Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions

In systems involving quantitative data, such as probabilistic, fuzzy, or metric systems, behavioural distances provide a more fine-grained comparison of states than two-valued notions of behavioural equivalence or behaviour inclusion. Like in the two-valued case, the wide variation found in system types creates a need for generic methods that apply to many system types at once. Approaches of this kind are emerging within the paradigm of universal coalgebra, based either on lifting pseudometrics along set functors or on lifting general real-valued (fuzzy) relations along functors by means of fuzzy lax extensions. An immediate benefit of the latter is that they allow bounding behavioural distance by means of fuzzy (bi-)simulations that need not themselves be hemi- or pseudometrics; this is analogous to classical simulations and bisimulations, which need not be preorders or equivalence relations, respectively. The known generic pseudometric liftings, specifically the generic Kantorovich and Wasserstein liftings, both can be extended to yield fuzzy lax extensions, using the fact that both are effectively given by a choice of quantitative modalities. Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities. For nonexpansive fuzzy lax extensions, this allows for the extraction of quantitative modal logics that characterize behavioural distance, i.e. satisfy a quantitative version of the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a quantitative version of Moss' coalgebraic logic. All our results explicitly hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative simulation.


Introduction
Branching-time equivalences on reactive systems are typically governed by notions of bisimilarity [Par81,Mil89]. For systems involving quantitative data, such as transition probabilities, fuzzy truth values, or labellings in metric spaces, it is often appropriate to use more finegrained, quantitative measures of behavioural similarity, arriving at notions of behavioural distance. Distance-based approaches in particular avoid the problem that small quantitative deviations in behaviour will typically render two given systems inequivalent under two-valued notions of equivalence, losing information about their similarity. We note in passing that behavioural distances are typically pseudometrics, i.e. distinct states can have distance 0 if their behaviours are exactly equivalent.
Behavioural distances serve evident purposes in system verification, allowing as they do for a reasonable notion of a specification being satisfied up to an acceptable margin of deviation (e.g. [Gav18]). Applications have also been proposed in differential privacy [CGPX14] and conformance testing of hybrid systems [KM15]. Like their two-valued counterparts, behavioural distances have been introduced for quite a range of system types, such as various forms of probabilistic labelled transition systems or labelled Markov processes [GJS90,vBW05,Des99,DGJP04]; systems combining nondeterministic and probabilistic branching variously known as nondeterministic probabilistic transition systems [CGT16], probabilistic automata [DCPP06], and Markov decision processes [FPP04]; weighted automata [BGP17]; fuzzy transition systems [CSWC13] and fuzzy Kripke models [Fan15]; and various forms of metric transition systems [dAFS09,FLT11,FL14], which are nondeterministic transition systems with additional quantitative information, e.g. a metric on the labels and/or the states. Besides symmetric notions of behavioural distance, there are asymmetric variants, which correspond to quantitative notions of simulation, e.g. for rational- [CHR12], real- [TFL10], and lattice-weighted transition systems [PLC15].
This range of variation creates a need for unifying concepts and methods. The present work contributes to developing such a unified view within the framework of universal coalgebra, which is based on abstracting a wide range of system types (including all the mentioned ones) as set functors. Specifically, we work with a generic notion of quantitative simulation via the key notion of nonexpansive (fuzzy) lax extension of a functor. Fuzzy and quantale-valued generalizations of lax extensions have been studied in the past [Gav18,HST14]; we identify a new criterion for such lax extensions to be nonexpansive (equivalently strong in the sense of Gavazzo [Gav18]) that allows us to relate lax extensions to fuzzy logics featuring nonexpansive modalities via a Hennessy-Milner thoerem. Given a fuzzy lax extension, behavioural distance is defined as the greatest quantitative simulation; in general, behavioural distance is a hemimetric, i.e. obeys the usual axioms of a pseudometric except symmetry, or equivalently a generalized metric space in the sense of Lawvere [Law73].
For instance, on weighted transition systems with labels in a finite metric space (M, d M ) Larsen et al. [LFT11] consider a simulation distance defined as the least fixed point of the equation where 0 ≤ λ < 1 is a discount factor. We shall later see that this simulation distance arises via a nonexpansive fuzzy lax extension and thus forms an instance of this framework. For lax extensions obeying a suitable symmetry axiom, quantitative simulations are in fact quantitative bisimulations in the sense that their relational converse is also a simulation, and the induced behavioural distance is symmetric, i.e. forms a pseudometric. Existing coalgebraic approaches to behavioural pseudometrics rely on pseudometric liftings of functors [BBKK18], and in particular lift only pseudometrics; contrastingly, fuzzy lax extensions act on unrestricted quantitative relations. Hence, quantitative (bi-)simulations need not themselves be hemi-or pseudometrics, in analogy to classical bisimulations not needing to be equivalence relations, and thus may serve as small certificates for low behavioural distance. We show that two known systematic constructions of functor liftings from chosen sets of modalities, the generic Wasserstein and Kantorovich liftings, both extend to yield nonexpansive fuzzy lax extensions (it is essentially known that the Wasserstein lifting yields a fuzzy lax extension [Hof07]). As our main result, we then establish that every fuzzy lax extension of a finitary functor is a Kantorovich extension induced by a suitable set of modalities, the so-called Moss modalities. Notably, the definition of the Moss modalities involves application of the given lax extension to the quantitative elementhood relation, and hence centrally relies on lifting quantitative relations that fail to be hemi-or pseudometrics.
This result may be seen as a quantitative version of previous results asserting the existence of separating sets of two-valued modalities for finitary functors [Sch08, KL09,MV15], which allow for generic Hennessy-Milner-type theorems stating that states in finitely branching systems (coalgebras) are behaviourally equivalent iff they satisfy the same modal formulae [Pat04,Sch08]. Indeed, for nonexpansive lax extensions our main result similarly allows extracting characteristic quantitative modal logics from given behavioural hemi-or pseudometrics, where a logic is characteristic or expressive if the induced logical distance of states coincides with behavioural distance. This result may equivalently be phrased as expressiveness of a quantitative version of Moss' coalgebraic logic [Mos99], which provides a coalgebraic generalization of the classical relational ∇-modality (which e.g. underlies the a → Ψ notation used in Walukiewicz's µ-calculus completeness proof [Wal95]). We relax the standard requirement of finite branching, i.e. use of finitary functors, to an approximability condition called finitary separability, and hence in particular cover countable probabilistic branching. Moreover, we emphasize that we obtain characteristic logics also for asymmetric distances, i.e. notions of quantitative simulation.
Organization. We recall basic concepts on hemi-and pseudometrics, coalgebraic bisimilarity, and coalgebraic logic in Section 2. The central notion of (nonexpansive) fuzzy lax extension is introduced in Section 3, and the arising principle of quantitative (bi-)simulation in Section 4. The generic Kantorovich and Wasserstein liftings are discussed in Sections 5 and 6, respectively. Our central result showing that every lax extension is a Kantorovich lifting is established in Section 7. In Section 8, we show how our results amount to extracting characteristic modal logics from given nonexpansive lax extensions.
Related Work. Probabilistic quantitative characteristic modal logics go back to Desharnais et al. [DGJP04]; they relate to fragments of quantitative µ-calculi [HK97,MS17,MM97]. A further well-known class of quantitative modal logics are fuzzy modal and description logics (e.g. [Mor79,Fit91,Str98,LS08]). Van Breugel and Worrell [vBW05] prove a Hennessy-Milner theorem for quantitative probabilistic modal logic. Quantitative Hennessy-Milner-type theorems have since been established for fuzzy modal logic with Gödel semantics [Fan15], for systems combining probability and non-determinism [DDG16], and for Heyting-valued modal logics [EKN12] as introduced by Fitting [Fit91]. König and Mika-Michalski [KMM18] provide a quantitative Hennessy-Milner theorem in coalgebraic generality for the case where behavioural distance is induced by the pseudometric Kantorovich lifting defined by the same set of modalities as the logic, a result that we complement by showing that in fact all fuzzy lax extensions are Kantorovich. The assumptions of König and Mika-Michalski's theorem require that behavioural distance be approximable in ω steps. We give a sufficient criterion for this property: The predicate liftings need to be nonexpansive, and the given lax extension needs to be finitarily separable (as mentioned above). Again, we remove any assumption of symmetry, obtaining an expressiveness criterion for characteristic logics of quantitative simulations; in this sense, our work relates also to (coalgebraic and specific) results on characteristic logics for two-valued notions of similarity, e.g. [vG90,Bal00,Cîr06,KKV12,FMS21].
Fuzzy lax extensions are a quantitative version of lax extensions [MV15, Thi96, Lev11, BdBH + 91], which in turn belong to an extended strand of research on relation liftings [HJ04,Thi96,Lev11]. They appear to go back to work on monoidal topology [HST14], and have been used in work on applicative bisimulation [Gav18]; as indicated above, Hofmann [Hof07] effectively already introduces the generic Wasserstein lax extension (without using the term but proving the relevant properties, except nonexpansiveness). Our notion of nonexpansive lax extension, which is central to the connection with characteristic logics, appears to be new, but as indicated above it can be seen to relate to a condition involving the strength of the underlying functor as considered by Gavazzo [Gav18]. Our method of extracting quantitative modalities from fuzzy lax extensions generalizes the construction of two-valued Moss liftings for (two-valued) lax extensions [KL09,MV15].
This paper is an extended and revised version of a previous conference publication [WS20]. Besides containing additional discussion and full proofs, the present version generalizes the overall technical treatment including the main results to the asymmetric setting, thus covering not only quantitative notions of bisimulation but also quantitative notions of simulation.

Preliminaries
We recall basic notions on metrics, pseudometrics (where distinct points may have distance 0), and hemimetrics (where additionally distance is not required to be symmetric). Moreover, we give a brief introduction to universal coalgebra [Rut00] and the generic treatment of two-valued bisimilarity. Basic knowledge of category theory (e.g. [AHS90]) will be helpful.
Hemimetric Spaces. For the present purposes, we are interested only in bounded distance functions, and then normalize distances to lie in the unit interval. Thus, a (1-bounded) hemimetric on a set X is a function d∶ X × X → [0, 1] satisfying d(x, x) = 0 (reflexivity), and d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) for x, y, z ∈ X. If additionally d(x, y) = d(y, x) for all x, y ∈ X (symmetry), then d is a pseudometric. If moreover for all x, y ∈ X, d(x, y) = 0 implies x = y, then d is a metric. The pair (X, d) is a hemimetric space, or respectively a (pseudo-)metric space if d is a (hemi-/pseudo-)metric. We write ⊖ for truncated subtraction on the unit interval, i.e. x ⊖ y = max(x − y, 0) for x, y ∈ [0, 1]. Then Universal Coalgebra is a uniform framework for a broad range of state-based system types. It is based on encapsulating the transition type of a system as an (endo-)functor, for the present purposes on the category of sets: A functor T assigns to each set X a set T X, and to each map f ∶ X → Y a map T f ∶ T X → T Y , preserving identities and composition. We may think of T X as a parametrized datatype; e.g. the (covariant) powerset functor P assigns to each set X its powerset PX, and to f ∶ X → Y the direct image map Pf ∶ PX → PY, A ↦ f [A]; and the distribution functor D maps each set X to the set of discrete probability distributions on X. Recall that a discrete probability distribution on X is given by a probability mass function µ∶ X → [0, 1] such that ∑ x∈X µ(x) = 1 (implying that the support {x ∈ X µ(x) > 0} of µ is at most countable); we abuse µ to denote also the induced probability measure, Systems of a transition type T are then cast as T -coalgebras (A, α), consisting of a set A of states and a transition function α∶ A → T A, thought of as assigning to each state a structured collection of successors. E.g. a P-coalgebra α∶ A → PA assigns to each state a a set α(a) of successors, so is just a (non-deterministic) transition system. Similarly, a D-coalgebra assigns to each state a distribution over successor states, and thus is a probabilistic transition system or a Markov chain.
where ○ denotes the usual (applicative) composition of functions; e.g. morphisms of P-coalgebras are functional bisimulations, also known as p-morphisms or bounded morphisms.
A functor T is finitary if for each set X and each t ∈ T X, there exists a finite subset Y ⊆ X such that t = T i(t ′ ) for some t ′ ∈ T Y , where i∶ Y → X is the inclusion map (this is equivalent to the more categorically phrased condition that T preserves directed colimits). Intuitively, T is finitary if every element of T X mentions only finitely many elements of X. Every set functor T has a finitary part T ω given by E.g. P ω , the finite powerset functor, maps a set to the set of its finite subsets, and D ω , the finite distribution functor, maps a set X to the set of discrete probability distributions on X with finite support. Coalgebras for finitary functors generalize finitely branching systems, and hence feature in Hennessy-Milner type theorems, which typically fail under infinite branching.
Bisimilarity and Lax Extensions. Coalgebras come with a canonical notion of observable equivalence: States a ∈ A, b ∈ B in T -coalgebras (A, α), (B, β) are behaviourally equivalent if there exist a coalgebra (C, γ) and morphisms f ∶ (A, α) → (C, γ), g∶ (B, β) → (C, γ) such that f (a) = g(b). Behavioural equivalence can often be characterized in terms of bisimulation relations, which may provide small witnesses for behavioural equivalence of states and in particular need not form equivalence relations. The most general known way of treating bisimulation coalgebraically is via lax extensions L of the functor T , which map relations R ⊆ X × Y to LR ⊆ T X × T Y subject to a number of axioms (monotonicity, preservation of relational converse, lax preservation of composition, extension of function graphs) [MV15]; L preserves diagonals if L∆ X = ∆ T X for each set X, where for any set Y , ∆ Y denotes the diagonal {(y, y) y ∈ Y }. The Barr extension T of T [ Bar70,Trn80] is defined by where π 1 ∶ R → X and π 2 ∶ R → Y are the projections; T preserves diagonals, and is a lax extension if T preserves weak pullbacks. E.g., the Barr extension P of the powerset functor P is the well-known Egli-Milner extension, given by for R ⊆ X × Y , V ∈ P(X), W ∈ P(Y ). An L-bisimulation between T -coalgebras (A, α), (B, β) is a relation R ⊆ A × B such that (α(a), β(b)) ∈ LR for all (a, b) ∈ R; e.g. for L = P, L-bisimulations are precisely Park/Milner bisimulations on transition systems. If a lax extension L preserves diagonals, then two states are behaviourally equivalent iff they are related by some L-bisimulation [MV15]. Coalgebraic Logic serves as a generic framework for the specification of state-based systems [CKP + 11]. For our present purposes, we are primarily interested in its simplest incarnation as a modal next-step logic, dubbed coalgebraic modal logic, and its role as a characteristic logic for behavioural equivalence in generalization of Hennessy-Milner logic [HM85]. We briefly recall the syntax and semantics of coalgebraic modal logic, as well as basic results. The framework is based on interpreting custom modalities of given finite arity over coalgebras for a functor T as n-ary predicate liftings, which are families of maps λ X ∶ (2 X ) n → 2 T X (subject to a naturality condition) where 2 = { , ⊺} and for any set Y , 2 Y is the set of 2-valued predicates on Y . We do not distinguish notationally between modalities and the associated predicate liftings. Satisfaction of a formula of the form λ(φ 1 , . . . , φ n ) (in some ambient logic) in a state a ∈ A of a T -coalgebra (A, α) is then defined inductively by where for any formula ψ, ψ = {c ∈ A c ⊧ ψ}. E.g. the standard diamond modality is interpreted over the powerset functor P by the predicate lifting X (Y ) = {Z ∈ P(X) ∃x ∈ Z. Y (x) = ⊺}, which according to (2.1) induces precisely the usual semantics of over transition systems (P-coalgebras). The standard Hennessy-Milner theorem is generalized coalgebraically [Pat04,Sch08] as saying that two states in T -coalgebras are behaviourally equivalent iff they satisfy the same Λ-formulae, provided that T is finitary (which corresponds to the usual assumption of finite branching) and Λ is separating, i.e. for any set X, every t ∈ T X is uniquely determined (within T X) by the set {(λ, Y 1 , . . . , Y n ) λ ∈ Λ n-ary, Y 1 , . . . , Y n ∈ 2 X , t ∈ λ(Y 1 , . . . , Y n )}.
For finitary T , a separating set of modalities always exists [Sch08].

Fuzzy Relations and Lax Extensions
We next introduce the central notion of the paper, concerning extensions of fuzzy (or realvalued ) relations along a set functor T , which we fix for the remainder of the paper. We begin by fixing basic concepts and notation on fuzzy relations. Hemimetrics can be viewed as particular fuzzy relations, forming a quantitative analogue of preorders; correspondingly, pseudometrics may be seen as a quantitative analogue of equivalence relations.
Definition 3.1. Let A and B be sets. A fuzzy relation between A and B is a map R∶ A × B → [0, 1], also written R∶ A → + B. We say that R is crisp if R(a, b) ∈ {0, 1} for all a ∈ A, b ∈ B (and generally apply the term crisp to concepts that live in the standard two-valued setting). The converse relation R ○ ∶ B → + A is given by Convention 3.2. Crisp relations are just ordinary relations. However, since we are working in a metric setting, it will be more natural to use the convention that elements a ∈ A, b ∈ B are related by a crisp relation R if R(a, b) = 0, in which case we write aRb.
Convention 3.3 (Composition). We write composition of fuzzy relations diagrammatically, using ';'. Explicitly, the composite R 1 ; R 2 ∶ A → + C of R 1 ∶ A → + B and R 2 ∶ B → + C is defined by where ⊕ denotes Lukasiewicz disjunction: x ⊕ y = min(x + y, 1). Note that given our previous convention on crisp relations, the restriction of this composition operator to crisp relations is precisely the standard relational composition. We reserve the applicative composition operator ○ for composition of functions. In particular, R∶ A → + B is viewed as a function A × B → [0, 1] whenever ○ is applied to R. Throughout the paper, we use the fact that composition is monotone, that is, for and Gr ,f (a, b) = 1 otherwise. The -graph of the identity function id A is also called the -diagonal of A, and denoted by ∆ ,A . We refer to Gr 0,f simply as the graph of f , also denoted Gr f , and to ∆ 0,A as the diagonal of A, which we continue to denote as ∆ A .
The following is now straightforward.
Using the notation assembled, we can rephrase the definition of hemimetric and pseudometric as follows.
Lemma 3.6. Let d∶ X → + X be a fuzzy relation.
(2) d is a pseudometric iff it is a hemimetric and additionally d ○ = d (symmetry).
We now introduce our central notion of nonexpansive lax extension: (2) We say that L is a (fuzzy) lax extension if it satisfies examples of this will be shown in Example 3.14. This necessitates the addition of the second clause in Axiom (L3) (which of course is implied by the first clause in presence of (L0)).
Axiom (L4) has no two-valued analogue; its role and the terminology are explained by Lemma 3.9 below. The axioms (L1)-(L3) imply the following basic property [HST14, Corollary III.1.4.4]: Lemma 3.8 (Naturality). Let L be a fuzzy lax extension of T , let R∶ A ′ → + B ′ be a fuzzy relation, and let Proof. We need to show two inequalities. For '≤', we have (Lemma 3.5.1) Using Lemma 3.8, we can prove the following characterization of Axiom (L4), which is an important prerequisite for the Hennessy-Milner theorem.
Lemma 3.9. Let L be a fuzzy lax extension of T . Then the following are equivalent: (1) L satisfies Axiom (L4) (i.e. is nonexpansive).
(2) For all functions f ∶ A → B and all > 0, LGr ,f ≤ Gr ,T f .
(3) For all sets A, B, the map R ↦ LR is nonexpansive w.r.t. the supremum metric on A→ + B. Proof.
Remark 3.10. As stated in the introduction, nonexpansiveness of lax extensions relates to conditions on certain lax extensions for strong monads introduced by Gavazzo [Gav18], called L-continuous V -relators (for V a quantale). Specifically, as T is a set functor, it has a tensorial strength Instantiating to the unit interval and using our notation, the axioms of an L-continuous [0, 1]-relator L require that strength is nonexpansive, i.e. that for all sets A, B, X and Y and all fuzzy relations R∶ A → + B and S∶ X → + Y we have where ⊕ is taken pointwise. We say that L is strong if it satisfies (3.1) and show that L is strong iff it is nonexpansive: Let a ∈ A, b ∈ B and let p∶ X → A × X and q∶ Y → B × Y be the maps x ↦ (a, x) and y ↦ (b, y), respectively. Then, unfolding the definition of str and applying Lemma 3.8, we have, for t 1 ∈ T X and t 2 ∈ T Y , Put ∶= R(a, b). Then we have, for all x ∈ X and y ∈ Y , Similarly, for a and b fixed like this we have that (R ⊕ LS)((a, t 1 ), (b, t 2 )) = (∆ ,T X ; LS)(t 1 , t 2 ). Thus, (3.1) is equivalent to the requirement that L(∆ ,X ; S) ≤ ∆ ,T X ; LS (3.2) for all sets X, Y , all S∶ X → + Y and all ≥ 0. Finally, we show that (3.2) is equivalent to (L4): and (L3) in the inequalities and neutrality of ∆ in the equalities. • '⇐': By (L2) and (L4), we have L(∆ ,X ; S) ≤ L∆ ,X ; LS ≤ ∆ ,T X ; LS.
As indicated previously, many existing approaches to behavioural metrics (e.g. [vBW05,BBKK18]) are based on lifting functors to pseudometric spaces. In the present framework, every lax extension induces a functor lifting to hemimetric spaces; or to pseudometric spaces if the lax extension preserves converse: Lemma 3.11. Let L be a fuzzy lax extension.
(1) Let d∶ X → + X be a hemimetric. Then Ld is a hemimetric on T X. If d is a pseudometric and L preserves converse, then Ld is a pseudometric as well. Proof.
(1) Using Lemma 3.6 and the laws of lax extensions, we have Ld ≤ L∆ X ≤ ∆ T X and Ld ≤ L(d; d) ≤ Ld; Ld, so Ld is a hemimetric. If L preserves converse and d is a pseudometric, then (Ld) ○ = L(d ○ ) = Ld, so that Ld is a pseudometric.
Then T f is nonexpansive as well, by naturality (Lemma 3.8) and monotonicity: As a consequence of Lemma 3.11, every fuzzy lax extension of T ∶ Set → Set gives rise to a functor T ∶ HMet → HMet on the category HMet of hemimetric spaces and nonexpansive maps that lifts T in the sense that U ○ T = T ○ U , where U ∶ HMet → Set is the functor that forgets the hemimetric. Similarly, every converse-preserving fuzzy lax extension induces a lifting of T to the category of pseudometric spaces. Much of the development will be based on finitary functors; for instance, we need a finitary functor so we can give an explicit syntax for the characterizing logic of a lax extension. We can capture a broader class of functors, specifically those functors that are suitably approximated by their finitary parts in the sense that the finitary part forms a dense subset of the unrestricted functor.
This notion of density for hemimetrics coincides with an existing one for quantale-valued distances [FK97]. In particular, it is essential to require both inequalities in Definition 3.12, as otherwise certain pathological cases of dense subsets may occur. For instance, if we left out the second inequality from the above definition, then the singleton set {1} would be a dense subset of the unit interval [0, 1] under the hemimetric d ⊖ (Section 2).
Equipped with this definition of density, we proceed to introduce the following condition which allows for the treatment of lax extensions of certain non-finitary functors.
Definition 3.13. A fuzzy lax extension L of the functor T is finitarily separable if for every set X, T ω X is a dense subset of T X w.r.t. the hemimetric L∆ X .
Clearly, any lax extension of a finitary functor is finitarily separable. The prototypical example of a finitarily separable lax extension of a non-finitary functor is the Kantorovich lifting of the discrete distribution functor D (Example 5.11.1); that is, every discrete distribution can be approximated, under the usual Kantorovich metric, by finitely supported distributions.
We conclude the section with a basic example of a nonexpansive lax extension, deferring further examples to the sections on systematic constructions of such extensions (Sections 5 and 6): Example 3.14 (Hausdorff lifting). The Hausdorff lifting is the relation lifting H for the powerset functor P, defined for fuzzy relations R∶ A → + B by The Hausdorff lifting can be viewed as a quantitative analogue of the Egli-Milner extension (Section 2), where sup replaces universal quantification and inf replaces existential quantification. It is shown already in [HST14] that H is a fuzzy lax extension. Indeed, it is easy to see that H is also converse-preserving and nonexpansive. These properties will also follow from the results of Section 6, where we show that H is in fact an instance of the Wasserstein lifting. H is not finitarily separable, because for every set X we have H∆ X = ∆ PX . We may also consider asymmetric versions of the Hausdorff lifting by simply omitting one of the two terms in the definition, putting and Both H ← and H → are nonexpansive fuzzy lax extensions, but neither of them preserves converse.

Quantitative Simulations
We next identify a notion of simulation based on a lax extension L of the functor T ; similar concepts appear in work on quantitative applicative bisimilarity [Gav18]. We define behavioural distance based on this notion, and show coincidence with the distance defined via the pseudometric lifting induced by L according to Lemma 3.11.
Definition 4.1. Let L be a lax extension of T , and let α∶ A → T A and β∶ B → T B be coalgebras. ( Remark 4.2. Putting Definition 4.1 in other words, an L-simulation is precisely a prefix point for the map F (R) = LR ○ (α × β). Note that F is monotone by (L1). This means that, according to the Knaster-Tarski fixpoint theorem, d L α,β is itself a prefix point (i.e. an L-simulation), and also the least fixpoint of F , i.e. d L α,β = Ld L α,β ○ (α × β). In particular, the infimum in Definition 4.1.3 is always a minimum.
Example 4.3. The weighted transition systems discussed in the introduction can be modelled as coalgebras for the functor P ω (M × −), and the simulation distance given there is then L-behavioural distance for the fuzzy lax extension L defined for R∶ A → + B by To ensure that all values of LR lie in the unit interval [0, 1], we require that d M (m, n) ≤ 1 − λ for all m, n ∈ M . If M is finite (as is the case in [LFT11]) this can always be achieved by rescaling.
We note the following facts about L-simulations: Lemma 4.4. Let L be a fuzzy lax extension, and let α∶ A → T A, β∶ B → T B and γ∶ C → T C be coalgebras. Then (1) ∆ A is an L-simulation.
(2) For any L-simulations R∶ A → + B and S∶ B → + C, R; S is an L-simulation. Proof. For Item (1), we have Gr α ≤ ∆ A by (L3) and both parts of Lemma 3.5. For Item (2), we compute For converse-preserving lax extensions, this notion of simulation is actually one of bisimulation, more precisely: Lemma 4.5. If L preserves converse, then every L-simulation is an L-bisimulation.
Proof. Let α∶ A → T A and β∶ B → T B be coalgebras and let R be an L-simulation. Then by (L0) we have As L-behavioural distance is the least L-simulation, we have In the converse-preserving case, we additionally have that (d L α ) ○ is an L-simulation by Lemma 4.5, and therefore d L α ≤ (d L α ) ○ , making d L α a pseudometric by Lemma 3.6. Remark 4.7. As announced above and as we show next, existing generic notions of behavioural distance defined via functor liftings [BBKK18] agree with the one given above (when both apply). Specifically, when applied to the functor lifting induced by a conversepreserving lax extension L of T according to Lemma 3.11, the definition of behavioural distance via functor liftings amounts to taking the same least fixpoint as in Definition 4.1 but only over pseudometrics instead of over fuzzy relations [BBKK18, Lemma 6.1]. Now let (A, α) be a coalgebra and denote the behavioural distance on A according to the definition in [BBKK18] byd α . We claim thatd α = d L α , with d L α defined according to Definition 4.1. Indeed, '≥' is trivial sinced α is, by definition, an L-bisimulation, and '≤' is immediate from d L α being a pseudometric (Lemma 4.6). Remark 4.8. Every converse-preserving fuzzy lax extension L induces a crisp lax extension L c , where for any crisp relation R, It is easily checked that L c preserves diagonals (Section 2) iff L∆ A is a metric for each set A. (4.1) By results on lax extensions cited in Section 2, L c -bisimilarity coincides with behavioural equivalence in this case, i.e. if L satisfies (4.1), then L characterizes behavioural equivalence: Two states a ∈ A and b ∈ B in coalgebras (A, α) and (B, β) are behaviourally equivalent iff d L α,β (a, b) = 0. Example 4.9 (Small bisimulations). We give an example of a bisimulation for a lax extension of the functor T X = [0, 1] × PX. Coalgebras for T are Kripke frames where each state is labelled with a number from the unit interval. They are similar to the weighted transition systems from [LFT11], except that here the labels are on the states rather than on the transitions. This T has a converse-preserving nonexpansive lax extension L, defined for fuzzy relations R∶ A → + B by and H is the Hausdorff lifting (Example 3.14). The idea behind this definition is that the L-behavioural distance of two states is the supremum of the accumulated branching-time differences between state labels over all runs of a process starting at these states. The factor 1 2 ensures that the total distance is at most 1 by discounting the differences at later stages with exponentially decreasing factors. Now consider the T -coalgebras (A, α) and (B, β) below: We put R(a 1 , b 1 ) = 0.2, R(a 2 , b 3 ) = R(a 3 , b 2 ) = 0.1, and R(a i , b j ) = 1 in all other cases. We show that R is an L-bisimulation witnessing that d L α,β (a 1 , b 1 ) ≤ 0.2, even though it is clearly neither reflexive nor symmetric on the disjoint union of the systems (it is easy to come up with similar but slightly larger examples where R also fails to be transitive, i.e. to satisfy the triangle inequality).
Specifically, we need to show for each a i and b j that LR(α(a i ), β(b j )) ≤ R(a i , b j ). The cases with R(a i , b j ) = 1 are trivial; in the other cases we have: ((0.8, ∅), (0.7, ∅)) = 1 2 0.1 ≤ 0.1 = R(a 3 , b 2 ) As indicated previously, quantitative Hennessy-Milner theorems can only be expected to hold for nonexpansive lax extensions. The key observation is the following. By standard fixpoint theory, L-behavioural distance can be approximated from below by an ordinal-indexed increasing chain:  We show some basic properties of these sequences: Lemma 4.11. Let L be a lax extension of T , let (A, α), (B, β) be T -coalgebras, and let (d κ ∶ A → + B) κ be the sequence of approximants of d L α,β . Then: Proof.
This fact is essentially standard; we give a proof for the sake of completeness: Proof.
(ii) This is immediate from (i), as behavioural equivalence is witnessed by the coalgebra morphisms id∶ C → C and last∶ C + → C.
Proof of Theorem 4.12. By the fixpoint definition of d L α,β and Lemma 4.11.2, (ii) is immediate from (i). We prove (i), i.e. that Ld ω (α(a), β(b)) = d ω (a, b) for all a ∈ A, b ∈ B. We begin by assuming that T is finitary, and generalize to the non-finitary case later.
Since T is finitary, there exist finite subsets A 0 ⊆ A, B 0 ⊆ B and s ∈ T A 0 , t ∈ T B 0 such that α(a) = T i(s) and β(b) = T j(t), where i∶ A 0 → A and j∶ B 0 → B are the inclusion maps. We then have Ld ω (α(a), β(b)) = L(d ω ○ (i × j))(s, t) by naturality (Lemma 3.8). By Lemma 4.11.1, the maps d n ○ (i × j) converge to d ω ○ (i × j) pointwise, and therefore also under the supremum metric (i.e. uniformly), since A 0 × B 0 is finite. Since L is nonexpansive, it is also continuous w.r.t. the supremum metric by Lemma 3.9, so it follows that This covers the finitary case. In the general case, we make use of the unravellings (A + , α + ) and (B + , β + ), as well as the sequence (d + κ ∶ A + → + B + ) κ of approximants of d L α + ,β + . We can assume w.l.o.g. that A ≠ ∅ ≠ B; then the inclusions A m ↪ A + , B m ↪ B + (for m ≥ 1) are preserved by T , and for readability we assume in the following that T A m is in fact a subset of T A + ; similarly for B m and T ω , with naturality of L guaranteeing that the identification does not affect lifted distance. Now let > 0. As L is finitarily separable, we can construct T ω -coalgebras α ∶ A + → T ω A + and β ∶ B + → T ω B + approximating α + and β + respectively. Specifically, for everyā ∈ A m we have α + (ā) ∈ T A m+1 by definition, and as T ω A m+1 is dense in T A m+1 , we can choose an element α (ā) ∈ T ω A m+1 such that (4.2) Similarly, for eachb ∈ B m we choose β (b) ∈ T ω B m+1 such that We denote the sequence of approximants of d L α ,β by (d κ ∶ A + → + B + ) κ and show by induction that the d κ approximate the d + κ in the following sense: for all m ≥ 1 and allā ∈ A m ,b ∈ B m , for all ordinals κ. For κ = 0 this clearly holds. For the inductive step from κ to κ + 1, we note again that forā ∈ A m andb ∈ B m we have α + (ā) ∈ T A m+1 and β + (b) ∈ T B m+1 by definition . Therefore, by Lemma 3.9.3 and the inductive hypothesis, we have so that we compute: We can symmetrically derive d + κ+1 (ā,b) ≤ d κ+1 (ā,b) + ⋅ 3 1−m , this time using the other inequalities in (4.2) and (4.3), so (4.4) holds for κ + 1 as claimed. Finally, if κ is a limit ordinal, then (4.4) also follows inductively, as taking suprema is a nonexpansive operation.

The Kantorovich Lifting
As a pseudometric lifting, the Kantorovich lifting is standard in the probabilistic setting: Given a metric d on a set X, the Kantorovich distance Kd(µ 1 , µ 2 ) between discrete distributions µ 1 , µ 2 on X is defined by where E takes expected values and d E (x, y) = x − y is Euclidean distance. The coalgebraic generalization of the Kantorovich lifting, both in the pseudometric setting [KMM18] and in the present setting of fuzzy relations, is based on fuzzy predicate liftings, a quantitative analogue of two-valued predicate liftings (Section 2) that goes back to work on coalgebraic fuzzy description logics [SP11]. Fuzzy predicate liftings will feature in the generic quantitative modal logics that we extract from fuzzy lax extensions (Section 8).
Recall that the contravariant fuzzy powerset functor Q∶ Set op → Set is defined on sets X as QX = (X → [0, 1]) and on functions f ∶ X → Y as Qf ∶ QY → QX, Qf (h) = h ○ f . (1) An n-ary (fuzzy) predicate lifting is a natural transformation where the exponent n denotes n-fold cartesian product.
Before we can show that the Kantorovich lifting is a lax extension, we first need to generalize it so that it lifts arbitrary fuzzy relations instead of just pseudometrics. To this end, we introduce the notion of nonexpansive pairs (a similar idea appears already in [Vil08, Section 5]): This notion is compatible with our previous use of the term: When A = B and d∶ A → + A is a hemimetric, then f ∶ (A, d) → ([0, 1], d ⊖ ) is nonexpansive in the sense used so far (cf. Section 2) iff the pair (f, f ) is d-nonexpansive in the sense defined above. If d is a pseudometric, then this is moreover equivalent to f being nonexpansive as a map (A, d) → ([0, 1], d E ).
Given a function and a fuzzy relation, we can construct a nonexpansive companion: (recall from Section 2 that ⊖ denotes truncated subtraction).
We note some basic properties of nonexpansive pairs and nonexpansive companions. In particular, the nonexpansive companion of some function f is the least function (in pointwise order) forming a nonexpansive pair with f .
To show that the Kantorovich lifting is a lax extension, we need the following fact about nonexpansive pairs that will be used in the proof of the triangle inequality (L2).
rearranging. Similar rearranging also shows that choosing g(b) in this way ensures that (f, g) is R-nonexpansive and (g, h) is S-nonexpansive.
We are now ready to prove the central result of the section, stating that the Kantorovich lifting is always a fuzzy lax extension. In general, it does not preserve converse, but does if the set of predicate liftings contains all duals of predicate liftings.
Theorem 5.8. Let Λ be a set of monotone predicate liftings. The Kantorovich lifting K Λ is a lax extension. If Λ is closed under duals, then K Λ preserves converse. If all λ ∈ Λ are nonexpansive, then K Λ is nonexpansive as well.
Proof. For readability, we pretend that all λ ∈ Λ are unary although the proof works just as well for unrestricted arities, whose treatment requires no more than adding indices. We show the five properties one by one: Then every R 1 -nonexpansive pair is also R 2 -nonexpansive. Thus K Λ R 1 ≤ K Λ R 2 , because the supremum on the left side is taken over a subset of that on the right side. • (L2): Let R∶ A → + B, S∶ B → + C and t 1 ∈ T A, t 2 ∈ T B, t 3 ∈ T C. Let λ ∈ Λ and let (f, h) be (R; S)-nonexpansive. Let g be given by Lemma 5.7. Then it is enough to observe that: • (L3): Let h∶ A → B and t ∈ T A. We need to show that K Λ Gr h (t, T h(t)) = 0. Let λ ∈ Λ and let (f, g) be Gr h -nonexpansive, implying f ≤ g ○ h. Then by monotonicity and naturality of λ. The proof for Gr ○ h is analogous, noting that a pair • (L4): Let A be a set, t ∈ T A and > 0. We need to show that K Λ ∆ ,A (t, t) ≤ . Let λ ∈ Λ and let (f, g) be ∆ ,A -nonexpansive, implying f (a) − g(a) ≤ for all a ∈ A. By monotonicity of λ, we can restrict our attention to the case g(a) = f (a) ⊖ , so that we have f − g ∞ ≤ . In this case, is R-nonexpansive. Now, using that Λ is closed under duals, Remark 5.9 (Kantorovich for pseudometrics). On pseudometrics, the Kantorovich lifting K Λ as given by Definition 5.6 agrees with the usual Kantorovich distance − ↑T defined for pseudometrics [BBKK18,Definition 5.4]. If d∶ A → + A is a pseudometric, then Lemma 5.10. If Λ is closed under duals, then K Λ (d) = d ↑T for every pseudometric d. (1) The standard Kantorovich lifting K of the discrete distribution functor D is an instance of the generic one, for the single predicate lifting (f )(µ) = E µ (f ). We claim that K is finitarily separable. To see this, let µ ∈ DX and > 0. We need to find µ ∈ DX with finite support such that K∆ X (µ, µ ) ≤ . Note that a pair (f, g) is ∆ X -nonexpansive iff f ≤ g, so by monotonicity Because µ is discrete, there exists a finite set Y ⊆ X with µ(Y ) ≥ 1 − 2 . If Y = X, then we can just put µ = µ. Otherwise, let x 0 ∈ X ∖ Y . Then we define µ as follows: x ∈ Y , and µ (x) = 0 otherwise. In this case, Following Remark 4.8, we can also see that the Kantorovich lifting characterizes behavioural equivalence for probabilistic transition systems, i.e. probabilistic bisimilarity [LS91]: To see that K satisfies (4.1), by Lemma 3.11 it suffices to show that K∆ X (µ 1 , µ 2 ) > 0 for any µ 1 ≠ µ 2 ∈ DX. W.l.o.g. assume µ 1 (x) > µ 2 (x) for some x ∈ X and let f ∈ QX be such that f (x) = 1 and f (x ′ ) = 0 otherwise. Then, as (f, f ) is ∆ X -nonexpansive, we have K∆ X (µ 1 , µ 2 ) ≥ f (x)(µ 1 (x) − µ 2 (x)) > 0. • '≥': Let a ∈ U and let f ∈ QA be the indicator function of {a}, that is f ( Then, as (f, g) is R-nonexpansive (Lemma 5.5),

The Wasserstein Lifting
The other generic construction for lax extensions arises in a similar way, by generalizing the generic Wasserstein lifting for pseudometrics [BBKK18] to lift arbitrary fuzzy relations instead of just pseudometrics; our construction slightly generalizes one given by Hofmann [Hof07]. Like the Kantorovich lifting, the Wasserstein lifting is based on a choice of predicate liftings. Compared to the case of the Kantorovich lifting, where we needed to work with nonexpansive pairs, the generalization from lifting pseudometrics to lifting relations is much more direct for the Wasserstein lifting. In the same way as for the original construction of pseudometric Wasserstein liftings, additional constraints, both on the functor and the set of predicate liftings involved, are needed for the Wasserstein lifting to be a lax extension. Indeed, the Wasserstein lifting may be seen as a quantitative analogue of the two-valued Barr extension (Section 2), and like the latter works only for functors that preserve weak pullbacks. In particular, Wasserstein liftings are based on the central notion of coupling: Definition 6.1. Let t 1 ∈ T A, t 2 ∈ T B for sets A, B. The set of couplings of t 1 and t 2 is Cpl(t 1 , t 2 ) = {t ∈ T (A × B) T π 1 (t) = t 1 , T π 2 (t) = t 2 }.
The Wasserstein lifting uses predicate liftings in a quite different manner from the Kantorovich lifting, and in particular appears to make sense only for unary predicate liftings, so unlike elsewhere in the paper, the restriction to unary liftings in the next definition is not just for readability.
This construction is similar to [Hof07, Definition 3.4] except that we admit more than one modality. On pseudometrics, the Wasserstein lifting coincides with the pseudometric lifting − ↓T as defined in [BBKK18, Definition 5.12] (again up to the fact that we admit more than one modality). We will see that the following conditions ensure that the Wasserstein lifting is a fuzzy lax extension: Definition 6.3. Let λ be a unary predicate lifting.
(3) λ is standard if it is monotone, subadditive, and preserves the zero function. (1) The predicate lifting λ e is monotone.
Then it is easily checked that f ⊕ g ≥ π 2 and therefore λ e (π 2 ) ≤ λ e (f ⊕ g) ≤ λ e (f ) + λ e (g) = λ e (d E ) + λ e (π 1 ) by monotonicity and subadditivity of λ e , so λ e (π 1 ) − λ e (π 2 ) ≤ λ e (d E ). Similarly, we can show that λ e (π 2 ) − λ e (π 1 ) ≤ λ e (d E ) by swapping the roles of π 1 and π 2 . Similar conditions also feature in Hofmann's topological theories [Hof07, Definition 3.1], which consist of a monad acting on a quantale via an evaluation function and on which his generic Wasserstein extension is based. Explicitly, a topological theory is defined as a triple consisting of a monad T , a quantale V , and a map ξ∶ T V → V satisfying a number of axioms. We only consider the case of the quantale [0, 1] op , with the order given by ≥ and the monoid structure by ⊕. The first two axioms state that ξ is a T -algebra and can be ignored for our purposes. The remaining axioms instantiate as follows, where as usual λ ξ (f ) = ξ ○ T f is the predicate lifting associated with ξ: Using a similar idea as in Lemma 6.5, we see that (Q ⊗ ) is equivalent to subadditivity of λ ξ and (Q k ) is equivalent to preservation of the zero function. Finally note that [Hof07, Theorem 3.5 (d)] (which states that the Wasserstein lifting satisfies (L2)) requires that the functor satisfies the Beck-Chevalley condition, i.e. preserves weak pullbacks. If T preserves weak pullbacks, the following so-called gluing lemma holds [BBKK18, Lemma 5.18]: Lemma 6.6 (Gluing). Let A, B and C be sets, and let t 1 ∈ T A, t 2 ∈ T B, t 3 ∈ T C. Let t 12 ∈ Cpl(t 1 , t 2 ) and t 23 ∈ Cpl(t 2 , t 3 ). Then there exists t 123 ∈ Cpl(t 1 , t 2 , t 3 ) such that T ⟨π 1 , π 2 ⟩(t 123 ) = t 12 and T ⟨π 2 , π 3 ⟩(t 123 ) = t 23 , where the π j are the projections of the product A × B × C. Moreover, t 13 ∶= T ⟨π 1 , π 3 ⟩(t 123 ) ∈ Cpl(t 1 , t 3 ).
Using Lemma 6.6, we can now show that the Wasserstein lifting is a fuzzy lax extension. Unlike the Kantorovich lifting, the Wasserstein lifting always preserves converse, without any further restrictions on the set of predicate liftings.
Theorem 6.7. If T preserves weak pullbacks and Λ is a set of standard predicate liftings, then the Wasserstein lifting W Λ is a converse-preserving lax extension. If additionally all λ ∈ Λ are nonexpansive, then W Λ is nonexpansive as well.
(3) The convex powerset functor C, whose coalgebras combine probabilistic branching and nondeterminism [BSS17], maps a set X to the set of nonempty convex subsets of DX. The • '≤': Let Y ∈ Cpl P (U, V ) and > 0. It suffices to find Z ∈ Cpl C (µ 1 , µ 2 ) such that For every (µ 1 , µ 2 ) ∈ DA × DB there exists some µ ∈ Cpl D (U, V ) such that E µ (R) ≤ W R(µ 1 , µ 2 ) + . Let Z ′ be a set consisting of one such µ for every pair (µ 1 , µ 2 ) ∈ Y and put Z = conv(Z ′ ), where conv is convex hull. Then we have Here we made use of the fact that Dπ 1 is linear when considered as a map R A×B → R A , and linear maps preserve convex sets. We similarly have Cπ 2 (Z) = V , so that Z ∈ Cpl C (U, V ). Finally, we note that taking expected values is a linear operation, so if

Lax Extensions as Kantorovich Liftings
We proceed to establish the central result that every fuzzy lax extension is a Kantorovich lifting for some suitable set Λ of predicate liftings, and moreover we characterize the Kantorovich liftings induced by nonexpansive predicate liftings as precisely the nonexpansive lax extensions. For a given fuzzy lax extension L, the equality K Λ R = LR splits into two inequalities, one of which is characterized straightforwardly: Definition 7.1. An n-ary predicate lifting λ preserves nonexpansiveness if for all fuzzy relations R and all R-nonexpansive pairs (f 1 , g 1 ), . . . , (f n , g n ), the pair (λ A (f 1 , . . . , f n ), λ B (g 1 , . . . , g n )) is LR-nonexpansive. A set Λ of predicate liftings preserves nonexpansiveness if all λ ∈ Λ preserve nonexpansiveness. To motivate Definition 7.3, recall from Section 2 that in the two-valued setting a set Λ of predicate liftings (for simplicity, assumed to be unary) is separating if for t 1 , t 2 ∈ T A. Analogously, unfolding definitions in the inequality K Λ R ≥ LR (and again assuming unary liftings), we arrive at the condition that for all t 1 ∈ T A, t 2 ∈ T B, > 0, We are now ready to state our main result, which says that all lax extensions are Kantorovich: Theorem 7.4. If L is a finitarily separable lax extension of T , then there exists a set Λ of monotone predicate liftings that preserves nonexpansiveness and is separating for L, i.e. L = K Λ . Moreover, L is nonexpansive iff Λ can be chosen in such a way that all λ ∈ Λ are nonexpansive.
This result can be seen as a fuzzy version of the statements that every finitary functor has a separating set of two-valued modalities (and hence an expressive two-valued coalgebraic modal logic) [Sch08, Corollary 45], and that more specifically, every finitary functor equipped with a diagonal-preserving lax extension has a separating set of two-valued monotone predicate liftings [MV15,Theorem 14]. We will detail in Section 8 how Theorem 7.4 implies the existence of characteristic modal logics. The proof of Theorem 7.4 uses a quantitative version of the so-called Moss modalities [KL09,MV15]. The construction of these modalities relies on the fact that T ω can be presented by algebraic operations of finite arity: Definition 7.5. A finitary presentation of T ω consists of a signature Σ of operations with given finite arities, and for each σ ∈ Σ of arity n a natural transformation σ∶ (−) n ⇒ T ω such that every element of T ω X has the form σ X (x 1 , . . . , x n ) for some σ ∈ Σ.
For the remainder of this section, we fix a finitary presentation of T ω with signature Σ (such a presentation always exists [MV15, Example 21]) and assume a finitarily separable fuzzy lax extension L of T . To derive predicate liftings from the operations in Σ, we make use of the fuzzy elementhood relation ∈ X (indexed over arbitrary sets X), where ∈ X ∶ X → + QX is given by ∈ X (x, f ) = f (x).
It follows from naturality of σ and L (Lemma 3.8) that µ σ is indeed natural and therefore a predicate lifting, as shown next. Indeed, for any g∶ A → B, f 1 , . . . , f n ∈ QB and t ∈ T B we note that ∈ A ○ (id × Qg) = ∈ B ○ (g × id) by definition of ∈ A and ∈ B and thus We are now in a position to present the proof of Theorem 7.4: We take Λ to be the set of Moss liftings and show the required properties of Λ one by one.
Convention 7.7. Throughout this proof, all statements and proofs will be written for the case where all σ ∈ Σ (and therefore the induced Moss liftings) are unary. This is purely in the interest of readability; the general case requires only more indexing.
Monotonicity. The proof is based on the following auxiliary fact about pairs of elements that are mapped to 0.
Proof. We make use of the fuzzy relation R∶ QX → + QX given by R(g, f ) = sup x∈X f (x)⊖g(x), which we claim to satisfy the following two useful properties: 3) The first property is clear; the second property amounts to showing that f (x) ≤ g(x)⊕R(g, f ) for all x ∈ X and all f, g ∈ QX and is easily shown by case analysis on the definition of ⊕. Let f, g ∈ QX with f ≤ g and let t ∈ T X. First, we note that by (7.2) we have R(g, f ) = 0 and thus LR(σ QX (g), σ QX (f )) = 0 by Lemma 7.8. Second, by (7.3) and the axioms of lax extensions we have L∈ X ≤ L(∈ X ; R) ≤ L∈ X ; LR. Therefore: ≤ L∈ X (t, σ QX (g)) ⊕ LR(σ QX (g), σ QX (f )) = µ σ X (g)(t).
Let f ∈ QA, g ∈ QB, and let a ∈ A, b ∈ B. We need to show that , then we have f (a) ⊖ R(a, b) ≤ g(b) by definition, and hence . Now let (f, g) be R-nonexpansive and let t 1 ∈ T A and t 2 ∈ T B. We need to show that and (7.5) follows by rearranging.
Separation. To show that Λ is separating for L, we need to make use of the fact that L is finitarily separable.
Lemma 7.11. Λ is separating for L, that is, L ≤ K Λ .
As before, we have L∈ B ≤ L∈ B ; L∆ QB . We also have σ QB (g) = T R[−](σ QA (f )) by naturality of σ. Therefore, by naturality of L: Breugel and Worrell's characteristic logic for behavioural distance of probabilistic transition systems [vBW05] does feature essentially the same propositional operators as our grammar (8.1), if negation is defined as in Remark 8.2 below. Following [vBW05], we restrict truth constants in formulae to rational numbers, thus ensuring that the set of formulae is countable provided Λ is countable. This countability is not needed for any of our results, and they will still hold if the truth constants come from any dense subset of [0, 1] (including [0, 1] itself).
Remark 8.2. The logic as defined above does not include negation. This is to be expected, as already in the classical case the characteristic logic for similarity is negation-free modal logic with as the only modality [vG90]. However, if the set Λ of predicate liftings is closed under duals (and the corresponding Kantorovich lifting therefore preserves converse), then negation ¬φ can be defined recursively using De Morgan's laws for the propositional operators and duals for the modalities: . . . , φ n ) =λ(¬φ 1 , . . . , ¬φ n ) With negation defined like this, this version of real-valued coalgebraic modal logic is equivalent to the one in [WS20], which includes negation as a primitive. The latter logic does not explicitly include addition, but in the presence of subtraction and negation we can define it as φ ⊕ c = ¬(¬φ ⊖ c).
We note that and ◻ are dual, so that negation can be defined as in Remark 8.2. Hennessy-Milner-type results necessarily apply only to finitely branching models, i.e. coalgebras for F ω .
(2) Probabilistic modal logic: Take models to be probabilistic transition systems with possible deadlocks, i.e. coalgebras for the functor 1+D, where DA is the set of discrete probability distributions on A (Section 2); and Λ = { }, with A (f )( * ) = 0 for * ∈ 1, and A (f )(µ) = E µ (f ) = ∑ a∈A µ(a) ⋅ f (a). Probabilistic modal logic can be extended with negation by adding the dual ◻ of . As taking expected values is self-dual, ◻ only differs from on deadlocks, where ◻ A (f )( * ) = 1. When additionally extended with propositional atoms, this induces (up to restricting to discrete probabilities) van Breugel et al.'s contraction-free quantitative probabilistic modal logic [vBHMW07].
In the two-valued setting, modal logic is typically invariant under bisimulation, i.e. bisimilar states satisfy the same modal formulae. By contrast, under the asymmetric notion of similarity, the corresponding statement is that the fragment of modal logic that only uses Theorem 8.8 (Coalgebraic quantitative Hennessy-Milner theorem). Let L be a finitarily separable fuzzy lax extension, and let Λ be a separating set of monotone nonexpansive predicate liftings for L. Then d Λ = d L .
Proof. By Lemma 8.7 we have d ω = sup n<ω d n = d Λ and by Theorem 4.12 we have d ω = d L .
(1) Since we only require L to be finitarily separable (rather than T finitary), Example 5.11.1 implies that we recover expressiveness [vBHMW07,vBW05] of quantitative probabilistic modal logic over countably branching discrete probabilistic transition systems (Example 8.3.2) as an instance of Theorem 8.8. (2) Let L be the lax extension of T = P ω (M × −) from Example 4.3. As T is finitary, it follows by Theorem 7.4 that L = K Λ for the set Λ of Moss liftings of L and the logic L Λ is characteristic for simulation distance by Theorem 8.8.
Applying Lemma 8.6 and Theorem 8.8 to L = K Λ and using our result that all lax extensions are Kantorovich extensions for their Moss liftings (Theorem 7.4), which moreover are monotone and nonexpansive in case L is nonexpansive, we obtain expressive logics for finitarily separable nonexpansive lax extensions: Corollary 8.10. If L is a finitarily separable nonexpansive lax extension of a functor T , then d L = d Λ for the set Λ of Moss liftings.
We can see the coalgebraic modal logic of Moss liftings as concrete syntax for a more abstract logic where we incorporate functor elements into the syntax directly, as in Moss' coalgebraic logic [Mos99] and its generalization to lax extensions [MV15]. The set L L of formulae in the arising quantitative Moss logic is generated by the same propositional operators as above, and additionally by a modality ∆ that applies to Φ ∈ T L 0 for finite L 0 ⊆ L L , with semantics ∆Φ (a) = L∈ A (α(a), Φ).
The dual of ∆ is denoted ∇, and behaves like a quantitative analogue of Moss' two-valued ∇. From Corollary 8.10, it is immediate that this logic is expressive: Corollary 8.11 (Expressiveness of quantitative Moss logic). Let L be a finitarily separable nonexpansive lax extension of a functor T . Then L-behavioural distance d L coincides with logical distance in quantitative Moss logic, i.e. for all states a ∈ A, b ∈ B in coalgebras α∶ A → T A, β∶ B → T B, d L α,β (a, b) = sup{ φ (a) ⊖ φ (b) φ ∈ L L }. Example 8.12.
(2) Let C fg be the subfunctor of the convex powerset functor C given by the finitely generated convex sets of (not necessarily finite) discrete distributions, equipped with the Wasserstein lifting described in Example 6.8.3. Then ∇ applies to finite sets of finite distributions on formulae, understood as spanning a convex polytope. By Corollary 8.11, the arising instance of quantitative Moss logic is expressive for all C fg -coalgebras.