish between primal and dual witnesses depending on the type of game for which they are used. In order to clarify the notation, we decorate lattice elements with a dot in the dual case.

Definition 7 (witness).

Let $\mathbb{L}$ , $\mathbb{B}$ be two lattices with a Galois connection $\alpha\dashv\gamma$ and monotone functions $\operatorname{\mathsf{log}}\colon\mathbb{L}\to\mathbb{L}$ , $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ . Let $b\in\mathbb{B}$ .

A primal witness for $b$ is an element $a\in\mathbb{L}$ such that $a\ll\mu\operatorname{\mathsf{log}}$ and $b\ll\alpha(a)$ .

A dual witness for $\dot{b}$ is an element $a\in\mathbb{L}$ such that $a\ll\mu\operatorname{\mathsf{log}}$ and $\alpha(a)\not\sqsubseteq\dot{b}$ .

Intuitively a witness is a formula of the logic that is strong enough to show that either $b$ is a lower bound or $\dot{b}$ is not an upper bound for the least fixpoint of $\operatorname{\mathsf{beh}}$ .

Proposition 8.

Assume the setting of Definition 7. Let $\mathbb{L}$ be continuous with join basis $J_{\mathbb{L}}$ .

Assume that $\mathbb{B}$ has a join basis $J_{\mathbb{B}}$ consisting only of way-below irreducibles. Given $b\in J_{\mathbb{B}}$ , there exists a primal witness $a\in J_{\mathbb{L}}$ for $b$ iff $b\ll\mu\operatorname{\mathsf{beh}}$ .

Given $\dot{b}\in\mathbb{B}$ , there exists a dual witness $a\in J_{\mathbb{L}}$ for $\dot{b}$ iff $\mu\operatorname{\mathsf{beh}}\not\sqsubseteq\dot{b}$ .

Note that for primal witnesses we have to assume a basis of way-below irreducibles, which is a restriction. Because of the above proposition, we can focus our attention on the chosen (join or meet) basis and will typically assume that the witness $a\in\mathbb{L}$ is a basis element. The same holds for $b\in\mathbb{B}$ .

Existence of witnesses is rather easy to prove, however the construction of such witnesses by a recursive process is more challenging and will be treated in the next sections.

Example 9 (Bases & Witnesses).

We continue our running example and instantiate these concepts to the case of unlabelled transition systems and bisimilarity.

For a join basis of $\mathbb{L}=\mathcal{P}(\mathcal{P}(X))$ we consider singleton sets $\{Y\}$ where $Y\subseteq X$ .

For $\mathbb{B}=\mathcal{P}(X\times X)$ (ordered by $\subseteq$ ) we consider a join basis consisting of all singletons $\{(x_{1},x_{2})\}\subseteq X\times X$ (which are also irreducibles). As a meet basis we use all co-singletons $\overline{\{(x_{1},x_{2})\}}:=(X\times X)\backslash\{(x_{1},x_{2})\}$ for $x_{1},x_{2}\in X$ .

If we reverse the order to $\supseteq$ , the join basis becomes a meet basis and vice versa and all join basis elements are irreducible. We will now continue working wrt. the inverse inclusion.

It holds that $\overline{\{(x_{1},x_{2})\}}\ll\mu\operatorname{\mathsf{beh}}\iff\mu% \operatorname{\mathsf{beh}}\subseteq\overline{\{(x_{1},x_{2})\}}\iff(x_{1},x_{% 2})\not\in\mu\operatorname{\mathsf{beh}}\iff x_{1}\not\sim x_{2}$ . A primal witness is then a predicate $a=\{X^{\prime}\}\subseteq\mu\operatorname{\mathsf{log}}$ (with $X^{\prime}\subseteq X$ ) that is obtained by evaluating a logical formula, for which $\overline{\{(x_{1},x_{2})\}}\ll\alpha(\{Y\})\iff\alpha(\{Y\})\subseteq% \overline{\{(x_{1},x_{2})\}}\iff\lnot(x_{1}\,\alpha(\{Y\})\,x_{2})\iff(x_{1}% \in Y\Leftrightarrow x_{2}\not\in Y)$ . And that is exactly the notion of a distinguishing formula.

In the dual case $\mu\operatorname{\mathsf{beh}}\not\sqsubseteq\{(x_{1},x_{2})\}\iff\mu% \operatorname{\mathsf{beh}}\not\supseteq\{(x_{1},x_{2})\}\iff(x_{1},x_{2})\not% \in\mu\operatorname{\mathsf{beh}}\iff x_{1}\not\sim x_{2}$ . As in the primal case a witness is a predicate obtained from a logical formula that distinguishes $x_{1},x_{2}$ .

We will now define the notion of (co-)degree, which is a ordinal that gives the least number of iterations needed to cover (or reach) a certain element.

Definition 10.

Let $\operatorname{\mathsf{log}}\colon\mathbb{L}\to\mathbb{L}$ . The degree and co-degree of an element $a\in\mathbb{L}$ (wrt. $\operatorname{\mathsf{log}}$ ) are defined as follows (where we assume that $\min\emptyset$ is undefined):

\operatorname{\mathsf{deg}}_{\operatorname{\mathsf{log}}}(a)=\min\{i\in\mathsf% {Ord}\mid a\ll\operatorname{\mathsf{log}}^{i}(\bot)\}\qquad\operatorname{% \mathsf{cdeg}}_{\operatorname{\mathsf{log}}}(a)=\min\{i\in\mathsf{Ord}\mid% \operatorname{\mathsf{log}}^{i}(\bot)\not\sqsubseteq a\}

Note that the degree (co-degree) of an element $a$ is defined whenever $a\ll\mu\operatorname{\mathsf{log}}$ ( $\mu\operatorname{\mathsf{log}}\not\sqsubseteq a$ ). It holds that $\operatorname{\mathsf{deg}}(\bot)=0$ , otherwise the (co-)degree is always a successor ordinal. We will often omit the subscript and simply write $\operatorname{\mathsf{deg}}$ or $\operatorname{\mathsf{cdeg}}$ if the function ( $\operatorname{\mathsf{log}}$ or $\operatorname{\mathsf{beh}}$ ) is clear from the context.

4 Games

In this section we will assume a monotone function on a complete lattice and study fixpoint games [3] that characterize the least fixpoint of such functions. Our aim is in particular to single out cases where finite strategies are sufficient and eventually connect game strategies with witnesses. In the following we spell out the games for a function $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ , but we will also play the primal way below game for the “other” function $\operatorname{\mathsf{log}}\colon\mathbb{L}\to\mathbb{L}$ .

We first introduce the game as it is presented in [3], in the following also called the primal game. Let $\mathbb{B}$ be a continuous lattice with a join basis $J_{\mathbb{B}}$ such that $\bot\not\in J_{\mathbb{B}}$ . Let $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ be monotone. The game starts with $b\in J_{\mathbb{B}}$ and players $\exists$ and $\forall$ play according to the following rules:

Position	Player	Moves
$b\in J_{\mathbb{B}}$	$\exists$	$d\in\mathbb{B}$ , such that $b\sqsubseteq\operatorname{\mathsf{beh}}(d)$
$d\in\mathbb{B}$	$\forall$	$b^{\prime}\in J_{\mathbb{B}}$ such that $b^{\prime}\ll d$

If a player cannot move, the opponent wins. Infinite games are won by $\forall$ .

It holds that $b\sqsubseteq\mu\operatorname{\mathsf{beh}}$ iff the $\exists$ player has a winning strategy from $b$ [3]. The intuitive reason for requiring the way-below order for the answer of $\forall$ is that – whenever $b\sqsubseteq\mu\operatorname{\mathsf{beh}}$ – $\exists$ can ensure that the game positions descend the chain of ordinals and $\forall$ eventually runs out of answering moves.

Furthermore $b\sqsubseteq\nu\operatorname{\mathsf{beh}}$ iff $\exists$ has a strategy in the same game with the only modification that $\exists$ wins infinite games.

4.1 Primal Way-Below Game

There is also a “way-below” version of the primal game where the aim of $\exists$ is to show a strict lower bound, i.e. prove that $b\ll\mu\operatorname{\mathsf{beh}}$ . To our knowledge this version of the game is original.

Definition 11.

Let $\mathbb{B}$ be a continuous lattice with a join basis $J_{\mathbb{B}}$ such that $\bot\not\in J_{\mathbb{B}}$ . Let $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ be monotone. We define a game between players $\exists$ and $\forall$ that play according to the following rules starting with $b\in J_{\mathbb{B}}$ :

Position	Player	Moves
$b\in J_{\mathbb{B}}$	$\exists$	$d\in\mathbb{B}$ , such that $b\ll\operatorname{\mathsf{beh}}(d)$
$d\in\mathbb{B}$	$\forall$	$b^{\prime}\in J_{\mathbb{B}}$ such that $b^{\prime}\ll d$

If a player cannot move, the opponent wins. Infinite games are won by $\forall$ .

In the context of the primal way-below game, we will also call the $\exists$ player the attacker and the $\forall$ player the defender.

Note that since in parity games memoryless winning strategies are always enough [8], it suffices to consider positional winning strategies. We can also show that under some circumstances formulas are “finitely constructed”, i.e., strategies are finitary.

Proposition 12.

Let $b\in J_{\mathbb{B}}$ . Player $\exists$ has a winning strategy in the game in Definition 11 iff $b\ll\mu\operatorname{\mathsf{beh}}$ .

Whenever $\operatorname{\mathsf{beh}}$ is a continuous function, then $\exists$ has a finitary winning strategy. Finitary means that $d=\bigsqcup F$ where $F$ is a finite subset of $J_{\mathbb{B}}$ and $\operatorname{\mathsf{deg}}(b^{\prime})<\operatorname{\mathsf{deg}}(b)$ for each $b^{\prime}\in F$ . In particular $\exists$ can win in $\operatorname{\mathsf{deg}}(b)$ steps.

Since it is guaranteed that the degrees decrease the game will terminate eventually and $\exists$ wins. Such a finitary winning strategy for $\exists$ (the attacker) is denoted by $S_{p,\operatorname{\mathsf{beh}}}^{\exists}$ and assigns a suitable move $F\subseteq J_{\mathbb{B}}$ to $b\in J_{\mathbb{B}}$ . Then $\exists$ plays $\bigsqcup S_{p,\operatorname{\mathsf{beh}}}^{\exists}(b)$ . $S_{p,\operatorname{\mathsf{beh}}}^{\exists}(b)$ is undefined if there is no winning strategy from $b$ .

Example 13 (Strategy for the Primal Game).

We spell out the primal game for our running example. The join basis for $(\mathcal{P}(X\times X),\supseteq)$ are the co-singletons (cf. Example 9). Now, when the initial situation is $\overline{\{(x_{1},x_{2})\}}$ , the game proceeds as follows:

$\blacksquare$

$\exists$ (attacker) plays $R\subseteq X\times X$ such that $\operatorname{\mathsf{beh}}(R)\subseteq\overline{\{(x_{1},x_{2})\}}$ (i.e., $(x_{1},x_{2})\not\in\operatorname{\mathsf{beh}}(R)$ ).

More concretely $\exists$ plays $R$ satisfying: there exists $y_{1}\in X$ with $x_{1}\to y_{1}$ such that for all $y_{2}\in X$ with $x_{2}\to y_{2}$ it holds that $(y_{1},y_{2})\not\in R$ or vice versa.
$\blacksquare$

$\forall$ (defender) chooses $R\subseteq\overline{\{(y_{1},y_{2})\}}$ (i.e., $(y_{1},y_{2})\not\in R$ ) and wins infinite games.

If $x_{1}\not\sim x_{2}$ , a possible winning strategy of $\exists$ (attacker) is to play

S_{p,\operatorname{\mathsf{beh}}}^{\exists}(\overline{\{(x_{1},x_{2})\}})=% \overline{\{y_{1}\}\times\mathit{succ}(x_{2})}=\bigcap\nolimits_{y\in\mathit{% succ}(x_{2})}\overline{\{(y_{1},y)\}}

if $x_{1}\stackrel{{\scriptstyle a}}{{\to}}y_{1}$ is a winning move in the traditional game (or vice versa). By deriving the strategy from the fixpoint iteration it is always possible to choose $y_{1}$ such that for all $y\in\mathit{succ}(x_{2})$ , $\overline{\{(y_{1},y)\}}$ has a smaller degree than $\overline{\{(x_{1},x_{2})\}}$ . Then $\forall$ must pick $(y_{1},y)$ , which is equivalent to making an answering move in the traditional game. Hence we obtain a game that is close to the game in [35] where a defender mimics the moves made by the attacker.

In Section 3 we considered (primal) witnesses as monolithic objects. But in fact, in order to use witnesses to derive strategies, we have to be able to deconstruct them into subformulas. This is exactly what this game does for the logic function $\operatorname{\mathsf{log}}$ (as defined in Example 4). Given $a\in J_{\mathbb{L}}$ with $a\ll\mu\operatorname{\mathsf{log}}$ (a basis element, representing to a formula of the logic), $\exists$ is obliged to play a finite $A\subseteq J_{\mathbb{L}}$ with $a\sqsubseteq\operatorname{\mathsf{log}}(\bigsqcup A)$ . That is, $\exists$ exhibits the subformulas, from which $a$ can be constructed by applying $\operatorname{\mathsf{log}}$ . Then $\forall$ can pick $a^{\prime}\ll\bigsqcup A$ (in the case of powerset lattices and a basis of singletons this amounts to choosing $a^{\prime}\in A$ ) and continue the game from there, asking $\exists$ to spell out why $a^{\prime}$ is also a logical formula.

If the lattice $\mathbb{B}$ is not continuous, the game can fail, in particular $\exists$ may be able to win although $b\not\ll\mu\operatorname{\mathsf{beh}}$ . It could however be the case that $\mathbb{B}$ is co-continuous and then we can use the game presented next.

4.2 Dual Game

We adjust the game in [3] for greatest fixpoints to the dual setting by flipping the order and obtain another game characterizing the least fixpoint. This gives us an alternative perspective on games and witness generation.

Definition 14.

We consider the co-continuous lattice $\mathbb{B}$ with a meet basis $M_{\mathbb{B}}$ such that $\top\not\in M_{\mathbb{B}}$ and a monotone function $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ . The dual game on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ follows the following rules.

Position	Player	Moves
$\dot{b}\in M_{\mathbb{B}}$	$\exists$	$\dot{d}\in\mathbb{B}$ , such that $\operatorname{\mathsf{beh}}(\dot{d})\sqsubseteq\dot{b}$
$\dot{d}\in\mathbb{B}$	$\forall$	$\dot{b^{\prime}}\in M_{\mathbb{B}}$ such that $\dot{d}\operatorname{\triangleleft\!\triangleleft}\dot{b^{\prime}}$

If a player cannot move, the opponent wins. Infinite games are won by $\exists$ .

Here $\exists$ takes the role of the defender, and $\forall$ the role of the attacker. It holds that $\mu\operatorname{\mathsf{beh}}\sqsubseteq\dot{b}$ iff $\exists$ has a winning strategy when starting from $\dot{b}$ .

We show that $\forall$ has a finitary winning strategy that is in some sense independent on the move of $\exists$ . For this we require that $\operatorname{\mathsf{beh}}$ is co-proper wrt. to the basis.

Proposition 15.

Assume that $\mathbb{B}$ is co-continuous and that $\operatorname{\mathsf{beh}}$ is co-proper wrt. $M_{\mathbb{B}}$ .

Fix $\dot{b}\in M_{\mathbb{B}}$ with $\mu\operatorname{\mathsf{beh}}\not\sqsubseteq\dot{b}$ . Then there exists a finite set $F\subseteq M_{\mathbb{B}}$ such that – for every move $\dot{d}$ of $\exists$ – $\forall$ can always choose some $\dot{b^{\prime}}\in F$ to win the game. In particular $\dot{b^{\prime}}\operatorname{\triangleright\!\triangleright}\dot{d}$ and $\operatorname{\mathsf{cdeg}}(\dot{b^{\prime}})<\operatorname{\mathsf{cdeg}}(% \dot{b})$ for all $\dot{b^{\prime}}\in F$ . We denote $F$ by $S_{d,\operatorname{\mathsf{beh}}}^{\forall}(\dot{b})$ . Note that $\forall$ can win in $\operatorname{\mathsf{cdeg}}(\dot{b})$ steps.

Example 16 (Strategy for the Dual Game).

We consider again our running example, i.e., the case of bisimilarity. We take the function $\operatorname{\mathsf{beh}}$ from Example 4 and the lattice $(\mathcal{P}(X\times X),\supseteq)$ . The dual game on $\operatorname{\mathsf{beh}}$ corresponds to a coupling game [4]. Given a basis element $\dot{b}=\{(x_{1},x_{2})\}$ , $\exists$ produces a “coupling” $R\subseteq X\times X$ such that $(x_{1},x_{2})\in\operatorname{\mathsf{beh}}(R)$ , which means every successor of $x_{1}$ must be paired with some successor of $x_{2}$ and vice versa. Then $\forall$ picks one such pair in $R$ , claims that it is not bisimilar and the game continues. Note whenever $x_{1}\not\sim x_{2}$ $\forall$ can always precompute a finitary winning strategy $S_{d,\operatorname{\mathsf{beh}}}^{\forall}(\dot{b})=\{\{(y_{1},y)\}\mid y\in% \mathit{succ}(x_{2})\}$ whenever we choose $y_{1}$ as in Example 13 as some state that has no bisimilar partner in $\mathit{succ}(x_{2})$ (or vice versa).

5 Transforming Witnesses and Winning Strategies

5.1 Auxiliary Functions

Before we start to transform witnesses into winning strategies for the attacker and vice versa, we first define some auxiliary functions.

We can show the existence of the following three functions specified in the table below. For the first two lines we assume a monotone function $f\colon\mathbb{L}\to\mathbb{L}$ and a finite set $A\subseteq J_{\mathbb{L}}$ . We will in particular instantiate to $f=\mathsf{id}$ and $f=\operatorname{\mathsf{log}}$ . We write $W_{p}$ , $W_{d}$ instead of $W_{p,\mathsf{id}}$ , $W_{d,\mathsf{id}}$ . For $f=\operatorname{\mathsf{log}}$ note that $\alpha(\operatorname{\mathsf{log}}(\bigsqcup A)))=\operatorname{\mathsf{beh}}(% \alpha(\bigsqcup A))=\operatorname{\mathsf{beh}}(\bigsqcup\alpha[A])$ .

function parameters output $W_{p,f}$ $b\in J_{\mathbb{B}}$ , $A\subseteq_{\mathit{fin}}J_{\mathbb{L}}$ , $b\ll\alpha(f(\bigsqcup A))$ $W_{p,f}(b,A)=a\in J_{\mathbb{L}}$ s.t. $a\ll f(\bigsqcup A)$ , $b\ll\alpha(a)$ $W_{d,f}$ $\dot{b}\in M_{\mathbb{B}}$ , $A\subseteq_{\mathit{fin}}J_{\mathbb{L}}$ , $\alpha(f(\bigsqcup A))\not\sqsubseteq\dot{b}$ $W_{d,f}(b,A)=a\in J_{\mathbb{L}}$ s.t. $a\ll f(\bigsqcup A)$ , $\alpha(a)\not\sqsubseteq\dot{b}$ $Z$ $\dot{e},\dot{d}\in\mathbb{B}$ , $\dot{e}\not\sqsubseteq\dot{d}$ $Z(\dot{e},\dot{d})=\dot{b^{\prime}}\in M_{\mathbb{B}}$ s.t. $\dot{b^{\prime}}\operatorname{\triangleright\!\triangleright}\dot{d}$ and $\dot{e}\not\sqsubseteq\dot{b^{\prime}}$

We assume that $\mathbb{L}$ is continuous lattice with join basis $J_{\mathbb{L}}$ , additionally in the first line the join basis $J_{\mathbb{B}}$ contains only irreducibles, in the second line $M_{\mathbb{B}}$ is a meet basis and in the third line $\mathbb{B}$ must additionally be co-continuous.

The first two functions have the task of picking a suitable witness from $\mathbb{L}$ from a set of joint witnesses. The last auxiliary functions chooses a basis element of $\mathbb{B}$ that explains why a given inequality does not hold.

5.2 Primal Case: Transforming Strategies along the Galois Connection

We now spell out how to use primal witnesses to obtain winning strategies for $\exists$ in the primal way-below game (on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ ), see Section 4.1. We furthermore show how such winning strategies can be used to construct primal witnesses. In this subsection we assume that $\mathbb{L}$ , $\mathbb{B}$ are both continuous lattices with join bases $J_{\mathbb{L}}$ , $J_{\mathbb{B}}$ . Furthermore $J_{\mathbb{L}}$ must contain only way-below irreducibles.

Proposition 17.

Let $\operatorname{\mathsf{log}}$ be a continuous function, which implies that there is a finitary winning strategy $S_{p,\operatorname{\mathsf{log}}}^{\exists}$ for $\exists$ in the primal way-below game on $\operatorname{\mathsf{log}}\colon\mathbb{L}\to\mathbb{L}$ (cf. Proposition 12).

Given $b\in J_{\mathbb{B}}$ and a primal witness $a$ for $b$ ( $b\ll\alpha(a)$ ), we choose the move of $\exists$ in the primal way-below game on $\operatorname{\mathsf{beh}}:\mathbb{B}\to\mathbb{B}$ as

d=\alpha(\bigsqcup S_{p,\operatorname{\mathsf{log}}}^{\exists}(a)).

Let $b^{\prime}\ll d$ be an answering move of $\forall$ . Then $a^{\prime}=W_{p}(b^{\prime},S_{p,\operatorname{\mathsf{log}}}^{\exists}(a))$ is a primal witness for $b^{\prime}$ and $\operatorname{\mathsf{deg}}(a^{\prime})<\operatorname{\mathsf{deg}}(a)$ . We continue from $b^{\prime}$ and $a^{\prime}$ and obtain a winning strategy for $\exists$ in the primal way-below game on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ (cf. Definition 11) for all $b\in J_{\mathbb{B}}$ that have witnesses.

Now we treat witness construction and assume that there exists a finitary winning strategy $S_{p,\operatorname{\mathsf{beh}}}^{\exists}$ for $\exists$ in the primal way-below game on $\operatorname{\mathsf{beh}}$ (cf. Definition 11) which assigns to each $b\in J_{\mathbb{B}}$ a set $S_{p,\operatorname{\mathsf{beh}}}^{\exists}(b)\subseteq J_{\mathbb{B}}$ . From such a strategy we construct a witness $\mathit{wit}(b)$ for a given $b\ll\mu\operatorname{\mathsf{beh}}$ .

From the construction one can also extract a finitary winning strategy for the witness in the game on $\operatorname{\mathsf{log}}$ .

Theorem 18.

Assume that $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ is continuous, which implies the existence of a finitary winning strategy $S_{p,\operatorname{\mathsf{beh}}}^{\exists}$ for $\exists$ in the primal way-below game on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ (cf. Proposition 12).

Given $b\in J_{\mathbb{B}}$ with $b\ll\mu\operatorname{\mathsf{beh}}$ , we can compute the witness of $b$ inductively as follows:

\mathit{wit}(b)=W_{p,\operatorname{\mathsf{log}}}(b,\mathit{wit}[S_{p,% \operatorname{\mathsf{beh}}}^{\exists}(b)]).

Then $\operatorname{\mathsf{deg}}(\mathit{wit}(b))\leq\operatorname{\mathsf{deg}}(b)$ and this is a well-defined inductive definition with base case $S_{p,\operatorname{\mathsf{beh}}}^{\exists}(b)=\emptyset$ .

Example 19 (Witness Construction – Primal).

We study the construction of witnesses, also known as distinguishing formulas, in the running example. We are again working on $\mathbb{B}=\mathit{Rel}(X)$ (ordered by $\supseteq$ ) with a join basis consisting of co-singletons (Example 9, primal case).

Assume that $x_{1}\not\sim x_{2}$ for $x_{1},x_{2}\in X$ and let $b=\overline{\{(x_{1},x_{2})\}}$ . We can choose some finitary strategy $S_{p,\operatorname{\mathsf{beh}}}^{\exists}$ for the primal way-below game on $\operatorname{\mathsf{beh}}$ (see e.g. Example 13). Let $A=\mathit{wit}[S_{p,\operatorname{\mathsf{beh}}}^{\exists}(b)]$ , i.e., we compute witnesses recursively. We then obtain $\mathit{wit}(b)=W_{p,\operatorname{\mathsf{log}}}(b,A)$ , where the auxiliary function $W_{p,\operatorname{\mathsf{log}}}$ is defined as follows:

$\blacksquare$

If $A=\emptyset$ (which holds if $\operatorname{\mathsf{deg}}(b)=1$ , hence $S_{p,\operatorname{\mathsf{beh}}}^{\exists}(b)=\emptyset$ and $\exists$ can play $\bigsqcup\emptyset=\bigcap\emptyset=X\times X$ ), then define $W_{p,\operatorname{\mathsf{log}}}(b,A)=\Diamond\mathit{true}\in\operatorname{% \mathsf{log}}(\emptyset)$ (where $\mathit{true}$ is the empty conjunction).
$\blacksquare$

Otherwise we know by construction that

$\displaystyle\alpha(\operatorname{\mathsf{log}}(\bigsqcup A))=\operatorname{% \mathsf{beh}}(\alpha(A))\gg b$ $\displaystyle\iff\operatorname{\mathsf{beh}}(\alpha(A))\subseteq\overline{\{(x% _{1},x_{2})\}}$

$\displaystyle\iff(x_{1},x_{2})\not\in\operatorname{\mathsf{beh}}(\alpha(A))$

Since $\alpha(A)$ suffices to distinguish $x_{1},x_{2}$ (after applying $\operatorname{\mathsf{beh}}$ ), it must be the case that there exists $y_{1}\in\mathit{succ}(x_{1})$ that is not related via $\alpha(A)$ to any state in $y_{2}\in\mathit{succ}(x_{2})$ (or vice versa). Hence for $y_{1}$ and each such $y_{2}$ there exists a predicate $P^{\prime}\in A$ that separates them and we can define

$W_{p,\operatorname{\mathsf{log}}}(b,A)=\Diamond\big(\bigcap_{P^{\prime}\in A,y% _{1}\in P^{\prime}}P^{\prime}\cap\bigcap_{P^{\prime}\in A,y_{1}\not\in P^{% \prime}}\lnot P^{\prime}\big),$

which contains $x_{1}$ (which has a successor $y_{1}$ satisfying all predicates) but not $x_{2}$ (where for each successor $y_{2}$ there exists a predicate not containing $y_{2}$ ).

5.3 Primal/Dual Case: Transforming Strategies along the Galois Connection

We now explain how to transform dual witnesses into winning strategies for $\forall$ in the dual game on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ . In this subsection we assume that $\mathbb{L}$ is a continuous lattice with join basis $J_{\mathbb{L}}$ and $\mathbb{B}$ is a co-continuous lattice with meet basis $M_{\mathbb{B}}$ .

Proposition 20.

Assume that $\operatorname{\mathsf{log}}$ is continuous, which implies the existence of a finitary winning strategy $S_{p,\operatorname{\mathsf{log}}}^{\exists}$ for $\exists$ in the primal way-below game on $\operatorname{\mathsf{log}}\colon\mathbb{L}\to\mathbb{L}$ (cf. Proposition 12).

Let $\dot{b}\in M_{\mathbb{B}}$ and let $a$ be a dual witness for $\dot{b}$ ( $\alpha(a)\not\sqsubseteq\dot{b}$ ). We let $A=S_{p,\operatorname{\mathsf{log}}}^{\exists}(a)$ . Given a move $\dot{d}\in\mathbb{B}$ by $\exists$ , the $\forall$ player plays

\dot{b^{\prime}}=Z(\alpha(\bigsqcup A),\dot{d}),

which has a dual witness $a^{\prime}=W_{d}(\dot{b^{\prime}},A)$ with $\operatorname{\mathsf{deg}}(a^{\prime})<\operatorname{\mathsf{deg}}(a)$ . We continue from $\dot{b^{\prime}}$ and $a^{\prime}$ and obtain a winning strategy for $\forall$ in the dual game on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ (see Definition 14).

We will now explain how to construct a witness from a winning strategy for $\forall$ in the dual game on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ , using the strategy from Proposition 15. From the construction one can also extract a winning strategy for the witness in the game on $\operatorname{\mathsf{log}}$ .

Theorem 21.

Assume that $\operatorname{\mathsf{beh}}$ is co-proper wrt. $M_{\mathbb{B}}$ , which implies the existence of a finitary winnning strategy $S_{d,\operatorname{\mathsf{beh}}}^{\forall}$ for $\forall$ in the dual game on $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ . (cf. Proposition 15).

Given $\dot{b}\in M_{\mathbb{B}}$ with $\mu\operatorname{\mathsf{beh}}\not\sqsubseteq\dot{b}$ , we can compute the witness of $\dot{b}$ inductively as:

\mathit{wit}(\dot{b})=W_{d,\operatorname{\mathsf{log}}}(\dot{b},\mathit{wit}[S% _{d,\operatorname{\mathsf{beh}}}^{\forall}(\dot{b})]).

Then $\operatorname{\mathsf{deg}}(\mathit{wit}(\dot{b}))\leq\operatorname{\mathsf{% deg}}(\dot{b})$ and this is a well-defined inductive definition with base case $S_{d,\operatorname{\mathsf{beh}}}^{\forall}(\dot{b})=\emptyset$ .

Example 22 (Witness Construction – Dual).

In the running example, we choose a meet basis (for the order $\supseteq$ ) for $\mathit{Rel}(X)$ with singletons as in Example 9.

Now fix $\dot{b}=\{(x_{1},x_{2})\}$ with $x_{1}\not\sim x_{2}$ and we choose some finitary strategy $S_{d,\operatorname{\mathsf{beh}}}^{\forall}$ for the dual game on $\operatorname{\mathsf{beh}}$ (see e.g. Example 16). By construction $\mathit{wit}(\dot{b})=W_{d,\operatorname{\mathsf{log}}}(\dot{b},A)$ , where $A=\mathit{wit}[S_{d,\operatorname{\mathsf{beh}}}^{\forall}(\dot{b})]$ and $W_{d,\operatorname{\mathsf{log}}}$ is defined as follows:

$\blacksquare$

If $A=\emptyset$ (which holds if $\operatorname{\mathsf{cdeg}}(\dot{b})=1$ ), we again set $W_{d,\operatorname{\mathsf{log}}}(\dot{b},A)=\Diamond\mathit{true}$ .
$\blacksquare$

Otherwise

$\alpha(\operatorname{\mathsf{log}}(\bigsqcup A))=\operatorname{\mathsf{beh}}(% \alpha(A))\not\sqsubseteq\dot{b}\iff\operatorname{\mathsf{beh}}(\alpha(A))\not% \supseteq\{(x_{1},x_{2})\}\iff(x_{1},x_{2})\not\in\operatorname{\mathsf{beh}}(% \alpha(A))$

As in the primal case we determine a state $y_{1}\in\mathit{succ}(x_{1})$ that is not related (via $\alpha(A)$ ) to any state in $\mathit{succ}(x_{2})$ (or vice versa) and define:

$W_{d,\operatorname{\mathsf{log}}}(\dot{b},A)=\Diamond\big(\bigcap_{P^{\prime}% \in A,y_{1}\in P^{\prime}}P^{\prime}\cap\bigcap_{P^{\prime}\in A,y_{1}\not\in P% ^{\prime}}\lnot P^{\prime}\big).$

6 Case Studies

6.1 Behavioural Metrics

We now consider the case of behavioural metrics, where we measure the distance of two states. Here we focus on probabilistic transition systems. The construction of distinguishing formulas in this setting was presented previously in [29], here we show how a similar construction arises as a special case of our theory. Note that the instantiation of probabilistic transition systems to the Galois connection approach is new, it was not studied in [7].

We consider labelled Markov chains $(X,\delta,\ell)$ consisting of a finite state space $X$ , a probabilistic transition function $\delta\colon X\to\mathcal{D}(X)$ (where $\mathcal{D}(X)$ is the set of probability distributions over $X$ ) and a labelling function $\ell\colon X\to\Lambda$ .

Lattices, Functions and Galois Connection.

Behaviour: On the behaviour side we fix the lattice $\mathbb{B}=(\mathit{Dist}(X),\leq)$ (where $\mathit{Dist}(X)$ denotes the set of distance functions over $X$ , i.e., functions of the form $d\colon X\times X\to[0,1]$ ). The way-below and way-above relations $\ll$ , $\operatorname{\triangleleft\!\triangleleft}$ are the pointwise $<$ -orders (in addition $0\ll 0$ , $1\operatorname{\triangleleft\!\triangleleft}1$ ), see Example 1. The behaviour function $\operatorname{\mathsf{beh}}$ is characterized as: given $d\in\mathit{Dist}(X)$ it holds that

\displaystyle\operatorname{\mathsf{beh}}(d)(x_{1},x_{2})

\displaystyle=

\displaystyle\begin{cases}\mathcal{K}(d)(\delta(x_{1}),\delta(x_{2}))&\text{if $\ell(x_{1})=\ell(x_{2})$}\\ 1&\text{otherwise}\end{cases}

where $\mathcal{K}$ is the (price-function based) Kantorovich lifting [40, 2] that transform a distance on $X$ into a distance on $\mathcal{D}(X)$ . In terms of the Galois connection this is defined as $\mathcal{K}=\alpha\circ\bigcirc\circ\gamma$ . Here $\bigcirc f(x)=\mathbb{E}_{\delta(x)}[f]$ where $\mathbb{E}_{p}[f]=\sum_{x\in X}f(x)\cdot p(x)$ determines the expectation of random variable $f$ under probability distribution $p$ . The function $\operatorname{\mathsf{beh}}$ is obviously monotone (see also [29]).

Hence $\mu\operatorname{\mathsf{beh}}(x_{1},x_{2})$ denotes the behavioural distance of $x_{1},x_{2}$ , based on the Kantorovich lifting.

Logic: The lattice on the logic side is $\mathbb{L}=(\mathcal{P}([0,1]^{X}),\subseteq)$ (sets of random variables). The way-below relation for powerset lattices is defined in Example 1. The logic function is based on the operators of [29]. We define $\operatorname{\mathsf{log}}\colon\mathcal{P}([0,1]^{X})\to\mathcal{P}([0,1]^{X})$ with

\operatorname{\mathsf{log}}(\mathcal{F})=\{[a]\mid a\in\Lambda\}\cup\{\bigcirc f% \mid f\in\mathit{cl}(\mathcal{F})\}

where $[a]\colon X\to[0,1]$ is defined as $[a](x)=1$ if $\ell(x)=a$ and $0$ otherwise. Furthermore $\mathit{cl}(\mathcal{F})$ is the closure of the set $\mathcal{F}\subseteq[0,1]^{X}$ under the operators $1-f$ , $f\ominus q$ ( $q\in\mathbb{Q}\cap[0,1]$ , $\ominus$ is the modified subtraction) and $\max(f,g)$ for functions $f,g\colon X\to[0,1]$ .

Galois Connection: The Galois connection is given as follows:

\alpha(\mathcal{F})=(x_{1},x_{2})\mapsto\sup_{f\in\mathcal{F}}|f(x_{1})-f(x_{2% })|\qquad\gamma(d)=\{f\colon X\to[0,1]\mid\text{$f$ non-expansive wrt. $d$}\}

Given $d\in\mathit{Dist}(X)$ , a non-expansive function $f\colon X\to[0,1]$ wrt. $d$ must satisfy $|f(x_{1})-f(x_{2})|\leq d(x_{1},x_{2})$ .

Primal Case.

We now construct witnesses in the primal case. We first observe that $\operatorname{\mathsf{beh}}$ is indeed continuous.

Basis: As join basis of $\mathbb{L}$ (consisting of way-below irreducibles) we consider singleton sets $\{f\}$ where $f\colon X\to[0,1]$ , while the join basis of $\mathbb{B}$ contains all distance functions $d_{x_{1},x_{2}}^{c}$ (for $x_{1}\neq x_{2}$ , $c>0$ ) that are defined as

d_{x_{1},x_{2}}^{c}(y_{1},y_{2})=\begin{cases}c&\text{if $(x_{1},x_{2})=(y_{1}% ,y_{2})$}\\ 0&\text{otherwise}\end{cases}

Strategy computation (cf. Prop. 12): Let $b=d_{x_{1},x_{2}}^{c}$ be a basis element that is way-below $\mu\operatorname{\mathsf{beh}}$ (i.e., $c<\mu\operatorname{\mathsf{beh}}(x_{1},x_{2})$ ), for which we determine the strategy of $\exists$ . Let $k=\operatorname{\mathsf{deg}}(b)$ and define $d_{k}=\operatorname{\mathsf{beh}}^{k}(0)$ as the $k$ -th iterate in the Kleene iteration.

Whenever $k=1$ we have $\ell(x_{1})\neq\ell(x_{2})$ and $\exists$ can play $\emptyset$ .

Otherwise we rely on the coupling characterization of the Kantorovich lifting [40]. Given two probability distributions $p,q\in\mathcal{D}(X)$ , a coupling of $p, q$ is a probability distribution $C\in\mathcal{D}(X\times X)$ with $p, q$ as marginals, i.e., for all $x_{1}\in X$ : $\sum_{x_{2}\in X}C(x_{1},x_{2})=p(x_{1})$ and for all $x_{2}\in X$ : $\sum_{x_{1}\in X}C(x_{1},x_{2})=q(x_{2})$ . We denote the set of couplings of $p, q$ as $\Gamma(p,q)$ . Then – for a pseudometric $d$ – we can spell out the Kantorovich lifting as

	$\displaystyle\mathcal{K}(d)(x_{1},x_{2})$	$\displaystyle=$	$\displaystyle\inf\{\sum_{y_{1},y_{2}\in X}C(y_{1},y_{2})\cdot d(y_{1},y_{2})% \mid C\in\Gamma(\delta(x_{1}),\delta(x_{2}))\}$
		$\displaystyle=$	$\displaystyle\inf\{\mathbb{E}_{C}[d]\mid C\in\Gamma(\delta(x_{1}),\delta(x_{2}% ))\}.$

The set of all couplings form a polytope and – as is well-known in linear programming – the infimum above is a minimum and is achieved in one of the (finitely many) vertices of the coupling polytope. We denote the set of vertices by $\Gamma_{V}(\delta(x_{1}),\delta(x_{2}))$ and can replace $\Gamma(\delta(x_{1}),\delta(x_{2}))$ in the equation above by $\Gamma_{V}(\delta(x_{1}),\delta(x_{2}))$ and $\inf$ by $\min$ .

Let $\mathit{succ}(x_{i})$ be the states reachable from $x_{i}$ with non-zero probability. For each pair $y_{1}\in\mathit{succ}(x_{1}),y_{2}\in\mathit{succ}(x_{2})$ determine constants $c_{y_{1},y_{2}}\in[0,1]$ – which may equal $0$ – that satisfy the following inequalities for some coupling $C$ of the successor sets:

\sum_{y_{1},y_{2}}C(y_{1},y_{2})\cdot c_{y_{1},y_{2}}>c\qquad\qquad d_{k-1}(y_% {1},y_{2})>c_{y_{1},y_{2}}\mbox{ or }c_{y_{1},y_{2}}=0.

We also have to include inequalities characterizing pseudometrics (reflexivity, symmetry, triangle inequality) so that we obtain a pseudometric, ensuring that the coupling-based definition coincides with the Kantorovich lifting and we obtain a valid strategy.

Then collect all basis elements $d_{y_{1},y_{2}}^{c_{y_{1},y_{2}}}$ (for $c_{y_{1},y_{2}}\neq 0$ ) as the finitary strategy $S_{p,\operatorname{\mathsf{beh}}}^{\exists}(d_{x_{1},x_{2}}^{c})$ .

This is reminiscent of the game introduced in [41] where couplings are used as policies explaining the distance.

Auxiliary functions (cf. Sct. 5.1): Let $b=d_{x_{1},x_{2}}^{c}$ and $A\subseteq[0,1]^{X}$ . We define:

$\blacksquare$

$W_{p}(b,A)=f$ , where $f\in A$ such that $|f(x_{1})-f(x_{2})|>c$ .
$\blacksquare$

$W_{p,\operatorname{\mathsf{log}}}(b,A)$ : if $\ell(x_{1})\neq\ell(x_{2})$ choose $[a]\in\operatorname{\mathsf{log}}(A)$ . Otherwise, since

$c<\operatorname{\mathsf{beh}}(\bigsqcup\alpha[A])(x_{1},x_{2})=\mathcal{K}(% \bigsqcup\alpha[A])(x_{1},x_{2})=\alpha\circ\bigcirc\circ\gamma(\bigsqcup% \alpha[A])(x_{1},x_{2})$

there exists a price function $f$ that is non-expansive wrt. $\bigsqcup\alpha[A]$ and $|\bigcirc f(x_{1})-\bigcirc f(x_{2})|>c$ . Now approximate $f$ from $A$ by using the logical operators and apply $\bigcirc$ (for more details see [39, 24, 29]) to obtain a logical formula in $\operatorname{\mathsf{log}}(\bigsqcup A)$ that witnesses that the distance of $x_{1},x_{2}$ is at least $c$ .

Combined this gives us a construction similar to [29]. There the price function is chosen directly while we first determine a strategy. In [29] it has been observed that one can not always construct a formula that witnesses the exact distance between $x_{1},x_{2}$ . This problem is avoided here, since we are interested in certifying strict lower bounds.

Dual Case.

Basis: In the dual case we choose a meet basis for $\mathit{Dist}(X)$ that contains all distance functions $\dot{d}_{x_{1},x_{2}}^{c}$ (for $x_{1}\neq x_{2}$ , $c<1$ ) that are defined as

\dot{d}_{x_{1},x_{2}}^{c}(y_{1},y_{2})=\begin{cases}c&\text{if $(x_{1},x_{2})=% (y_{1},y_{2})$}\\ 1&\text{otherwise}\end{cases}

It can be shown that $\operatorname{\mathsf{beh}}$ is co-proper wrt. the basis.

Strategy computation (cf. Prop. 12): The computation of the strategy $S_{d,\operatorname{\mathsf{beh}}}^{\forall}(\dot{d}_{x_{1},x_{2}}^{c})$ in the dual case works analogously to the primal case. This is due to the fact that $d_{x_{1},x_{2}}^{c}\ll\mu\operatorname{\mathsf{beh}}$ iff $\mu\operatorname{\mathsf{beh}}\not\sqsubseteq\dot{d}_{x_{1},x_{2}}^{c}$ and the orders $\ll$ , $\operatorname{\triangleleft\!\triangleleft}$ coincide (on elements different from $0,1$ ). Furthermore the requirement that for all moves $d$ of $\exists$ with $\operatorname{\mathsf{beh}}(d)\sqsubseteq b$ there exists a basis element $\dot{b^{\prime}}$ of the chosen finitary strategy such that $d\operatorname{\triangleleft\!\triangleleft}\dot{b^{\prime}}$ can also be ensured via the inequality involving couplings as above.

Auxiliary functions (cf. Sct. 5.1): The auxiliary functions $W_{d}$ , $W_{d,\operatorname{\mathsf{log}}}$ can also be defined analogously to the primal case. In addition:

$\blacksquare$

$Z(\dot{e},\dot{d})$ : $\dot{e},\dot{d}\in\mathit{Dist}(X)$ with $\dot{e}\not\sqsubseteq\dot{d}$ , i.e., there exists $x_{1},x_{2}\in X$ with $\dot{e}(x_{1},x_{2})>\dot{d}(x_{1},x_{2})$ . In this case choose $c$ such that $\dot{e}(x_{1},x_{2})>c>\dot{d}(x_{1},x_{2})$ and return $\dot{d}_{x_{1},x_{2}}^{c}$ .

6.2 Termination Probabilities in Markov Chains

In this section, we consider unlabelled Markov chains [21, 1] and witness termination probabilities of states. We fix a Markov chain $(X,T,\delta)$ which has a finite state space $X$ , a subset of terminal states $T\subseteq X$ (which do not have outgoing transitions) and a probabilistic transition function $\delta\colon X\backslash T\to\mathcal{D}(X)$ , where $\mathcal{D}(X)$ is a set of discrete probability distributions. The termination probability of a state $x\in X$ is the probability that a run starting from $x$ will eventually terminate in a state in $T$ .

Lattices, Functions and Galois Connections

Behaviour: On the behaviour side we use the lattice $\mathbb{B}=[0,1]^{X}$ , with function $\operatorname{\mathsf{beh}}\colon\mathbb{B}\to\mathbb{B}$ defined as:

\operatorname{\mathsf{beh}}(f)(x)=\begin{cases}1&\text{if $x\in T$}\\ \sum_{y\in X}\delta(x)(y)\cdot f(y)&\text{otherwise}.\end{cases}

This function is clearly monotone and its least fixpoint $\mu\operatorname{\mathsf{beh}}$ assigns to each state its termination probability.

Logic: Witnesses are trees, where a tree is either a terminal node $t\in T$ , or, of the form $x\rightarrow\mathit{Tr}_{1},\dots,\mathit{Tr}_{k}$ for $x\in X\setminus T$ and where the $\mathit{Tr}_{i}$ are trees. Every tree has a root, defined by $\mathit{root}(t)=t$ , for $t\in T$ , and $\mathit{root}(x\rightarrow\mathit{Tr}_{1},\dots,\mathit{Tr}_{k})=x$ . We require that in a tree $x\rightarrow\mathit{Tr}_{1},\dots,\mathit{Tr}_{k}$ the children $\mathit{Tr}_{1},\dots,\mathit{Tr}_{k}$ all have different roots. The set of all trees will be denoted as $\mathit{Trees}$ . For a set of trees $\mathcal{T}\subseteq\mathit{Trees}$ , we write $\mathcal{T}_{x}=\{\mathit{Tr}\in\mathcal{T}\mid\mathit{root}(\mathit{Tr})=x\}$ . The degree of a tree corresponds to its height, where the height of $t\in T$ is $1$ .

We consider a map $\mathit{pt}:\mathit{Trees}\rightarrow[0,1]$ , where $\mathit{pt}(\mathit{Tr})$ under-estimates the termination probability from $\mathit{root}(\mathit{Tr})$ , based on the paths in $\mathit{Tr}$ .

\mathit{pt}(\mathit{Tr})=\begin{cases}1&\mbox{if $\mathit{Tr}=t\in T$}\\ \sum_{i=1}^{k}\delta(x)(\mathit{root}(\mathit{Tr}_{i}))\cdot pt(\mathit{Tr}_{i% })&\mbox{if $\mathit{Tr}=x\rightarrow\mathit{Tr}_{1},\dots,\mathit{Tr}_{k}$}% \end{cases}

On the logic side we use the lattice $\mathbb{L}=\mathcal{P}(\mathit{Trees})$ (with inclusion order) with functions $\operatorname{\mathsf{log}}\colon\mathbb{L}\rightarrow\mathbb{L}$ :

\operatorname{\mathsf{log}}(\mathcal{T})=\{x\rightarrow\mathit{Tr}_{1},\dots,% \mathit{Tr}_{k}\mid x\in X\backslash T,\mathit{Tr}_{i}\in\mathcal{T},\mathit{% root}(\mathit{Tr}_{i})\not=\mathit{root}(\mathit{Tr}_{j})\text{ for }i\not=j\}\cup\{t\in T\},

the least fixpoint of which is the set of all trees. Since the trees are finitely branching, it can easily be seen that $\operatorname{\mathsf{log}}$ is continuous, the same holds for $\operatorname{\mathsf{beh}}$ .

Galois Connection: The Galois connection is given as follows:

\alpha(\mathcal{T})=\lambda x.\sup_{\mathit{Tr}\in\mathcal{T}_{x}}\mathit{pt}(% \mathit{Tr})\qquad\gamma(f)=\{\mathit{Tr}\in\emph{Trees}\mid\mathit{pt}(% \mathit{Tr})\leq f(\mathit{root}(\mathit{Tr}))\}.

Here, $\alpha$ maps a set of trees $\mathcal{T}$ to a function that provides a lower bound for the termination probability of a node $x$ , based on the trees in $\mathcal{T}_{x}$ . On the other hand, $\gamma$ maps a function to all trees $\mathit{Tr}$ which induce a lower value for the respective assignment.

We now construct witnesses that certify termination probabilities for the primal and for the dual case.

Primal case

Basis: for the join basis of $\mathbb{L}$ we take the singletons $\{\mathit{Tr}\}$ , where $\mathit{Tr}\in\mathit{Trees}$ is a tree. For the join basis of $\mathbb{B}$ we take all functions $f^{c}_{x}$ (for $x\in X$ , $c>0$ ) where

f^{c}_{x}(y)=\begin{cases}c&\text{ if }x=y\\ 0&\text{otherwise}\end{cases}

Strategy computation (cf. Prop. 12): let $k=\operatorname{\mathsf{deg}}(f_{x}^{c})$ . If $k=1$ , $\exists$ plays the empty set. Otherwise we have to solve the following inequalities (where $d_{k}=\operatorname{\mathsf{beh}}^{k}(0)$ ) in order to obtain values $c_{i}\in[0,1]$ for the successors $x_{1},\dots,x_{n}$ of $x$ :

\sum_{i=1}^{n}\delta(x)(x_{i})\cdot c_{i}>c\qquad\qquad d_{k-1}(x_{i})>c_{i}% \mbox{ or }c_{i}=0

Then collect all such $f_{x_{i}}^{c_{i}}$ where $c_{i}\neq 0$ as finitary strategy $S_{p,\operatorname{\mathsf{beh}}}^{\exists}(f_{x}^{c})$ .

Auxiliary functions (cf. Sct. 5.1): Let $b=f_{x}^{c}$ and $A\subseteq\mathit{Trees}$ . We define:

$\blacksquare$

$W_{p}(b,A)=\mathit{Tr}$ where $\mathit{Tr}\in A$ is a tree such that $\mathit{pt}(\mathit{Tr})>c$ .
$\blacksquare$

$W_{p,\operatorname{\mathsf{log}}}(b,A)$ : if $A=\emptyset$ it must hold that $x\in T$ , hence choose $x$ . Otherwise choose $\{\mathit{Tr}_{1},\dots,\mathit{Tr}_{n}\}\subseteq A$ such that each tree $\mathit{Tr}_{i}$ witnesses a maximal termination probability for one of the successors of $x$ and all trees have different roots. Then return $x\to\mathit{Tr}_{1},\dots,\mathit{Tr}_{n}$ .

For a concrete example of the computation of the strategy and how to obtain the witness from this strategy, we refer to Example 23 in the appendix.

Dual Case

Basis: For the dual case, a meet basis of $\mathbb{L}$ is given by the co-singletons $\overline{\{\mathit{Tr}\}}=\mathit{Trees}\backslash\{\mathit{Tr}\}$ , and for $\mathbb{B}$ , we define the basis elements to be all functions $\dot{f}^{c}_{x}$ ( $x\in X$ , $c<1$ ) where

\dot{f}^{c}_{x}(y)=\begin{cases}c&\text{if }x=y\\ 1&\text{otherwise}\end{cases}

Strategy computation (cf. Prop. 12): $S_{d,\operatorname{\mathsf{beh}}}^{\exists}(\dot{f}^{c}_{x})$ is determined analogous to the primal case.

Auxiliary functions (cf. Sct. 5.1): The auxiliary functions $W_{d}$ , $W_{d,\operatorname{\mathsf{log}}}$ can also be defined analogously to the dual case. In addition:

$\blacksquare$

$Z(\dot{e},\dot{d})$ : $\dot{e},\dot{d}\in[0,1]^{X}$ with $\dot{e}\not\sqsubseteq\dot{d}$ , i.e., there exists $x\in X$ with $\dot{e}(x)>\dot{d}(x)$ . In this case choose $c$ such that $\dot{e}(x)>c>\dot{d}(x)$ and return $\dot{b^{\prime}}=\dot{f}_{x}^{c}$ .

7 Conclusion

We have shown how to generate witnesses, generalizing the construction of distinguishing formulas [10, 29] for explaining non-bisimilarity or certifying lower bounds. We concentrated in particular on guarantees for obtaining finitary strategies, which can then be transformed into witnesses. To summarize the results, on the logic side we have both witnesses and strategies $S_{p,\operatorname{\mathsf{log}}}^{\exists}$ proving that a witness is generated by the logic function. Proposition 17 explains how to transform witnesses into a winning strategy $S_{p,\operatorname{\mathsf{beh}}}^{\exists}$ of the primal way-below game, while Theorem 18 states the other direction. Analogously for strategies $S_{d,\operatorname{\mathsf{beh}}}^{\forall}$ of the dual game, where the connection arises from Proposition 20 and Theorem 21. Existence of finitary strategies in the various games is guaranteed by Propositions 12 and 15.

In order to obtain formal computability results, it would in addition be necessary to make assumptions on the decidability of the order relation, computability of joins and auxiliary functions, etc. [33].

Instantiating this framework to the case of bisimilarity yields a game reminiscent of the bisimulation game in [35] and a witness construction similar to [10]. We also rediscover a known construction for behavioural metrics and study a new example in the context of Markov chains.

Another general framework for bisimilarity and behavioural metrics is coalgebra [30] in which sound and complete logics have been studied [32, 28], giving rise to a Hennessy-Milner theorem. While there is work on the construction of distinguishing formulas in a qualitative coalgebraic setting [25], there is – to the best of our knowledge – no general construction in the quantitative coalgebraic case. Furthermore our framework offers the flexibility of arbitrary lattices.

The current work could be extended in several directions, such as exploring characteristic formulas (a characteristic formula of a given state characterizes all states that are in a preorder relation to the original state [34]) instead of distinguishing formulas or studying the use of the dual game on the logic side.

We also plan to investigate further examples, where witnesses generated from the strategies of the primal and dual game might be completely different. Potential application areas are dataflow analysis and abstract interpretation where least and greatest fixpoints play a major role [12, 27].

Furthermore we are interested in the connection to the codensity game [23]. This game uses predicates in the “behaviour game” and might give rise to games that are simultaneously played on both lattices. In general we believe that the lattice-based approach can be used to classify and categorize various types of behavioural games that have been presented in the literature, such as [35, 41, 17, 15] for concrete types of transition systems and [24, 6, 18, 23] in the coalgebraic setting.

In addition we will further investigate the connection to apartness [19] in particular to [38]. In the latter paper proof systems are used to obtain witnesses for lower bounds with the connection to games mentioned as future work.

References

[1] Christel Baier and Joost-Pieter Katoen. Principles of Model Checking. MIT Press, 2008.
[2] Paolo Baldan, Filippo Bonchi, Henning Kerstan, and Barbara König. Coalgebraic behavioral metrics. Logical Methods in Computer Science, 14(3), 2018. Selected Papers of the 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). doi:10.23638/LMCS-14(3:20)2018.
[3] Paolo Baldan, Barbara König, Christina Mika-Michalski, and Tommaso Padoan. Fixpoint games on continuous lattices. Proc. ACM Program. Lang., 3(POPL):26:1–26:29, 2019. doi:10.1145/3290339.
[4] A. Baltag. Truth-as-simulation: Towards a coalgebraic perspective on logic and games. Technical Report SEN-R9923, Centrum voor Wiskunde en Informatica (CWI), November 1999.
[5] Richard Bellman. A Markovian decision process. Journal of Mathematics and Mechanics, 6(5):679–684, 1957.
[6] Harsh Beohar, Chase Ford, Barbara König, Stefan Milius, and Lutz Schröder. Graded monads and behavioural equivalence games. In Proc. of LICS ’22. IEEE, 2022.
[7] Harsh Beohar, Sebastian Gurke, Barbara König, and Karla Messing. Hennessy-Milner Theorems via Galois Connections. In 31st EACSL Annual Conference on Computer Science Logic, CSL 2023, February 13-16, 2023, Warsaw, Poland, volume 252 of LIPIcs, pages 12:1–12:18. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2023. doi:10.4230/LIPIcs.CSL.2023.12.
[8] Julian Bradfield and Igor Walukiewicz. The mu-calculus and model checking. In Edmund M. Clarke, Thomas A. Henzinger, Helmut Veith, and Roderick Bloem, editors, Handbook of Model Checking, pages 871–919. Springer, 2018. doi:10.1007/978-3-319-10575-8_26.
[9] Valentina Castiglioni, Daniel Gebler, and Simone Tini. Logical characterization of bisimulation metrics. In Proc. of QAPL ’16, 2016. EPTCS 227.
[10] Rance Cleaveland. On automatically explaining bisimulation inequivalence. In Proc. of CAV ’90, pages 364–372. Springer, 1990. LNCS 531. doi:10.1007/BFB0023750.
[11] Anne Condon. The complexity of stochastic games. Information and Computation, 96(2):203–224, 1992. doi:10.1016/0890-5401(92)90048-K.
[12] Patrick Cousot and Radhia Cousot. Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In Proc. of POPL ’77 (Los Angeles, California), pages 238–252. ACM, 1977. doi:10.1145/512950.512973.
[13] Luca de Alfaro, Marco Faella, and Mariëlle Stoelinga. Linear and branching system metrics. IEEE Transactions on Software Engineering, 35(2):258–273, 2009. doi:10.1109/TSE.2008.106.
[14] Josée Desharnais, Vineet Gupta, Radha Jagadeesan, and Prakash Panangaden. Metrics for labelled Markov processes. Theoretical Computer Science, 318:323–354, 2004. doi:10.1016/J.TCS.2003.09.013.
[15] Josée Desharnais, Fran cois Laviolette, and Mathieu Tracol. Approximate analysis of probabilistic processes: Logic, simulation and games. In Proc. of QEST ’08, pages 264–273. IEEE, 2008.
[16] Uli Fahrenberg, Axel Legay, and Claus Thrane. The quantitative linear-time–branching-time spectrum. In Proc. of FSTTCS ’11, volume 13 of LIPIcs, pages 103–114. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2011. doi:10.4230/LIPIcs.FSTTCS.2011.103.
[17] Nathanaël Fijalkow, Bartek Klin, and Prakash Panangaden. Expressiveness of Probabilistic Modal Logics, Revisited. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 105:1–105:12. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2017. doi:10.4230/LIPIcs.ICALP.2017.105.
[18] Jonas Forster, Lutz Schröder, and Paul Wild. Conformance games for graded semantics. In Proc. of LICS ’25, pages 555–567. IEEE, 2025. doi:10.1109/LICS65433.2025.00048.
[19] Herman Geuvers and Bart Jacobs. Relating apartness and bisimulation. Logical Methods in Computer Science, 17(3):15:1–15:35, 2021. doi:10.46298/LMCS-17(3:15)2021.
[20] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer Berlin Heidelberg, 1980.
[21] Charles Grinstead and Laurie Snell. Markov chains. In Introduction to Probability, chapter 11, pages 405–470. American Mathematical Society, second edition, 1997.
[22] Matthew Hennessy and Robin Milner. Algebraic laws for nondeterminism and concurrency. Journal of the ACM, 32:137–161, 1985. doi:10.1145/2455.2460.
[23] Yuichi Komorida, Shin-ya Katsumata, Nick Hu, Bartek Klin, and Ichiro Hasuo. Codensity games for bisimilarity. In Proc. of LICS ’19, pages 1–13. ACM, 2019. URL: https://dl.acm.org/doi/10.5555/3470152.3470184.
[24] Barbara König and Christina Mika-Michalski. (Metric) bisimulation games and real-valued modal logics for coalgebras. In Proc. of CONCUR ’18, volume 118 of LIPIcs, pages 37:1–37:17. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2018. doi:10.4230/LIPIcs.CONCUR.2018.37.
[25] Barbara König, Christina Mika-Michalski, and Lutz Schröder. Explaining non-bisimilarity in a coalgebraic approach: Games and distinguishing formulas. In Proc. of CMCS ’20, pages 133–154. Springer, 2020. LNCS 12094. doi:10.1007/978-3-030-57201-3_8.
[26] Barbara König and Karla Messing. Witnesses for fixpoint games on lattices, 2026. doi:10.48550/arXiv.2603.11908.
[27] Flemming Nielson, Hanne Riis Nielson, and Chris Hankin. Principles of Program Analysis. Springer-Verlag, 1999.
[28] Dirk Pattinson. Coalgebraic modal logic: soundness, completeness and decidability of local consequence. Theoretical Computer Science, 309(1):177–193, 2003. doi:10.1016/S0304-3975(03)00201-9.
[29] Amgad Rady and Franck van Breugel. Explainability of probabilistic bisimilarity distances for labelled markov chains. In Orna Kupferman and Pawel Sobocinski, editors, Proc. of FoSSaCS ’23, pages 285–307. Springer, 2023. LNCS 13992. doi:10.1007/978-3-031-30829-1_14.
[30] J.J.M.M. Rutten. Universal coalgebra: a theory of systems. Theoretical Computer Science, 249:3–80, 2000. doi:10.1016/S0304-3975(00)00056-6.
[31] Davide Sangiorgi. Introduction to Bisimulation and Coinduction. Cambridge University Press, 2011.
[32] Lutz Schröder. Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical Computer Science, 390:230–247, 2008. doi:10.1016/J.TCS.2007.09.023.
[33] M.B. Smyth. Effectively given domains. Theoretical Computer Science, 5:257–274, 1977. doi:10.1016/0304-3975(77)90045-7.
[34] Bernhard Steffen. Characteristic formulae. In Proc. of ICALP ’89, pages 723–732. Springer, 1989. LNCS. doi:10.1007/BFB0035794.
[35] Colin Stirling. Bisimulation, modal logic and model checking games. Logic Journal of the IGPL, 7(1):103–124, 1999. doi:10.1093/JIGPAL/7.1.103.
[36] Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5:285–309, 1955.
[37] Ruben Turkenburg, Harsh Beohar, Clemens Kupke, and Jurriaan Rot. Proving behavioural apartness. In Barbara König and Henning Urbat, editors, Proc. of CMCS ’20, pages 156–173. Springer, 2024. LNCS 14617. doi:10.1007/978-3-031-66438-0_8.
[38] Ruben Turkenburg, Harsh Beohar, Franck van Breugel, Clemens Kupke, and Jurriaan Rot. Constructing witnesses for lower bounds on behavioural distances. In Proc. of CSL ’26, volume 363 of LIPIcs, pages 25:1–25:22. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026. doi:10.4230/LIPIcs.CSL.2026.25.
[39] Franck van Breugel and James Worrell. A behavioural pseudometric for probabilistic transition systems. Theoretical Computer Science, 331:115–142, 2005. doi:10.1016/J.TCS.2004.09.035.
[40] Cédric Villani. Optimal Transport – Old and New, volume 338 of A Series of Comprehensive Studies in Mathematics. Springer, 2009.
[41] Emily Vlasman, Anto Nanah Ji, James Worrell, and Franck van Breugel. Explainability is a game for probabilistic bisimilarity distances. In Proc. of CONCUR ’25, volume 348 of LIPIcs, pages 36:1–36:20. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2025. doi:10.4230/LIPIcs.CONCUR.2025.36.

Appendix A Example: Termination Probability for Markov Chains

Example 23.

Consider the Markov chain below with states $x, y$ , where $y$ is a terminating state, and $x$ can transition to both $x$ and $y$ with probability $\nicefrac{{1}}{{2}}$ . The smallest fixpoint of the behaviour function $\mu\operatorname{\mathsf{beh}}$ maps $x$ and $y$ to one, i.e. both states terminate almost surely. We want to witness that the probability for $x$ is strictly greater than $\nicefrac{{1}}{{2}}$ .

We will denote a function $f:x\mapsto a,y\mapsto b$ by $(a,b)$ . Basis elements are tuples $f^{a}_{x}=(a,0)$ or $f^{b}_{y}=(0,b)$ , where $a,b>0$ .

Hence we start with the basis element $f^{\nicefrac{{1}}{{2}}}_{x}$ and observe that $\operatorname{\mathsf{deg}}(f^{\nicefrac{{1}}{{2}}}_{x})=3$ , i.e. $i=3$ is the first index for which $beh^{i}((0,0))(x)>\nicefrac{{1}}{{2}}$ holds.

Strategy computation: The computation of the strategies involves solving the following inequalities, and thereby distributes the required probability of $\nicefrac{{1}}{{2}}$ to the successors of $x$ :

	$\displaystyle\delta(x)(x)\cdot c_{1}+\delta(x)(y)\cdot c_{2}=\nicefrac{{1}}{{2% }}\cdot c_{1}+\nicefrac{{1}}{{2}}\cdot c_{2}>\nicefrac{{1}}{{2}}$
	$\displaystyle\operatorname{\mathsf{beh}}^{2}((0,0))(x)=\nicefrac{{1}}{{2}}>c_{1}$
	$\displaystyle\operatorname{\mathsf{beh}}^{2}((0,0))(y)=1>c_{2}$

Player $\exists$ can choose any $c_{1}=\nicefrac{{1}}{{2}}-\varepsilon$ , $c_{2}=1-\delta$ ( $\varepsilon,\delta>0$ ) that satisfy the first inequality. Setting $c_{1}=0$ or $c_{2}=0$ is not an option, as then the first inequality is not satisfied. Assume that $\exists$ chooses $c_{1}=\nicefrac{{1}}{{4}}$ and $c_{2}=\nicefrac{{7}}{{8}}$ . Hence the (finitary) strategy of $\exists$ is to play $d=\bigsqcup F$ with $F=\{f_{x}^{\nicefrac{{1}}{{4}}},f_{y}^{\nicefrac{{7}}{{8}}}\}$ . Note that $\operatorname{\mathsf{deg}}(f^{\nicefrac{{1}}{{4}}}_{x})=2$ and $\operatorname{\mathsf{deg}}(f^{\nicefrac{{7}}{{8}}}_{y})=1$ .

Now, the probability of $c_{1}=\nicefrac{{1}}{{4}}$ needs to be distributed to the successors of $x$ . Here we assume that $\exists$ chooses values $0,\nicefrac{{7}}{{8}}$ , resulting in the strategy $F^{\prime}=\{f_{y}^{\nicefrac{{7}}{{8}}}\}$ . Finally, for $f_{y}^{\nicefrac{{7}}{{8}}}$ it is sufficient to play the bottom element $(0,0)$ , i.e., the empty strategy.

Playing the game: Starting with $f^{\nicefrac{{1}}{{2}}}_{x}\ll\mu\operatorname{\mathsf{beh}}$ , player $\exists$ plays $\bigsqcup F=(\nicefrac{{1}}{{4}},\nicefrac{{7}}{{8}})$ as determined above. Now $\forall$ has to answer either with $(\nicefrac{{1}}{{4}}-\varepsilon,0)$ or $(0,\nicefrac{{7}}{{8}}-\delta)$ . As detailed above, $\exists$ can win in the latter case with the empty strategy, in the former case $\exists$ plays $\bigsqcup F^{\prime}=(0,\nicefrac{{7}}{{8}})$ , winning the game in the next step, since by assumption $\forall$ can not answer with $\bot=(0,0)$ .

Constructing the witness: Now we construct witnesses recursively from the strategy. More concretely, we obtain:

	$\displaystyle\mathit{wit}(f_{y}^{\nicefrac{{7}}{{8}}})$	$\displaystyle=W_{p,\operatorname{\mathsf{log}}}(f_{y}^{\nicefrac{{7}}{{8}}},% \emptyset)=y,\qquad\text{(base case)}$
	$\displaystyle\mathit{wit}(f_{x}^{\nicefrac{{1}}{{4}}})$	$\displaystyle=W_{p,\operatorname{\mathsf{log}}}(f_{x}^{\nicefrac{{1}}{{4}}},% \mathit{wit}[\{f_{y}^{\nicefrac{{7}}{{8}}}\}])=W_{p,\operatorname{\mathsf{log}% }}(f_{x}^{\nicefrac{{1}}{{4}}},\{y\})=x\rightarrow y$
	$\displaystyle\mathit{wit}(f_{x}^{\nicefrac{{1}}{{2}}})$	$\displaystyle=W_{p,\operatorname{\mathsf{log}}}(f_{x}^{\nicefrac{{1}}{{2}}},% \mathit{wit}[\{f_{x}^{\nicefrac{{1}}{{4}}},f_{y}^{\nicefrac{{7}}{{8}}}\}])=W_{% p,\operatorname{\mathsf{log}}}(f_{x}^{\nicefrac{{1}}{{2}}},\{x\rightarrow y,y% \})=x\rightarrow(x\rightarrow y),y$

Thus, the termination probability of $x$ being greater than $\nicefrac{{1}}{{2}}$ is witnessed by the tree $x\rightarrow(x\rightarrow y),y$ , where the subtree $y$ witnesses the termination probability of $y$ , which is greater than $\nicefrac{{7}}{{8}}$ , and the subtree $x\rightarrow y$ witnesses the termination probability of $x$ of at least $\nicefrac{{1}}{{4}}$ . In fact, the resulting tree even gives the value $\nicefrac{{3}}{{4}}$ .

[bib.bib1] [1] Christel Baier and Joost-Pieter Katoen. Principles of Model Checking. MIT Press, 2008.

[bib.bib2] [2] Paolo Baldan, Filippo Bonchi, Henning Kerstan, and Barbara König. Coalgebraic behavioral metrics. Logical Methods in Computer Science, 14(3), 2018. Selected Papers of the 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). doi:10.23638/LMCS-14(3:20)2018.

[bib.bib3] [3] Paolo Baldan, Barbara König, Christina Mika-Michalski, and Tommaso Padoan. Fixpoint games on continuous lattices. Proc. ACM Program. Lang., 3(POPL):26:1–26:29, 2019. doi:10.1145/3290339.

[bib.bib4] [4] A. Baltag. Truth-as-simulation: Towards a coalgebraic perspective on logic and games. Technical Report SEN-R9923, Centrum voor Wiskunde en Informatica (CWI), November 1999.

[bib.bib5] [5] Richard Bellman. A Markovian decision process. Journal of Mathematics and Mechanics, 6(5):679–684, 1957.

[bib.bib6] [6] Harsh Beohar, Chase Ford, Barbara König, Stefan Milius, and Lutz Schröder. Graded monads and behavioural equivalence games. In Proc. of LICS ’22. IEEE, 2022.

[bib.bib7] [7] Harsh Beohar, Sebastian Gurke, Barbara König, and Karla Messing. Hennessy-Milner Theorems via Galois Connections. In 31st EACSL Annual Conference on Computer Science Logic, CSL 2023, February 13-16, 2023, Warsaw, Poland, volume 252 of LIPIcs, pages 12:1–12:18. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2023. doi:10.4230/LIPIcs.CSL.2023.12.

[bib.bib8] [8] Julian Bradfield and Igor Walukiewicz. The mu-calculus and model checking. In Edmund M. Clarke, Thomas A. Henzinger, Helmut Veith, and Roderick Bloem, editors, Handbook of Model Checking, pages 871–919. Springer, 2018. doi:10.1007/978-3-319-10575-8_26.

[bib.bib9] [9] Valentina Castiglioni, Daniel Gebler, and Simone Tini. Logical characterization of bisimulation metrics. In Proc. of QAPL ’16, 2016. EPTCS 227.

[bib.bib10] [10] Rance Cleaveland. On automatically explaining bisimulation inequivalence. In Proc. of CAV ’90, pages 364–372. Springer, 1990. LNCS 531. doi:10.1007/BFB0023750.

[bib.bib11] [11] Anne Condon. The complexity of stochastic games. Information and Computation, 96(2):203–224, 1992. doi:10.1016/0890-5401(92)90048-K.

[bib.bib12] [12] Patrick Cousot and Radhia Cousot. Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In Proc. of POPL ’77 (Los Angeles, California), pages 238–252. ACM, 1977. doi:10.1145/512950.512973.

[bib.bib13] [13] Luca de Alfaro, Marco Faella, and Mariëlle Stoelinga. Linear and branching system metrics. IEEE Transactions on Software Engineering, 35(2):258–273, 2009. doi:10.1109/TSE.2008.106.

[bib.bib14] [14] Josée Desharnais, Vineet Gupta, Radha Jagadeesan, and Prakash Panangaden. Metrics for labelled Markov processes. Theoretical Computer Science, 318:323–354, 2004. doi:10.1016/J.TCS.2003.09.013.

[bib.bib15] [15] Josée Desharnais, Fran cois Laviolette, and Mathieu Tracol. Approximate analysis of probabilistic processes: Logic, simulation and games. In Proc. of QEST ’08, pages 264–273. IEEE, 2008.

[bib.bib16] [16] Uli Fahrenberg, Axel Legay, and Claus Thrane. The quantitative linear-time–branching-time spectrum. In Proc. of FSTTCS ’11, volume 13 of LIPIcs, pages 103–114. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2011. doi:10.4230/LIPIcs.FSTTCS.2011.103.

[bib.bib17] [17] Nathanaël Fijalkow, Bartek Klin, and Prakash Panangaden. Expressiveness of Probabilistic Modal Logics, Revisited. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 105:1–105:12. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2017. doi:10.4230/LIPIcs.ICALP.2017.105.

[bib.bib18] [18] Jonas Forster, Lutz Schröder, and Paul Wild. Conformance games for graded semantics. In Proc. of LICS ’25, pages 555–567. IEEE, 2025. doi:10.1109/LICS65433.2025.00048.

[bib.bib19] [19] Herman Geuvers and Bart Jacobs. Relating apartness and bisimulation. Logical Methods in Computer Science, 17(3):15:1–15:35, 2021. doi:10.46298/LMCS-17(3:15)2021.

[bib.bib20] [20] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer Berlin Heidelberg, 1980.

[bib.bib21] [21] Charles Grinstead and Laurie Snell. Markov chains. In Introduction to Probability, chapter 11, pages 405–470. American Mathematical Society, second edition, 1997.

[bib.bib22] [22] Matthew Hennessy and Robin Milner. Algebraic laws for nondeterminism and concurrency. Journal of the ACM, 32:137–161, 1985. doi:10.1145/2455.2460.

[bib.bib23] [23] Yuichi Komorida, Shin-ya Katsumata, Nick Hu, Bartek Klin, and Ichiro Hasuo. Codensity games for bisimilarity. In Proc. of LICS ’19, pages 1–13. ACM, 2019. URL: https://dl.acm.org/doi/10.5555/3470152.3470184.

[bib.bib24] [24] Barbara König and Christina Mika-Michalski. (Metric) bisimulation games and real-valued modal logics for coalgebras. In Proc. of CONCUR ’18, volume 118 of LIPIcs, pages 37:1–37:17. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2018. doi:10.4230/LIPIcs.CONCUR.2018.37.

[bib.bib25] [25] Barbara König, Christina Mika-Michalski, and Lutz Schröder. Explaining non-bisimilarity in a coalgebraic approach: Games and distinguishing formulas. In Proc. of CMCS ’20, pages 133–154. Springer, 2020. LNCS 12094. doi:10.1007/978-3-030-57201-3_8.

[bib.bib26] [26] Barbara König and Karla Messing. Witnesses for fixpoint games on lattices, 2026. doi:10.48550/arXiv.2603.11908.

[bib.bib27] [27] Flemming Nielson, Hanne Riis Nielson, and Chris Hankin. Principles of Program Analysis. Springer-Verlag, 1999.

[bib.bib28] [28] Dirk Pattinson. Coalgebraic modal logic: soundness, completeness and decidability of local consequence. Theoretical Computer Science, 309(1):177–193, 2003. doi:10.1016/S0304-3975(03)00201-9.

[bib.bib29] [29] Amgad Rady and Franck van Breugel. Explainability of probabilistic bisimilarity distances for labelled markov chains. In Orna Kupferman and Pawel Sobocinski, editors, Proc. of FoSSaCS ’23, pages 285–307. Springer, 2023. LNCS 13992. doi:10.1007/978-3-031-30829-1_14.

[bib.bib30] [30] J.J.M.M. Rutten. Universal coalgebra: a theory of systems. Theoretical Computer Science, 249:3–80, 2000. doi:10.1016/S0304-3975(00)00056-6.

[bib.bib31] [31] Davide Sangiorgi. Introduction to Bisimulation and Coinduction. Cambridge University Press, 2011.

[bib.bib32] [32] Lutz Schröder. Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical Computer Science, 390:230–247, 2008. doi:10.1016/J.TCS.2007.09.023.

[bib.bib33] [33] M.B. Smyth. Effectively given domains. Theoretical Computer Science, 5:257–274, 1977. doi:10.1016/0304-3975(77)90045-7.

[bib.bib34] [34] Bernhard Steffen. Characteristic formulae. In Proc. of ICALP ’89, pages 723–732. Springer, 1989. LNCS. doi:10.1007/BFB0035794.

[bib.bib35] [35] Colin Stirling. Bisimulation, modal logic and model checking games. Logic Journal of the IGPL, 7(1):103–124, 1999. doi:10.1093/JIGPAL/7.1.103.

[bib.bib36] [36] Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5:285–309, 1955.

[bib.bib37] [37] Ruben Turkenburg, Harsh Beohar, Clemens Kupke, and Jurriaan Rot. Proving behavioural apartness. In Barbara König and Henning Urbat, editors, Proc. of CMCS ’20, pages 156–173. Springer, 2024. LNCS 14617. doi:10.1007/978-3-031-66438-0_8.

[bib.bib38] [38] Ruben Turkenburg, Harsh Beohar, Franck van Breugel, Clemens Kupke, and Jurriaan Rot. Constructing witnesses for lower bounds on behavioural distances. In Proc. of CSL ’26, volume 363 of LIPIcs, pages 25:1–25:22. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2026. doi:10.4230/LIPIcs.CSL.2026.25.

[bib.bib39] [39] Franck van Breugel and James Worrell. A behavioural pseudometric for probabilistic transition systems. Theoretical Computer Science, 331:115–142, 2005. doi:10.1016/J.TCS.2004.09.035.

[bib.bib40] [40] Cédric Villani. Optimal Transport – Old and New, volume 338 of A Series of Comprehensive Studies in Mathematics. Springer, 2009.

[bib.bib41] [41] Emily Vlasman, Anto Nanah Ji, James Worrell, and Franck van Breugel. Explainability is a game for probabilistic bisimilarity distances. In Proc. of CONCUR ’25, volume 348 of LIPIcs, pages 36:1–36:20. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2025. doi:10.4230/LIPIcs.CONCUR.2025.36.

	$\displaystyle\alpha(\operatorname{\mathsf{log}}(\bigsqcup A))=\operatorname{% \mathsf{beh}}(\alpha(A))\gg b$	$\displaystyle\iff\operatorname{\mathsf{beh}}(\alpha(A))\subseteq\overline{\{(x% _{1},x_{2})\}}$
		$\displaystyle\iff(x_{1},x_{2})\not\in\operatorname{\mathsf{beh}}(\alpha(A))$