Expressive Logics for Coinductive Predicates

The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata.


Introduction
The deep connection between bisimilarity and modal logic manifests itself in the Hennessy-Milner theorem: two states of an image-finite labelled transition system (LTS) are behaviourally equivalent iff they satisfy the same formulas in a certain modal logic [HM85]. From left to right, this equivalence is sometimes referred to as adequacy of the logic w.r.t. bisimilarity, and from right to left as expressiveness. By stating both adequacy and expressiveness, the Hennessy-Milner theorem thus gives a logical characterisation of behavioural equivalence.
There are numerous variants and generalisations of this kind of result. For instance, a state x of an LTS is simulated by a state y if every formula satisfied by x is also satisfied by y, where the logic only has conjunction and diamond modalities; see [vG90] for this and many other related results. Another class of examples is logical characterisations of quantitative notions of equivalence, such as probabilistic bisimilarity and behavioural distances (e.g., [LS91,DGJP99,DEP02,vBW05,JS09,KM18,WSPK18,CFKP19]). In many such cases, including bisimilarity, the comparison between states is coinductive, and the problem is thus to characterise a coinductively defined relation (or distance) with a suitable modal logic.
Both coinduction and modal logic can be naturally and generally studied within the theory of coalgebra, which provides an abstract, uniform study of state-based systems [Rut00,Jac16]. Indeed, in the area of coalgebraic modal logic [KP11] there is a rich literature on deriving expressive logics for behavioural equivalence between state-based systems, thus going well beyond labelled transition systems [Pat04,Sch08,Kli07]. However, such results focus almost exclusively on behavioural equivalence or bisimilarity-a coalgebraic theory of logics for characterising coinductive predicates other than bisimilarity is still missing. The aim of this paper is to accommodate the study of logical characterisation of coinductive predicates in a general manner, and provide tools to prove adequacy and expressiveness.
Our approach is based on universal coalgebra, to achieve results that apply generally to state-based systems. Central to the approach are the following two ingredients.
(1) Coinductive predicates in a fibration. To characterise coinductive predicates, we make use of fibrations-this approach originates from the seminal work of Hermida and Jacobs [HJ98]. The fibration is used to speak about predicates and relations on states. In this context, liftings of the type functor of coalgebras uniformly determine coinductive predicates and relations on such coalgebras. An important feature of this approach, advocated in [HKC18], is that it covers not only bisimilarity, but also other coinductive predicates including, e.g., similarity of labelled transition systems and other coalgebras [HJ04], behavioural metrics [BBKK18,BKP18,SKDH18], unary predicates such as divergence [BPPR17,HKC18], and many more.
(2) Coalgebraic modal logic via dual adjunctions. We use an abstract formulation of coalgebraic logic, which originated in [PMW06,Kli07], building on a tradition of logics via duality (e.g., [KKP04,BK05a]). This framework is formulated in terms of a contravariant adjunction, which captures the basic connection between states and theories, and a distributive law, which captures the one-step semantics of the logic. It covers classical modal logics of course, but also easily accommodates multi-valued logics, and, e.g., logics without propositional connectives, where formulas can be thought of as basic tests on state-based systems. This makes the framework suitable for an abstract formulation of Hennessy-Milner type theorems, where formulas play the role of tests on state-based systems.
To formulate adequacy and expressiveness with respect to general coinductive predicates, we need to know how to compare collections of formulas. For instance, if the coinductive predicate is similarity of LTSs, then the associated logical theories of one state should be included in the other, not necessarily equal. This amounts to stipulating a relation on truth values, that extends to a relation between theories. In the quantitative case, we need a logical distance between collections of formulas; this typically arises from a distance between truth values (which, in this case, will typically be an interval in the real numbers). The fibrational setting provides a convenient means for defining such an object for comparing theories.
With this in hand, we arrive at the main contributions of this paper: the formulation of adequacy and expressiveness of a coalgebraic modal logic with respect to a coinductive predicate in a fibration, and sufficient conditions on the semantics of the logic that guarantee adequacy and expressiveness. We exemplify the approach through a range of examples, including logical characterisations of a simple behavioural distance on deterministic automata, similarity of labelled transition systems, and a logical characterisation of a unary predicate: divergence, the set of states of an LTS which have an infinite path of outgoing τ -steps. The Jacobs, Shin-ya Katsumata and Yuichi Komorida for helpful discussions, comments and suggestions.

Preliminaries
The category of sets and functions is denoted by Set. The powerset functor is denoted by P : Set → Set, and the finite powerset functor by P ω . The diagonal relation on a set X is denoted by ∆ X = {(x, x) | x ∈ X}.
Let C be a category, and B : C → C a functor. A (B)-coalgebra is a pair (X, γ) where X is an object in C and γ : X → BX a morphism. A homomorphism from a coalgebra (X, γ) to a coalgebra (Y, θ) is a morphism h : X → Y such that θ • h = Bh • γ. An algebra for a functor L : D → D on a category D is a pair (A, α) of an object A in D and an arrow α : LA → A.
Example 2.1. A labelled transition system (LTS) over a set of labels A is a coalgebra (X, γ) for the functor B : Set → Set, BX = (PX) A . For states x, x ∈ X and a label a ∈ A, we sometimes write x a − → x for x ∈ γ(x)(a). Image-finite labelled transition systems are coalgebras for the functor BX = (P ω X) A . A deterministic automaton over an alphabet A is a coalgebra for the functor B : Set → Set, BX = 2 × X A . For many other examples of state-based systems modelled as coalgebras, see, e.g., [Jac16,Rut00].
2.1. Coinductive Predicates in a Fibration. We recall the general approach to coinductive predicates in a fibration, starting by briefly presenting how bisimilarity of Set coalgebras arises in this setting (see [HKC18,HJ98,Jac16] for details). Let Rel be the category where an object is a pair (X, R) consisting of a set X and a relation R ⊆ X × X on it, and a morphism from (X, R) to (Y, S) is a map f : X → Y such that x R y implies f (x) R f (y), for all x, y ∈ X. Below, we sometimes refer to an object (X, R) only by the relation R ⊆ X × X. Any set functor B : Set → Set gives rise to a functor Rel(B) : Rel → Rel, defined by relation lifting: (2.1) Bisimilarity is the greatest such relation, and equivalently, the greatest fixed point of the monotone map R → (γ × γ) −1 (Rel(B)(R)) on the complete lattice of relations on X, ordered by inclusion. The functor Rel(B) is a lifting of B: it maps a relation on X to a relation on BX. A first step towards generalisation beyond bisimilarity is obtained by replacing Rel(B) by an arbitrary lifting B : Rel → Rel of B. For instance, for BX = (P ω X) A one may take B(R) = {(t 1 , t 2 ) | ∀a ∈ A. ∀x ∈ t 1 (a). ∃y ∈ t 2 (a).(x, y) ∈ R} . (2.2) Then, for an LTS γ : X → (P ω X) A , the greatest fixed point of the monotone map R → (γ × γ) −1 • B(R) is similarity. In the same way, by varying the lifting B, one can define many different coinductive relations on Set coalgebras. Yet a further generalisation is obtained by replacing Set by a general category C, and Rel by a category of 'predicates' on C. A suitable categorical infrastructure for such predicates on C is given by the notion of fibration. This allows us, for instance, to move beyond (Boolean, binary) relations to quantitative relations (e.g., behavioural metrics) or unary predicates. Such examples follow in Section 4; also see, e.g., [HKC18,BPPR17].
To define fibrations, it will be useful to fix some associated terminology first. Let p : E → C be a functor. If p(R) = X, then we say R is above X, and similarly for morphisms. The collection of all objects R above a given object X and arrows above the identity id X form a category, called the fibre above X and denoted by E X .
Definition 2.2. A functor p : E → C is a (poset) fibration if • each fibre E X is a poset category (that is, at most one arrow between every two objects); the corresponding order on objects is denoted by ≤; • for every f : X → Y in C and object S above Y there is a Cartesian morphism f S : f * (S) → S above f , with the property that for every arrow g : Z → X, every object R above Z and arrow h : In this paper we only consider poset fibrations, and refer to them simply as fibrations. The usual definition of fibration is more general (e.g., [Jac99]): normally, fibres are not assumed to be posets. Poset fibrations have several good properties, mentioned below. In the application to coinductive predicates, it is customary to work with poset fibrations.
For a morphism f : X → Y , the assignment R → f * (R) gives rise to a functorf * : E Y → E X , called reindexing along f . (Note that functors between poset categories are just monotone maps.) We use a strengthening of poset fibrations, following [SKDH18, KKH + 19].
Definition 2.4. A poset fibration p : E → C is called a CLat ∧ -fibration if (E X , ≤) is a complete lattice for every X, and reindexing preserves arbitrary meets.
Any poset fibration p is split: we have (g • f ) * = f * • g * for any morphisms f, g that compose. Further, p is faithful. This captures the intuition that morphisms in E are morphisms in C with a certain property; e.g., relation-preserving, or non-expansive (Examples 2.5, 2.6). We note that CLat ∧ -fibrations are instances of topological functors [Her74]. We use the former, in line with existing related work [HKC18,KKH + 19]. This also has the advantage of keeping our results amenable to possible future extensions to a wider class of examples.
Example 2.5. Consider the relation fibration p : Rel → Set, where p(R ⊆ X × X) = X. Reindexing is given by inverse image: for a map f : X → Y and a relation S ⊆ Y × Y , we have f * (S) = (f × f ) −1 (S). The functor p is a CLat ∧ -fibration.
Closely related is the predicate fibration p : Pred → Set. An object of Pred is a pair (X, Γ) consisting of a set X and a subset Γ ⊆ X, and an arrow from (X, Γ) to (Y, Θ) is a map f : X → Y such that x ∈ Γ implies f (x) ∈ Θ. The functor p is given by p(X, Γ) = X, reindexing is given by inverse image, and p is a CLat ∧ -fibration as well.
In the relation fibration, we sometimes refer to an object (X, R ⊆ X 2 ) simply by R, and similarly in the predicate fibration.
Example 2.6. Let V be a complete lattice. Define the category Rel V as follows: an object is a pair (X, d) where X is a set and a function d : X × X → V, and a morphism from (X, d) For V = 2 = {0, 1} with the usual order 0 ≤ 1, Rel V coincides with Rel. Another example is given by the closed interval V = [0, 1], with the reverse order. Then, a morphism from (with ≤ the usual order, i.e., where 0 is the smallest). This instance will be denoted by Liftings and Coinductive Predicates. Let p : E → C be a fibration, and B : In that case, B restricts to a functor B X : E X → E BX , for any X in C.
A lifting B of B gives rise to an abstract notion of coinductive predicate, as follows. For any B-coalgebra (X, γ) there is the functor, i.e., monotone function defined by γ * •B X : E X → E X . We think of post-fixed points of γ * • B X as invariants, generalising bisimulations. If p is a CLat ∧ -fibration, then γ * • B X has a greatest fixed point ν(γ * • B X ), which is also the greatest post-fixed point. It is referred to as the coinductive predicate defined by B on γ.
Example 2.7. First, for a Set functor B : Set → Set, recall the lifting Rel(B) of B defined in the beginning of this section. We refer to Rel(B) as the canonical relation lifting of B. For a coalgebra (X, γ), a post-fixed point of the operator γ * • Rel(B) X is a bisimulation, as explained above. The coinductive predicate ν(γ * • Rel(B) X ) defined by Rel(B) is bisimilarity. Another example is given by the lifting B for similarity defined in the beginning of this section, which we further study in Section 4. In that section we also define a unary predicate, divergence, making use of the predicate fibration. Coinductive predicates in the fibration Rel [0,1] can be thought of as behavioural distances, providing a quantitative analogue of bisimulations, measuring the distances between states. A simple example on deterministic automata is studied in Section 4.1.
Remark 2.8. In quantitative examples one often works in a category with more structure, e.g., by replacing Rel [0,1] by the category of pseudo-metrics and non-expansive maps. Similarly, one can replace Rel by the category of equivalence relations. Defining liftings then requires slightly more work, and since we use fibrations to define coinductive predicates, this is not needed. Therefore, we do not use such categories in our examples.
We sometimes need the notion of fibration map: if B is a lifting of B, the pair (B, B) is called a fibration map if (Bf ) * • B Y = B X • f * for any arrow f : X → Y in C. If B preserves weak pullbacks, then (Rel(B), B) is a fibration map [Jac16] in the relation fibration (Example 2.5). 2.2. Coalgebraic Modal Logic. We recall a general duality-based approach to coalgebraic modal logic where we work in the context of a contravariant adjunction [PMW06,Kli07,JS09] in contrast to earlier work [KKP04,BK05b] that assumed a dual equivalence. We assume the following setting, involving an adjunction P Q and a natural transformation δ : BQ ⇒ QL: In this context, a logic for B-coalgebras is a pair (L, δ) as above. The functor L : D → D represents the syntax of the modalities. It is assumed to have an initial algebra α : LΦ ∼ = → Φ, which represents the set (or other structure) of formulas of the logic. The natural transformation δ gives the one-step semantics. It can equivalently be presented in terms of its mate δ : LP ⇒ P B, which is perhaps more common in the literature. However, we will formulate adequacy and expressiveness in terms of the current presentation of δ.
Let (X, γ) be a B-coalgebra. The semantics of a logic (L, δ) arises by initiality of α, making use of the mate δ, as the unique map making the diagram on the left below commute.
The theory map th : X → QΦ is defined as the transpose of , i.e., th = Q • η X where η : Id → QP is the unit of the adjunction P Q. It is the unique map making the diagram on the right above commute.
Example 2.9. Let C = D = Set, P = Q = 2 − the contravariant powerset functor, and BX = 2 × X A . We define a simple logic for B-coalgebras, where formulas are just words over A. To this end, let LX = A × X + 1. The initial algebra of L is the set A * of words. Define δ : BQ ⇒ QL on a component X as follows: For a coalgebra o, t : X → 2 × X A , the associated theory map th : X → 2 A * is given by th(x)(ε) = o(x) and th(x)(aw) = th(t(x)(a))(w) for all x ∈ X, a ∈ A, w ∈ A * . This is, of course, the usual semantics of deterministic automata.
In the above example, the logic does not contain propositional connectives; this is reflected by the choice D = Set. Although it is possible to include propositional connectives into the functor L (cf. e.g. [Kli07]), one usually adds those connectives by choosing D to be a category of algebras. For instance, Boolean algebras are a standard choice for propositional logic, and in Section 4 we use the category of semilattices to represent conjunction. In fact, if one is only interested in defining the semantics of the logic, one can simply work with algebras for a signature; this is supported by the adjunctions presented in the next subsection. We outline in the next subsection how this can be used to represent the propositional part of a real-valued modal logic.

Contravariant Adjunctions.
In this subsection we discuss several adjunctions that we use for presenting coalgebraic logic as above, and will allow us in Section 4 to demonstrate that a large variety of concrete examples is covered by our framework. In all cases, the adjunctions that we use for the logic are generated by an object Ω of 'truth values'. In fact, we believe all of the dual adjunctions listed in this section are instances of the so-called concrete dualities from [PT91] where Ω is the dualising object inducing the adjunction.
For a simple but useful class of such adjunctions, let D be a category with products, and Ω an object in D. Then there is an adjunction where P X = Ω X and QX = Hom(X, Ω) , where Ω X is the X-fold product of Ω. This adjunction is instrumental for representing the semantics of a coalgebraic modal logic for B-coalgebras based on predicate liftings (cf. e.g. [KP11]) within the dual adjunction framework by defining a suitable category of L-algebras. In general, describing the category of L-algebras that precisely represents a given logic (i.e., where the initial algebra corresponds to the set of formulas modulo equivalence) is nontrivial. For studying expressiveness, however, it is sufficient to consider formulas and their semantics. This can be done as follows: We start by considering a set O of propositional operators, each o ∈ O associated with a certain finite arity ar(o) ∈ N and define the (propositional) signature functor The category Alg(Σ O ) of algebras for the functor Σ O will play the role of the category D in (2.4). We assume that we are given a set of truth values Ω together with a Σ O -algebra structure a Ω : Σ O Ω → Ω, which gives an interpretation of the propositional operators. As Ω is a Σ O -algebra we obtain functors P : Set → Alg(Σ O ) op and Q : Alg(Σ O ) op → Set as described in (2.4). An Ω-valued coalgebraic modal logic L(Λ) for a functor B : Set → Set is now given as a set Λ of modal operators where each λ ∈ Λ is an Ω-valued predicate lifting λ : P n ⇒ P B with ar(λ) = n the arity of λ.
where [λ](a 1 , . . . , a n ) should be understood as name of a generator and where T Σ O denotes the free (term) monad over Σ O . The action of L Λ on a given morphism f : . . , f (a n )).
It is now easy to see that the predicate liftings in Λ give rise to a natural transformation δ : L Λ P ⇒ P B where, for an arbitrary set X, the X component δ X : L Λ P X → P BX is the unique extension of the map for λ ∈ Λ, n = ar(λ) and u j ∈ Ω X for 1 ≤ j ≤ n. In other words, (L Λ , δ) with δ being the mate of δ is a logic for B-coalgebras in the sense of (2.3). We arrive at the following picture: Example 2.10. To illustrate the outlined approach, consider the real-valued coalgebraic modal logics from [KM18]. The set Φ of formulas of these logics is given by the following definition that is indexed by a set Λ of unary modal operators: where [0, ] is a closed interval of real numbers with denoting an arbitrary positive real number, is interpreted as truncated subtraction on [0, ] given by p q := max(p − q, 0), min is interpreted as minimum and negation on [0, ] is defined as ¬q := − q. Following the construction of L Λ as described above, we obtain the following dual adjunction: Here the operations on [0, ] are , min, ¬ and − q for q ∈ Q ∩ [0, ], thus To study expressiveness relative to a coinductive predicate in a fibration p : E → C we rely on a given dual adjunction P Q between C and D together with its lifted version P Q between E and D. In a large class of examples the fibration under consideration will be of type p : Rel V → Set with P Q being the dual adjunction between Set and Alg(Σ) described above. We will now provide a proposition that yields the required dual adjunction P Q between Rel V and Alg(Σ). To obtain this dual adjunction we need a number of assumptions. First we make some assumptions on the truth and distance values Ω and V: • V is a complete lattice of distance values, • Ω is a bounded poset of truth values, Furthermore we let ∆ : Set → Rel V be the diagonal functor given by ∆X = ∆ X where Proposition 2.11. Let Ω and V be sets of truth and distance values that satisfy the above assumptions and let Σ : Set → Set be a functor. Suppose furthermore that Σ has a lifting Σ : Proof. We first have to show that the functors that form the adjunction are well-defined. In the following we write α as abbreviation for an algebra (A, α). Throughout this proof we denote the least and the largest element of V by ⊥ and , respectively. Recall that the condition for a function f to be a Rel V -morphism is To see that the functors are well-defined on objects, first note that for each Σ-algebra (A, α), the set Hom(α, a Ω ) can be turned into a Rel V -object by defining Likewise, for each Rel V -object (X, R), the set Hom(R, R Ω ) carries a Σ-algebra structure a R : ΣHom(R, R Ω ) → Hom(R, R Ω ) given by the function that maps t : 1 → ΣHom(R, R Ω ) to the following composition of arrows: is the evaluation function. To see that the above is a well-defined Rel V -morphism we only have to check that ev ∈ Rel V as the other arrows are morphisms by our assumption that Σ has a lifting Σ : Rel V → Rel V such that st and a Ω become morphisms in Rel V . We now show that ev satisfies the Rel V morphism condition. Consider two pairs (x 1 , f 1 ), (x 2 , f 2 ) ∈ X × Hom(R, R Ω ). We distinguish cases: .
To see that the Hom-functors are well-defined on morphisms we first check that Hom( , a Ω ) maps algebra morphisms to morphisms in Rel V . To this aim consider an algebra morphism h : (A 1 , α 1 ) → (A 2 , α 2 ) and g 1 , g 2 ∈ Hom(α 2 , a Ω ). We calculate: We now check that the functor Hom( , R Ω ) is well-defined on morphisms as well. Let We calculate: where (*) holds as the following diagram can be easily seen to commute in Set: This finishes the argument that the functors are well-defined. We will now argue that they form an adjunction. We prove this by defining the unit and counit of the adjunction satisfying the triangle identities.
For (X, R) ∈ Rel V we define the unit map η R : R → Hom(Hom(R, R Ω ), a Ω ) by putting η R (x) := λf.f (x). Naturality of η can be easily checked (left to the reader), but welldefinedness is not obvious. For the latter we have to show that η R is a Rel V -morphism and that η R (x) is a Alg(Σ)-morphism for all (X, R) ∈ Rel V and all x ∈ X.
To see that η R is a Rel V -morphism, consider x 1 , x 2 ∈ X.
For the counit of the (dual) adjunction we define α : (A, α) → Hom(Hom(α, a Ω ), R Ω ) by putting α (a) := λg.g(a) for all (A, α) ∈ Alg(Σ) and all a ∈ A. Again we leave it to the reader to convince themselves that is natural. We have to check well-definedness, i.e, we need to check that α (a) is a Rel V -morphism and that α is an Alg(Σ)-morphism.
To see that α (a) is a Rel V -morphism we consider g 1 , g 2 ∈ Hom(α, a Ω ): To check that α is an Alg(Σ)-morphism we calculate: where (+) is an easy consequence of (ev(g, ) • α )(a) = g(a) for all a ∈ A. This finishes the definition of unit and counit of the adjunction -checking the triangle equalities is a straightforward exercise.
The following remark is obvious, but at the same time useful for concrete examples. Remark 2.12. Let C be a full subcategory of Rel V and D a full subcategory of Alg(Σ) such that Hom(−, a Ω ) and Hom(−, R Ω ) restrict to functors of type D → C and of type C → D, respectively. Then the dual adjunction from Prop. 2.11 restricts to a dual adjunction between C and D.
The assumptions in Proposition 2.11 concerning existence of a suitable lifting of Σ are in particular met when Σ is a polynomial functor.
Corollary 2.13. Let Ω and V be sets of truth and distance values that satisfy the assumptions from Prop. 2.11, let Σ be a signature functor. Then Σ lifts to Proof. It is clear that the existence of the dual adjunction follows from Prop 2.11 once we establish that any polynomial functor Σ has a lifting to In the following we prove this claim not only for signature functors but for the collection of functors F generated by the following grammar: where J are arbitrary sets of indexes, A denotes the constant functor and Id : Set → Set denotes the identity functor. For a functor Σ ∈ F we now inductively define the action of its lifting Σ : Rel V → Rel V on objects while at the same time proving conditions (i) and (ii).
Case: Σ = A (constant functor). Then we put The conditions on ∆ and st are easy to check as in this case ∆AX = ∆ A = A∆X and as st X,Y (x, a) = a which clearly lifts to a suitable st R,S . Case: Σ = Id. Then Σ(R) = R, ∆ • Σ = ∆ = Σ • ∆ and the strength map is simply the identity.
where π j is the projection onto the j-th component of the product. For proving property (i) we consider an arbitrary set X, x 1 , x 2 ∈ ΣX and we calculate: To check that st : and consider two arbitrary pairs (x 1 , z 1 ), (x 2 , z 2 ) ∈ X × ΣY . We calculate: where for (+) we used that π j • st = st j • (id × π j ) as can be easily checked.
where the κ n denotes the n-th inclusion into the coproduct. As in the previous case we first verify (i): let X be a set and consider x 1 , x 2 ∈ ΣX. W.l.o.g. we assume there are j ∈ J and x 1 , x 2 ∈ Σ j X with x i = κ j (x i ) for i = 1, 2 -otherwise property (i) is trivially satisfied. Spelling out the definitions we get Let st j : X ×Σ j Y → Σ j (X ×Y ) be the strength maps of the components of Σ. Consider pairs (x 1 , κ j (y 1 )), (x 2 , κ j (y 2 )) ∈ X × ΣY where we assumed that the y i 's are from the same j-th component of ΣY -otherwise the strength condition is trivially true. We calculate: (R × ΣS)((x 1 , κ j (y 1 )), (x 2 , κ j (y 2 ))) where the last equality follows from the easily verifiable fact that This finishes the definition of Σ on objects. Our argument also shows that for polynomial functors Σ, the map st lifts to Rel V as required. Finally, we extend Σ to a functor Rel V → Rel V by putting Σf := Σf for all morphisms f : R → S ∈ Rel V . In order to see that Σ is well defined on morphisms one has to prove that Σf is a Rel V -morphism from Σ(R) to Σ(S) whenever f : R → S is a Rel V -morphism. This can be easily shown by induction on the structure of Σ. Functoriality of Σ is an immediate consequence of functoriality of Σ. R Then the results of this section can be summarised in the following diagram: In the next section we will see that adequacy of the logic L(Λ) follows if δ lifts to δ : B Q ⇒ QL Λ , while expressiveness is implied by an additional property of δ.

Abstract Framework: Adequacy & Expressiveness
In this section, we define when a logic is adequate and expressive with respect to a coinductive predicate, and provide sufficient conditions on the logic. Coinductive predicates are expressed abstractly via fibrations and functor lifting, and logic via a contravariant adjunction. Therefore, we make the following assumptions.
As explained in the introduction, to formulate adequacy and expressiveness, we need one more crucial ingredient: an object that stipulates how collections of formulas should be compared. In the abstract fibrational setting, we assume an object above QΦ; more systematically, a functor Q above Q.
Definition 3.2 (Adequacy and Expressiveness). Let Q : D op → E be a functor such that p • Q = Q. We say the logic (L, δ) is When we need to refer to the functors Q or B explicitly, we speak about adequacy and expressiveness via Q w.r.t. B. Examples follow in Section 3.2, where classical expressiveness and adequacy w.r.t. bisimilarity is recovered, and Section 4, where other instances are treated. Remark 3.3. Definition 3.2 can be generalised to arbitrary poset fibrations, not necessarily assuming complete lattice structure on the fibres, as follows. Adequacy means that for any B-coalgebra (X, γ), if R ≤ γ * • B X (R) then R ≤ th * (QΦ). Expressiveness means that for any B-coalgebra (X, γ), we have th * (QΦ) ≤ R for some R with R ≤ γ * • B X (R). In fact, with these definitions, if (L, δ) is both adequate and expressive then γ * • B X has a greatest fixed point, given by th * (QΦ). We prefer to work with CLat ∧ -fibrations, since the definition is slightly simpler, and it covers all our examples.
3.1. Sufficient conditions for expressiveness and adequacy. The results below give conditions on B, Q and primarily the one-step semantics δ that guarantee expressiveness (Theorem 3.7) and adequacy (Theorem 3.5). For simplicity we fix the functor Q.
Assumption 3.4. In the remainder of this section we assume a functor Q : For adequacy, the main idea is to require sufficient conditions to lift δ to a logic for B.
Proof. The first assumption yields a natural transformation δ : B Q ⇒ QL, defined on a component X by where the left arrow is the inclusion B QX ≤ δ * X (QLX), and the right arrow δ is the Cartesian morphism to QLX above δ X . It follows that δ X is above δ X . Further, naturality follows from p being faithful (as it is a poset fibration, see Section 2.1) and naturality of δ. Observe that we have thus established (L, δ) as a logic for B-coalgebras, via the adjunction P Q. Now let (X, γ) be a B-coalgebra, and R = ν(γ * • B X ). Then, in particular, R ≤ γ * • B X (R), which is equivalent to a coalgebra γ : R → BR above γ : X → BX. The logic (L, δ) gives us a theory map th of (R, γ) as the unique map making the following diagram commute.
Expressiveness requires the converse inequality of the one in Theorem 3.5, but only on one component: the carrier Φ of the initial algebra. Further, the conditions include that (B, B) is a fibration map. In particular, for the canonical relation lifting Rel(B) this means that B should preserve weak pullbacks; this case is explained in more detail in Section 3.2. Proof. Since p • Q = Q, we have that Q(ι −1 ) : QX → QY is above Q(ι −1 ), and hence QX ≤ Q(ι −1 ) * (QY ) by the latter's universal property. For the converse, consider the following composition, where the left-hand side is the Cartesian morphism: This is above the identity on QX: Proof. Let (X, γ) be a B-coalgebra, with th the associated theory map. We show that th * (QΦ) is a post-fixed point of γ * • B X : Expressiveness follows since ν(γ * • B X ) is the greatest post-fixed point.
Note that in the above theorem the reference to the initial algebra Φ could be avoided by requiring that the inequality in the assumption holds for arbitrary objects in D. We opted for the above formulation reflecting the fact that, whenever one is applying the theorem to concrete instances, it is useful that one is able to focus on the initial L-algebra only.
3.2. Adequacy and Expressiveness w.r.t. Behavioural Equivalence. In the setting of coalgebraic modal logic recalled in Section 2.2, Klin [Kli07] proved that (1) the theory map th of a coalgebra (X, γ) factors through coalgebra morphisms from (X, γ); (2) if δ has monic components, then th factors as a coalgebra morphism followed by a mono. The first item can be seen as adequacy w.r.t. behavioural equivalence (i.e., identification by a coalgebra morphism), and the second as expressiveness. 1 In the current section we revisit this result for Set functors, as a sanity check of Definition 3.2. To obtain the appropriate notion of adequacy and expressiveness, we need to compare collections of formulas for equality. Therefore, the functor Q in Definition 3.2 will be instantiated with QX = (QX, ∆ QX ) where ∆ QX denotes the diagonal. Then, for a coalgebra (X, γ), th * (QΦ) is the set of all pairs of states (x, y) such that th(x) = th(y). Adequacy then means that for every coalgebra (X, γ), behavioural equivalence is contained in th * (QΦ), i.e., if x is behaviourally equivalent to y then th(x) = th(y). Expressiveness is the converse implication. We start with an abstract result, where the functor Q assigns the equality relation (diagonal); thus this is specifically about capturing (behavioural) equivalence logically. To state and prove it, let ∆ : Set → Rel be the functor given by ∆(X) = ∆ X . This functor has a left adjoint Quot : Rel → Set, which maps a relation R ⊆ X × X to the quotient of X by the least equivalence relation containing R (cf. [HJ98]). This can be generalised to the notion of fibration with quotients, see [Jac99], but we stick to Set here. Proof. For adequacy, we use Theorem 3.5. By composition of adjoints, P • Quot is a left adjoint to ∆ • Q. It will be useful to simplify B • ∆ • QX and δ * using that B preserves diagonals in the first equality. The remaining hypothesis of Theorem 3.5 is that B • ∆ • QX ≤ δ * X (∆ • Q • LX) for all X, i.e., ∆ BQX ⊆ (δ X × δ X ) −1 (∆ QLX ), which is trivial.
For expressiveness, we use Theorem 3.7. By assumption, (B, B) is a fibration map. We need to prove that δ * But this is equivalent to injectivity of δ Φ .
The canonical lifting Rel(B) of a Set functor B always preserves diagonals, and if B preserves weak pullbacks, then it is a fibration map. Thus, we obtain expressiveness w.r.t. bisimilarity for weak pullback preserving functors, if δ has injective components.
In order to be able to cover a larger class of functors, and move to behavioural equivalence, we use the notion of lax extension preserving diagonals. Here for a function f : X → Y we denote by Gr(f ) its graph relation: The following key fact is an immediate consequence of the results in [MV15]. Fact 3.10. Let B be a set functor and let B be a lax extension of B preserving diagonals. Then on any coalgebra (X, γ) we have that behavioural equivalence is equal to ν(γ * • B X ).
Proof (Sketch). Monotonicity of B implies that γ * • B X is a monotone operator. The result now follows from Rel X being a complete lattice and behavioural equivalence being the greatest post-fixed point of γ * • B X . The latter is a consequence of [MV15,Prop. 9].
In particular, for a weak pullback preserving functor B, the canonical lifting Rel(B) is a lax extension preserving diagonals. But the results in [MV15] also show that non-weak pullback preserving set functors have such lax extensions. In fact, any finitary functor for which an expressive logic with "monotone" modalities exist, has a suitable lifting. Examples include the so-called ( ) 3 2 -functor , the functor P n that maps a set X to the collection P n X of subsets of X with less than n elements and the so-called monotone neighbourhood functor (cf. Example 7 in [MV15]). The following proposition establishes that the lax lifting B fits into the fibrational framework of our paper, and that Proposition 3.8 applies.
Proposition 3.11. Let B : Set → Set be a functor and let B be a lax lifting of B that preserves diagonals. Then B : Rel → Rel is a lifting of B along the relation fibration p : Rel → Set. In addition to that, ( B, B) is a fibration map.
Proof. In order to turn B into a functor Rel → Rel we define B(f ) := Bf -we will verify later in the proof that the functor is well-defined. Now note that for all relations R ⊆ X × X and functions f : where (*) is a well-known property of lax extensions (cf. e.g. Remark 4 in [MV15]) and the other equalities follow from the definition of reindexing. This implies that ( B, B) is a fibration map once we establish that B is a lifting of B along p : Rel → Set. For the latter we only need to verify that B is a functor on Rel. To avoid confusion, please note that [MV15] uses a different category Rel where the relations are morphisms whereas in our case the relations are objects. In order to see that B is well-defined on Rel-morphisms, consider relations R ⊆ X × X, S ⊆ Y × Y and a function f : R → S ∈ Rel. We need to show that B(f ) : B(R) → B(S). As B(f ) = Bf , we need to prove that Bf is a Rel-morphism from B(R) to B(S). Consider an arbitrary pair (t 1 , t 2 ) ∈ B(R). We have where the inclusion is a consequence of f being a Rel-morphism and monotonicity of B, and the equality is an instance of (*). Therefore (t 1 , t 2 ) ∈ B(R) implies (t 1 , t 2 ) ∈ (Bf × Bf ) −1 [ B(S)] which is in turn equivalent to (Bf (t 1 ), Bf (t 2 )) ∈ B(S). This shows that Bf : B(R) → B(S) as required. Functoriality now follows easily from the fact that Bf = Bf for all functions f .

Examples
In this section we instantiate the abstract framework to three concrete examples: a behavioural metric on deterministic automata (Section 4.1), captured by [0, 1]-valued tests; a unary predicate on transition systems (Section 4.2); and similarity of transition systems, captured by a logic with conjunction and diamond modalities (Section 4.3).
4.1. Shortest distinguishing word distance. We study a simple behavioural distance on deterministic automata: for two states x, y and a fixed constant c with 0 < c < 1, the distance is given by c n , where n is the length of the smallest word accepted from one state but not the other. Following [BKP18], we refer to this distance as the shortest distinguishing word distance, and, for an automaton with state space X, denote it by d sdw : Formally, fix a finite alphabet A, and consider the functor B : Set → Set, BX = 2 × X A of deterministic automata. We make use of the fibration p : Rel [0,1] → Set, and define the lifting B : The shortest distinguishing word distance d sdw on a deterministic automaton γ : X → 2×X A is the greatest fixed point ν(γ * • B X ) (recall that in Rel [0,1] we use the reverse order on [0, 1], see Example 2.6).
For an associated logic, we simply use words over A as formulas, and define a satisfaction relation which is weighted in [0, 1]. Consider the following setting.
The initial algebra of L is the set of words A * . The natural transformation δ is given by which is a quantitative, discounted version of the Boolean-valued logic in Example 2.9. The logic (L, δ) defines, for any deterministic automaton o, t : X → 2 × X A , a theory map th : X → [0, 1] A * , given by and for all x ∈ X, a ∈ A, w ∈ A * . We characterise the shortest distinguishing word distance with the above logic, by instantiating and proving adequacy and expressiveness. Define Technically, this functor is given by mapping a set X to the X-fold product of the object The functor Q yields a 'logical distance' between states x, y ∈ X, given by th * (QΦ). We abbreviate it by d log : X × X → [0, 1]. Explicitly, we have (4.1) Instantiating Definition 3.2, the logic (L, δ) is Here ≤ is the usual order on [0, 1], with 0 the least element (the order in Rel [0,1] is reversed).
To prove adequacy and expressiveness, we use Theorem 3.5 and Theorem 3.7. The functor Q has a left adjoint, as explained above. Further, (B, B) is a fibration map [BKP18]. We prove the remaining hypotheses of both propositions by showing the equality B QX = δ * X (QLX) for every object X in D. To this end, we compute (suppressing the carrier set BQX): Hence, the logic (L, δ) is adequate and expressive w.r.t. the shortest distinguishing word distance, i.e., d sdw coincides with the logical distance d log given in Equation 4.1.

Divergence of processes.
A state of an LTS is said to be diverging if there exists an infinite path of τ -transitions starting at that state. To model this predicate, let B : Set → Set, BX = (P ω X) A , where A is a set of labels containing the symbol τ ∈ A. Consider the predicate fibration p : Pred → Set, and define the lifting B : Pred → Pred by The coinductive predicate defined by B on a B-coalgebra (X, γ) is the set of diverging states: Now, we want to prove in our framework of adequacy and expressiveness that x is diverging iff for every n ∈ N there is a finite path of τ -steps starting in x, i.e., x |= τ n for every n. The proof relies on two main observations: • if x satisfies infinitely many formulas of τ n , then one of its τ -successors does, too; • if a state x satisfies τ n for some n, then x satisfies τ m for all 0 ≤ m ≤ n.
Combined, one can then give a coinductive proof, showing that if the current state satisfies all formulas of the form τ n , then one of its τ -successors also satisfies all these formulas. We make this argument precise by casting it into the abstract framework. First, for the logic, we have the following setting: Here Pos is the category of posets and monotone maps, and 2 = {0, 1} is the poset given by the order 0 ≤ 1. For a poset S, Hom(S, 2) is then the set of upwards closed subsets of S. The functor LS = S is defined on a poset S by adjoining a new top element , i.e., the carrier is S + { } and is strictly above all elements of S. The initial algebra Φ of L is the set of natural numbers, representing the formulas of the form τ n , linearly ordered, with 0 the top element. The choice of Pos means that the set Hom(Φ, 2) used to represent the theory of a state x ∈ X consists of upwards closed sets (so closed under lower natural numbers in the usual ordering), corresponding to the second observation above concerning the set of formulas satisfied by x.
Instantiating Definition 3.2, adequacy means that if x is diverging, then x |= τ n for all n; and expressiveness is the converse.
To this end, let t be an element of the left-hand side, and suppose towards a contradiction that for all φ with φ ∈ t(τ ), there is an element x φ ∈ Φ with φ(x φ ) = 0. Choosing an assignment φ → x φ of such elements, we get a finite set {x φ | φ ∈ t(τ )}. Let x φ be the smallest element of that set (w.r.t. the order of Φ, i.e., the largest natural number), and let ψ ∈ t(τ ) be such that ψ(x φ ) = 1; such a ψ exists by assumption on t. However, since x φ ≤ x ψ and ψ is upwards closed we have ψ(x ψ ) = 1, which gives a contradiction. Hence, the inclusion holds as required. The lifting (B, B) is a fibration map. We thus conclude from Theorem 3.7 that the logic is expressive.
4.3. Simulation of processes. Let A be a set, and define the functor B : Set → Set by BX = (P ω X) A . Let γ : X → (P ω X) A be B-coalgebra, i.e., a labelled transition system. Denote similarity by ⊆ X × X, defined more precisely below. Consider the logic with the following syntax: ϕ, ψ : where a ranges over A, with the usual interpretation x |= ϕ for states x ∈ X. A classical Hennessy-Milner theorem for similarity is: We show how to formulate and prove this result within our abstract framework. First, recall from Equation 2.2 in Section 2.1 the appropriate lifting B : Rel → Rel in the relation fibration p : Rel → Set. A simulation on a B-coalgebra (X, γ) is a relation R such that R ≤ γ * • B X (R), and similarity is the greatest fixed point of γ * • B X .
For the logic, to incorporate finite conjunction, we instantiate D with the category SL of bounded (meet)-semilattices, i.e., sets equipped with an associative, commutative and idempotent binary operator ∧ and a top element .
To add the modalities a for each a ∈ A, we proceed as follows. Let U : SL → Set be the forgetful functor. It has a left adjoint F : Set → SL, mapping a set X to the meet-semilattice P ω (X) with the top element given by ∅ and the meet by union. The functor L : SL → SL is given by LX = F(A × U X); its initial algebra Φ consists precisely of the language presented in Equation 4.2, quotiented by the semilattice equations 2 . For the adjunction, we use: which is an instance of Equation 2.4. Here 2 = {0, 1} is the meet-semilattice given by the order 0 ≤ 1. For a semilattice S, the set Hom(S, 2) of semilattice morphisms is isomorphic to the set of filters on S: subsets X ⊆ S such that ∈ X, and x, y ∈ X iff x ∧ y ∈ X.
To define the natural transformation δ S : (P ω (Hom(S, 2))) A → Hom(F(A × U S), 2) on a semilattice S, we use that for every map f : A × U S → 2 there is a unique semilattice homomorphism f : F(A × U S) → 2 extending it: For an LTS (X, γ), the associated theory map th : X → Hom(Φ, 2) maps a state to the formulas in (4.2) that it accepts, with the usual semantics.
2 To simplify the presentation we do not quotient with monotonicity axioms for the modal operators, i.e., we do not ensure that ϕ1 ≤ ϕ2 implies a ϕ1 ≤ a ϕ2. To recover (4.3), we need to relate logical theories appropriately. Define Then th * (QΦ) = {(x, y) | ∀ϕ ∈ Φ. th(x)(ϕ) ≤ th(y)(ϕ)}, i.e., it relates all (x, y) such that the set of formulas satisfied at x is included in the set of formulas satisfied at y. Thus, instantiating Definition 3.2, adequacy = ν(γ * • B X ) ≤ th * (QΦ) is the implication from left to right in Equation 4.3, and expressiveness is the converse. We prove adequacy and expressiveness. The functor Q has a left adjoint, given by P (X, R) = Hom((X, R), 2), where 2 = (2, {(x, y) | x ≤ y}). This follows by Corollary 2.13 with Remark 2.12, with SL as a full subcategory of the category of all algebras for the corresponding signature.
Remark 4.1. In fact, the expressiveness argument also goes through if we replace SL in the above argument with the category of algebras for the bounded semilattice signature. As pointed out in Sec. 2.3 this can be useful in cases where an axiomatisation of the class of algebras involved is not known. In the concrete case above we opted to work with the well-known category SL instead.

Finite-depth expressiveness and the Kleene fixed point theorem
In Section 3 we formulated expressiveness as an inequality ν(γ * • B X ) ≥ th * (QΦ) for all B-coalgebras (X, γ). The sufficient conditions formulated in Theorem 3.7 ensure that th * (QΦ) is a post-fixed point of γ * • B X , so that the desired inequality follows. Thereby, that approach relies on the Knaster-Tarski fixed point theorem, constructing the greatest fixed point as the largest post-fixed point.
In the current section we explore a different abstract technique for proving expressiveness, which instead relies on a technique for constructing greatest fixed points which is often This suggests a different route to expressiveness: we will formulate sufficient conditions to ensure that (γ * • B X ) i ( ) ≥ (th i ) * (QΦ i ) for all i ∈ N; here Φ i refers to formulas of modal depth at most i, made more precise below using the initial sequence of the functor L, and th i : X → QΦ i is the associated theory map. The above family of inequalities (indexed by i) can be thought of as finite-depth expressiveness: it states that the formulas of modal depth at most i are expressive with respect to the i-th approximation of the coinductive predicate defined by B. For instance, that logical equivalence w.r.t. formulas of depth at most i in Hennessy-Milner logic imply i-step bisimilarity.
These conditions are sufficient to ensure finite-depth expressiveness-if we then make the additional assumption that γ * • B X is cocontinuous, we obtain proper expressiveness. In the 'Knaster-Tarski' approach to expressiveness of Theorem 3.7, instead, no such assumption is explicitly formulated. So in that approach, cocontinuity is not explicitly assumed. This explains why in some of the examples-for instance similarity of labelled transition systemspart of the argument resembles a proof of cocontinuity.
A remark is in order here. The cocontinuity of γ * • B X , which is a functor on a fibre (hence, a monotone map between posets), is of course different from preservation of limits of chains by B or B. We refer to [HKC18] for a proper study of the relation between these different sequences. The current section is primarily about another the relation between these various sequences and the initial sequence of L.
Throughout this section we work again under Assumptions 3.1 and 3.4 concerning our overall categorical setting. We start by recalling the notion of initial and final sequence.
Definition 5.1 (Initial and final sequence). Suppose D has an initial object 0. The initial sequence of L is the chain (L i 0) i∈N , with connecting morphisms l i,j : L i 0 → L j 0 for all i ≤ j defined by l 0,j = ! L j 0 for all j and l i+1,j+1 = Ll i,j . Further, given an algebra α : LA → A, we inductively define a cocone α i : If C has an initial object 1, the final sequence of B is defined dually as (B i 1) i∈N , with the associated connecting morphisms b j,i : B j 1 → B i 1 for i ≤ j. Any coalgebra γ : X → BX defines a cone γ i : X → B i 1 by γ 0 = ! X and γ i+1 = Bγ i • γ.
If L preserves colimits of ω-chains, of which the initial sequence is an instance, then the colimit colim i<ω L i 0 carries an initial algebra [Adá74]. Dually if B preserves limits of ω-co-chains, lim i<ω B i 1 is a final coalgebra. In both cases, the elements of the respective sequences can be thought of as approximations of the initial algebra and final coalgebra, respectively.
For a coalgebra (X, γ), we define the cone as in Definition 5.1 from the algebra P γ • δ X : LP X → P X. Let be the transpose of i . The elements of L i 0 are thought of as modal formulas of rank at most i, and th i is the theory map of a coalgebra restricted to those formulas. It is easy to show that the th i maps satisfy the following properties: Lemma 5.2. For all i, the two triangles in the following diagram commute.
O O Furthermore we define a sequence (δ i : B i 1 → QL i 0) i∈N , which iterates δ on the final sequence of B, as follows: We use here that Q0 = 1 is a final object, as 0 is initial and Q a right adjoint. This enables us to relate th i and γ i .
Proof. By induction on i. The base case is trivial: th 0 = ! X = δ 0 • γ 0 . Suppose it holds for some i. Then: where the first equality holds by Lemma 5.2.
The following lemma shows that, for a coalgebra (X, γ), the elements of the final sequence of γ * • B X (in the fibre E X ) can be retrieved from the final sequence of B (in the total category E) by reindexing along the maps γ i : X → B i 1.
Lemma 5.4. Suppose (B, B) is a fibration map, and let (X, γ) be a B-coalgebra. Then for all i, Proof. By induction on i. The base case is easy, since reindexing in CLat ∧ -fibrations preserves top elements. For the inductive case, suppose it holds for some i. We compute: The following result now establishes a sufficient condition on δ for finite-depth expressiveness, formulated in terms of the final sequence of B and initial sequence of L.  Proof. If δ * i (QL i 0) ≤ B i 1, then: A natural way to move from the above result on finite-depth expressiveness to full expressiveness is to assume that the functors B X on the fibre preserve limits of ω-chains. Note that these are functors on fibres (that is, monotone functions), and there is no assumption on B or B preserving anything.
In particular, if B X preserves limits of ω-cochains, then the logic (L, δ) is expressive.
Example 5.7. We show finite-depth expressiveness (Prop. 5.5) via the above approach for the example of similarity of labelled transition systems (Section 4.3). The relevant endofunctors B = P ω (−) A , B, L, adjunctions P Q and P Q, and δ are all as defined there. Contrary to the treatment in Section 4.3, with the current approach it matters quite a bit whether A is finite or not. 3 For the moment, we will assume that A is finite, which significantly simplifies the matter. In Example 5.8 below we discuss the infinite case.
The final sequence of B is concretely described as a sequence of relations on B i 1, by just instantiating B for the inductive case:
and B 0 1 = {( * , * )} ⊆ 1 × 1. Thus, (t 1 , t 2 ) ∈ B i 1 iff t 1 is "i-step simulated" by t 2 , where both t 1 and t 2 are viewed as trees of height at most i. The initial sequence of L : SL → SL is characterised, once again by spelling out the definition, by L 0 0 = 0 = { } (the one-element semilattice, which is the initial object in SL) and LL i 0 = P ω (A × L i 0) (the free semilattice, see Section 4.3). Concretely, elements of L i 0 can be identified with formulas of depth at most i in the logic of Section 4.3 (diamond modalities and conjunction), quotiented by the semilattice equations. By the assumption that A is finite, each set L i 0 is finite.
We continue to prove the main hypothesis of Proposition 5.5 and Theorem 5.6: that for all i. Before doing so, we spell out δ * i (QL i 0) in some more detail. First, we characterise δ i+1 : BB i 1 → QLL i 0: The map δ i assigns to an element t ∈ B i 1 the formulas of modal depth at most i that hold for t, viewed as a tree.
We now prove (5.1) by induction on i. The base case is trivial. For the inductive case, assume (5.1) holds for some i. We have to prove that, for all (t 1 , t 2 ) ∈ δ * i+1 (QL i+1 0), a ∈ A and x ∈ t 1 (a), there exists y ∈ t 2 (a) such that (x, y) ∈ B i 1.
The above proof relies on the assumption that the set of labels A is finite. In the following example we show a way to adapt the proof to the case where this assumption is dropped.
Example 5.8. If A is not assumed to be infinite, the above proof does not work, as the meet Ψ may be infinite, which is not defined as we are working with semilattices. To remedy this, with Ψ defined as above, let Ψ 0 ⊆ Ψ 1 ⊆ Ψ 2 ⊆ . . . be an increasing sequence of finite subsets of Ψ such that i∈N Ψ i = Ψ. First note that, for each i, we have δ i (x)( ψ∈Ψ i ψ) = 1 using again the filter property. Now, consider for i ∈ N. Following the earlier reasoning, we get for each i an element y ∈ t 2 (a) such that δ i (y)( ψ∈Ψ i ψ) = 1. In fact, since t 2 (a) is finite, there exists an y such that δ i (y)( ψ∈Ψ i ψ) = 1 for infinitely many i (and as a consequence for all i). Since δ i (y) is a filter this means ψ∈Ψ i δ i (y)(ψ) = 1 for infinitely many i. Finally, since every ψ ∈ Ψ is contained in some of these sets Ψ i we obtain the desired result that δ(y)(ψ) = 1 for all ψ ∈ Ψ. The proof then concludes as above.

Future work
We proposed suitable notions of expressiveness and adequacy, connecting coinductive predicates in a fibration to coalgebraic modal logic in a contravariant adjunction. Further, we gave sufficient conditions on the one-step semantics that guarantee expressiveness and adequacy, and showed how to put these methods to work in concrete examples.
There are several avenues for future work. First, an intriguing question is whether the characterisation of behavioural metrics in [KM18,WSPK18] can be covered in the setting of this paper, as well as logics for other distances such as the (abstract, coalgebraic) Wasserstein distance. Those behavioural metrics are already framed in a fibrational setting [BKP18, SKDH18, BBKK18, KKH + 19]. While all our examples are for coalgebras in Set, the fibrational framework allows different base categories, which might be useful to treat, e.g., behavioural metrics for continuous probabilistic systems [vBW05].
A further natural question is whether we can automatically derive logics for a given predicate. As mentioned in the introduction, there are various tools to find expressive logics for behavioural equivalence. But extending this to the current general setting is non-trivial. Conversely, given a logic, one would like to associate a lifting to it, perhaps based on techniques related to Λ-bisimulations [GS13,BH17,Enq13].