A model-theoretic approach to descriptive general frames: the van Benthem characterization theorem

The celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a ‘continuous’ binary relation. The proof of our theorem generalizes Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem. 1


Introduction
Kripke models are relational structures that provide standard models for modal logic. Bisimulations are relations on Kripke models that are indistinguishable by modal logic in the sense that the truth of modal formulae is preserved under these relations. The van Benthem characterization theorem states that in fact, any first-order formula that is invariant under bisimulations must be equivalent to a modal formula [5,8]. This is often formulated more succinctly by saying that modal logic is the bisimulation-invariant fragment of first-order logic. The theorem has inspired many generalizations and alternative versions, including a similar characterization for intuitionistic logic [24,25], neighbourhood models [16] and numerous coalgebraic generalizations [29]. Another notable result is the Janin-Walukiewicz theorem [19], showing that the modal μ-calculus is the bisimulation-invariant fragment of monadic second-order logic.
A result that is of particular importance to this paper was given by Rosen in [27], showing that over finite models, too, modal logic is the bisimulation-invariant fragment of first-order logic. This is particularly remarkable because the compactness theorem of first-order logic features prominently in the proof of the original van Benthem characterization theorem, while the class of finite models crucially lacks the compactness property. The result for finite models has in turn been generalized to multiple other classes lacking the compactness property, including the classes of rooted finite models, rooted transitive models and well-founded transitive models [10].
We next recall the general frame semantics for modal logic. DEFINITION 2.1 A triplet g = (W , R, A) is a general frame if (W , R) is a Kripke frame and A is a field of sets 2 over W that is closed under the operation R (i.e. for every a ∈ A also R a ∈ A). The underlying frame (W , R) of g will be denoted by g # .
A quadruplet m = (W , R, A, V ) is a general model based on g = (W , R, A) over a set of propositional variables P if V : P → A is a function. The definition of · m is the same for general models as it is for Kripke models. This definition is motivated by the fact that now, the truth set ϕ m belongs to A for any general model m and formula ϕ.
General frames also have a natural structure of a topological space, with A acting as a basis of clopens on a universe W . This opens the subject for topological analysis, which has led in the past to many crucial insights into these structures.
Some specific types of general frames are of special importance. The following classes of important general frames will be relevant to this paper. DEFINITION 2.2 Let g = (W , R, A) be a general frame. Then • g is called differentiated if for all distinct w, v ∈ W , there exists an a ∈ A such that v ∈ a and w / ∈ a; • g is tight 3 if for all w, v ∈ W with (w, v) / ∈ R, there exists an a ∈ A such that v ∈ a and w / ∈ R a; • g is called compact if all collections A ⊆ A with empty intersection have a finite subcollection A 0 ⊆ A with empty intersection; • g is called image-compact if for any point w ∈ W , a collection A ⊆ A whose intersection is disjoint from R[w], the set of successors of w, has a finite subcollection A 0 ⊆ A whose intersection is also disjoint from R[w]; • g is called descriptive if it is differentiated, tight and compact. The class of models based on descriptive frames will be denoted by D.
Models based on such a general frame are called differentiated, tight, compact or descriptive models.
In the presence of these modifiers, the adjective 'general' will usually be omitted from both the frames and the models.
The term compactness is used because it corresponds to topological compactness. Differentiatedness is equivalent to the topological space being totally separated (i.e. every two distinct points being separated by a clopen set). Topological spaces that are compact and totally separated are called Stone spaces. Descriptive frames are then pairs (W , R) where W is a Stone space and R is such that • For every clopen set U ⊆ W , the set R U is also clopen.
We refer to [8,21] for a detailed discussion on the connection of these two approaches to descriptive frames. From now on, we will use the topological concept interchangeably with the subclass of general frames.

Duality
One of the many reasons descriptive frames have proven a valuable tool is their role in Jónsson-Tarski duality for modal algebras. This duality builds on the celebrated Stone duality between the category of Boolean algebras and Boolean algebra homomorphisms and the category of Stone spaces and continuous maps.
The category MA has as objects all modal algebras and as morphisms the Boolean morphisms f such that f (♦a) = ♦f (a) for all a.

DEFINITION 2.4
The category DF is the category with all descriptive frames as objects and as morphisms the continuous bounded morphisms. 4 The functors that establish the category duality will be referred to as (−) * and (−) * . We will brief ly recall their definition.
Let M = (B, ♦) be a modal algebra. Then let M * = (W , R, A) be the descriptive frame (see [8,Theorem 5.76]) with the collection of ultrafilters on B; Moreover, if f : M → M is a morphism of modal algebras, let then f * : M * → M * be the map given by f * (F ) = f −1 [F ] for each ultrafilter F ∈ M * . In [8,Proposition 5.80], this is shown to be a continuous bounded morphism. Let g = (W , R, A) be a descriptive frame. Then the associated modal algebra is g * = (A, R ). A continuous bounded morphism f : g → g is sent to the map f * : (g ) * → g * of modal algebras given by f * (a) = f −1 [a] for each a ∈ A. THEOREM 2.5 (Jónsson-Tarski duality [8,9,20,21]). The functors (−) * and (−) * provide a dual equivalence between the categories MA and DF.
It is important to note the use of the (−) * -functor is not restricted to descriptive frames only. For the entire class of general frames, one may define the exact same operation. Combining it with the other functor results in an operation that turns every general frame into a descriptive frame. This may be called the descriptive completion of the general frame. DEFINITION 2.6 Let g = (W , R, A) be a general frame. Then its descriptive completion is defined as (g * ) * =: The name 'descriptive completion' is justified by the following theorem. THEOREM 2.7 ([8, Theorem 5.76]) Let g be a general frame. Then In the special case that g is a Kripke frame, meaning that A = P(W ) is the powerset, then (g * ) * is known as the ultrafilter extension of g, see, e.g., [8,Section 2.5].
This completion construction will be used in Section 4 to develop the central tool in the proof for a van Benthem-type result.

Kripke and Vietoris bisimulations
An important notion in modal logic that is central to main theorem of this thesis is the concept of bisimulations. Intuitively, if two points w and v in models M and N respectively are bisimilar, they are impossible to distinguish by walking from w and v through the models and looking only at the propositional variables that are true at the points reached. In this section, we will recall the notions of Kripke and Vietoris bisimulations for descriptive models. DEFINITION 2.8 Let M and M be (general) models on frames (W , R) and (W , R ) with valuations V and V . A relation Z ⊆ W × W is called a (Kripke) bisimulation if for all (w, w ) ∈ Z the following three conditions hold: If two points w and v in models M and N are linked by a bisimulation, then we say that w and v are bisimilar. A model M with a fixed element w will be called a pointed model. If two points w and v in models M and N respectively are bisimilar, then we say that pointed models M, w and N, v are bisimilar.
As will be shown in Theorem 2.15, two points linked by a bisimulation satisfy the same modal formulae. DEFINITION 2.9 ([6,12]). Let m and m be general models and Z a Kripke bisimulation between them. If Z is closed in the product topology of the two associated topological spaces, then it is called a Vietoris bisimulation.
If two pointed models are linked by a Kripke bisimulation this is denoted by m, w ↔ m , w and if they are linked by a Vietoris bisimulation this will be written as m, w m , w .

REMARK 2.10
This definition is motivated by a coalgebraic perspective on descriptive frames, in which it is the coalgebraic definition of a bisimulation on descriptive frames as coalgebras for the Vietoris functor. However, on descriptive frames this is equivalent to the definition provided above. A detailed exploration of the coalgebraic notion of descriptive frames can be found in [6,22,31] and an extensive treatment of Vietoris bisimulations, including this equivalence, can be found in [6,12]. If m, w and n, v are two pointed image-compact general models, then the following are equivalent:

Finite bisimulations
Crucial for this paper will be the notion of finite (approximations to) bisimulations. Like the finitary modal language, the finite number of steps involved makes them much easier to deal with.

DEFINITION 2.12
Let k ∈ N be a natural number, P a set of propositional variables and M and M be two (general) models with frames (W , R) and (W , R ) and valuations V and V over P, respectively. Then a kbisimulation over P is a ⊆-decreasing (k + 1)-sequence (Z ) 0≤  If there exists a k-bisimulation (Z ) 0≤ ≤k with (w, v) ∈ Z k , this is denoted by M, w ↔ k N, v.
One could also define finite Vietoris bisimulations when all Z are closed, but this notion will not be useful for this paper.

REMARK 2.13
If Z is a Kripke bisimulation, then for any k ∈ N the sequence (Z) 0≤ ≤k is a k-bisimulation. As such, LEMMA 2.14 Let k ∈ N and M, w and N, v be two pointed (general) models and P a set of propositional variables. Then where k denotes agreement on all modal formulae on modal depth ≤ k. Moreover, if P is finite, then the converse implication holds as well.
PROOF. The proof for Kripke models can be found in e.g. [8,Proposition 2.31]. As general models satisfy the same formulae as their associated Kripke models 5 and the definition of k-bisimulations references only the Kripke model structure, this immediately extends the result to general models. THEOREM 2.15 Any two (Vietoris-)bisimilar pointed (general) models satisfy the same formulae.
PROOF. This follows immediately from Lemma 2.14 and Remark 2.13 for the Kripke case and additionally Theorem 2.11 for the Vietoris case.
An important note to make is that these are equivalence relations, in particular transitive.
PROOF. Let (Z m ) 0≤m≤ be an -bisimulation between M 0 , w 0 and M 1 , w 1 , and let ( Z n ) 0≤n≤k be a k-bisimulation between M 1 , w 1 and M 2 , w 2 . Then consider (Z m ; Z m ) 0≤m≤ as an -bisimulation between M 0 , w 0 and M 2 , w 2 , where ; denotes the composition of relations. Suppose that (v 0 , v 2 ) ∈ Z m ; Z m for some m ≤ . Then there exists a v 1 such that v 0 Z m v 1 and v 1 Z m v 2 . Consequently, v 0 satisfies the same propositional variables as v 1 , which in turn satisfies the same propositional variables as v 2 , verifying that v 0 and v 2 satisfy the same propositional variables.
For the forth condition, if v 0 has a successor x 0 , then there is a successor x 1 of v 1 such that x 0 Z n x 1 for all n < m. Then from v 1 Z m v 2 , it follows that there is a successor x 2 of v 2 such that x 1 Z n x 2 for all n < m. Therefore, (x 0 , x 2 ) ∈ (Z n ; Z n ) for all n < m. The back condition is identical. The final observation then follows from Remark 2.13.

Model-theoretic failures on descriptive models
In this section, we will study the basic model-theoretic properties of descriptive frames. We will show that, similarly to finite frames, many classical model-theoretic results fail on descriptive frames.
The next lemma demonstrates that a subclass of finite frames can be defined with a single firstorder sentence. 6 LEMMA 3.1 An infinite, irref lexive, linear order cannot be given the structure of a descriptive frame. Therefore, the subclass of finite, irref lexive, linear orders of the class of descriptive models can be defined by a single first-order sentence.
PROOF. Suppose g = (W , R, A) is a descriptive frame such that (W , R) is an infinite, irref lexive linear order. The ⊆-decreasing sequence of sets W ⊇ R[W ] ⊇ · · · ⊇ R n [W ] ⊇ · · · must be non-empty for all n, otherwise g would be a finite chain. As such, C := n R n [W ] is closed and non-empty, because the space is topologically compact by assumption, the R-image of a closed set is closed [8,Proposition 5.83iii] and induction thus grants that R n [W ] is closed for all n. By tightness and irref lexivity, for each x ∈ C there must be a clopen set a x such that x ∈ a x but x / ∈ R a x . As then Since the x i are linearly ordered, there must be a least one, say The fact that R a x 1 is clopen, hence compact, and the fact that Thus, no such descriptive frame exists.
The irref lexive linear orders are definable in the single first-order sentence λ given by total, irreflexive and antisymmetric Since all finite irref lexive linear orders can be given the structure of a descriptive frame (with the powerset as a collection of admissible sets), the subclass of finite, irref lexive linear orders in the class of descriptive models can be defined by a single first-order sentence.
This lemma immediately implies the failure of the compactness theorem for first-order logic.
The class of descriptive models is not compact over first-order logic.
PROOF. From Lemma 3.1, the compactness theorem fails almost immediately. Taking λ to be Formula 1 and letting ϕ n denote the existence of at least n distinct elements by Lemma 3.1 implies directly that {λ} ∪ {ϕ n } n∈N is not satisfiable on the class of descriptive models. However, every finite subset is satisfied on a finite, irref lexive linear order, which can always be given the structure of a descriptive model with the powerset as collection of admissible sets.
With the failure of the compactness theorem in mind, it is natural to wonder if some of its famous consequences fail, and indeed they do. THEOREM 3.3 (Failure of Beth definability theorem on descriptive models). Let R be a binary relation symbol, and let P be a unary predicate symbol. Then there exists a formula ϕ that implicitly defines 7 P over the class D of descriptive models such that there is no formula ψ ∈ FOL({R, P}, ∅) that explicitly defines P relative to ϕ 8 over D.
PROOF. Let λ be Formula 1 defining the finite, irref lexive linear orders over the class of descriptive models as per Lemma 3.1. Then take ϕ to be the formula saying that R is an irref lexive linear order, the R-minimum does not have P and any two immediate successors disagree on P. Formally, ϕ is The models of λ are precisely finite, irref lexive linear orders that assign P to exactly the even points. Suppose now that there were a formula ψ(x) ∈ FOL({R}, ∅) that explicitly defined P. Then the formula ε stating that R is an irref lexive linear order and the maximum has the property ψ, given explicitly by would be a formula in FOL({R}, ∅) that characterizes exactly the finite, even, irref lexive linear orders. However, sufficiently large linear orders are first-order equivalent as shown in [11, Example 2.3.6] with the Ehrenfeucht-Fraïssé method. It follows that no one formula can characterize the even linear orders and thus ψ cannot exist.
The Craig interpolation theorem fails on the class of descriptive models for essentially the same reason that the Beth definability theorem fails. Usually, the Beth definability theorem is stated as a consequence of the Craig interpolation theorem, so that the failure of former immediately implies failure of the latter, but we consider a direct proof to be informative. THEOREM 3.4 (Failure of Craig interpolation theorem on descriptive models). Let R be a binary relation symbol, let P 0 , P 1 be unary predicate symbols, and let D 0 , D 1 and D 01 denote the class of descriptive models over predicate sets {P 0 }, {P 1 } and {P 0 , P 1 }, respectively. Then there exist formulae ϕ ∈ FOL({R, P 0 }, ∅) and ψ ∈ FOL({R, P 1 }, ∅) such that ϕ | D 01 ψ, but there is no formula θ ∈ FOL({R}, ∅) such that ϕ | D 0 θ and θ | D 1 ψ.
PROOF. Let ϕ state that R is an irref lexive linear order, P 0 occurs only on the even points as in Formula 2 in the proof of Theorem 3.3, as well as that the maximum satisfies P 0 like Formula 3. Let ψ state that R is an irref lexive linear order, and if P 1 occurs only on the even points, then the maximum has P 1 .
The formula ϕ is true on the finite, even, irref lexive linear orders with P 0 on the even points. Similarly, for ψ to be true, a structure must be a finite, irref lexive linear order with P 1 true on an odd point or false on an even point or it must be even. Since all structures satisfying ϕ are even, it follows that ϕ | D 01 ψ.
Suppose now for contradiction that there is an interpolant θ ∈ FOL({R}, ∅) such that ϕ | D 0 θ and θ | D 1 ψ. Then in particular θ must be true on all finite, even, irref lexive linear R-orders. Again from [11,Example 2.3.6], this means that θ is true on all sufficiently large finite, irref lexive linear R-orders. But an odd linear order among these can be expanded with a predicate P 1 on the even points, from which it follows that θ | D 1 ψ. This is a contradiction.
These two results use Lemma 3.1 to reduce a model-theoretic problem for the class of descriptive models to the corresponding result in finite model theory. This reasoning can resolve model-theoretic problems on descriptive models that have an analogue on finite models but does not help when considering results that have no sensible corresponding statement in finite model theory.
For an example of such results, consider the celebrated upward Löwenheim-Skolem theorem. It states that any first-order theory T with an infinite model has arbitrarily large models. The upward Löwenheim-Skolem theorem has no meaningful interpretation in finite model theory because the hypothesis of the implication is vacuously false. However, as the theorem is an immediate consequence of the compactness theorem, it is natural to wonder if it holds on the class of descriptive models on which this the compactness property fails. Indeed, this class turns out to lack the upward Löwenheim-Skolem property. In fact, it even fails when the theorem is restricted to speak only about formulae.
(antisymmetric and total) First, consider the Kripke frame (ω + 1, R) with Rab if and only if a > b or a = b = ω, equipped with the collection of admissible sets given by the finite subsets of N and cofinite subsets of N with ω. See Figure 1 for the underlying Kripke frame. This is easily checked to be a descriptive frame. Then any model based on this frame satisfies ϕ, because the relation is a linear order, has a ref lexive minimum and no other ref lexive points. Now suppose that m = (W , R, A, V ) is a descriptive model of cardinality |W | > ℵ 0 whose underlying frame is a linear order with exactly one ref lexive point v that is also its minimum. Because Downloaded from https://academic.oup.com/logcom/advance-article/doi/10.1093/logcom/exaa040/5896992 by guest on 11 October 2020 of transitivity and cardinality reasons, there is at least one point w with infinitely many successors that is not the minimum: where the last equality follows from transitivity. So there is at least one In this section, we gave a number of negative results about model theory of descriptive frames and models. However, as Barwise and van Benthem [4] have suggested, 'failures' of classical results in a new area may be viewed as failures of one particular formulation of these results-while other formulations (equivalent for the original logic) might hold. For example, [4] shows this for interpolation theorems that fail in one sense, but hold in another for infinitary logics. This raises an interesting question whether there exist modified versions of the classical model-theoretic results that hold on descriptive frames and models.
For example, the proof for the failure of the upward Löwenheim-Skolem theorem relies very specifically on the unique properties of ℵ 0 , namely that every smaller cardinal is finite. The same proof would fail immediately, with no obvious remedy, when starting out with a model of size ℵ 1 , because such a model would have infinitely many limit points and thus at least as many ref lexive points. One might thus conjecture that any theory with a model of size at least ℵ 1 must have arbitrarily large models.
Moreover, as previously noted, descriptive models are natural generalizations of finite models, and their model theories are very similar. While most of classical model theory fails for finite models, there is a notable exception: Rossman's homomorphism preservation theorems [28, Theorems 1.6-1.9]. A natural question, then, is to ask if Rossman's theorem also holds for descriptive models. However, based on the complexity of the proof for Rossman's original results, we expect this to be highly non-trivial.

The van Benthem characterization theorem for descriptive models
In this section, we prove the van Benthem characterization theorem for the class of models over descriptive frames, stating that modal logic is the Kripke-bisimulation-invariant (or equivalently the Vietoris-bisimulation-invariant) fragment of first-order logic. 3. If m, w and n, v are two Vietoris-bisimilar pointed, descriptive models, then The strategy followed will be modelled after [27] and is visually represented in Figure 2.
That is, the unravelling tree will be modified in Definitions 4.12 to remain in the class of descriptive models, while still Kripke-bisimilar (and Vietoris-bisimilar) to the original model. On these tree-like structures, an Ehrenfeucht-Fraïssé-type argument (see [14] for more details on this technique) will show in Lemma 4.26 that bisimulation-invariance implies k-bisimulation-invariance for sufficiently large k. This will imply that any formula that is bisimulation-invariant on descriptive models is modally expressible. Assuming that α is bisimulation-invariant, m | α[w] and m, w↔ n, v for sufficiently large , the formula α can be followed clockwise around the diagram to conclude n | α [v]. The final conclusion of the proof can be found in Section 4.3 FIGURE 2 A visual representation of the argument that will be used to prove Theorem 4.1.

Unravelling for general frames
Towards proving the van Benthem theorem for descriptive models, it will be necessary to modify the notion of unravelling trees and unravelling forests to descriptive frames. To do this, first the unravelling forest of all finite paths needs to be given the structure of a general frame. The most canonical way of doing that, pulling back the admissible sets through the projection map π , does not offer a useful solution. The resulting frame would not inherit differentiatedness nor tightness. To accomplish this inheritance, R T -closure and the collection I of paths of length 0 will be added.  is the forest of all finite paths in F and with collection of admissible sets A T , a subalgebra of (P(W T ), R T ) defined through the surjective bounded morphism π : T(F) F by the following recursive schema: In a similar vein, for a general model m = (g, V ), the unravelling cover will be T(m) := (T(g), P op π • V ), where P op π(a) = π −1 [a]. For any point w ∈ W , the connected component of paths starting at w will be written as T w (g) or T w (m).
Despite these extra additions, the resulting frame is always a general frame.

PROOF. All to be checked is that A T is a field of sets and closed under the R T -operation. This is an elementary, but tedious induction. Closure under binary union and intersection is immediate. Closure under complements can be checked by induction on the construction of A T . The complement of the initial points I c = R T [W T ] is the set of points with a predecessor and is in
The final case, the R T -image of an admissible set, requires a minor observation about R T : distinct points have disjoint image sets. After all, two distinct paths cannot have the same extension. An alternative way of saying this is that each point has at most one R T -predecessor. As such, we have for all a ⊆ W T that because each point with no predecessor in a either has a its unique predecessor in a c or has no predecessor.
For closure under R T , note that every element in A T can be written as a finite union of finite intersections of elements of the form π −1 [a] for a ∈ A admissible, R T [b] for b ∈ A T or I. This is evident for elements of the forms (1), (2) or (3) and will follow by induction on the construction for elements of the form (4).
As R T distributes over unions, it is sufficient to show R T -closure for finite intersections. By induction on n, it will be shown that for b 1 , . . . , b n of the forms (1), (2) and (3), the set Moreover, R T (R T [b]), again because each point has at most one predecessor, is the subset of b given by points with at least one successor. After all, x ∈ R T (R T [b]) if and only if there is a y such that y ∈ R T [b] and xR T y. That is equivalent to y being a successor to x and having a predecessor in b, and since predecessors in this frame are unique, this predecessor must be x. So Now suppose that R T (b 1 ∩ · · · ∩ b n ) ∈ A T when all b i are of the forms (1), (2) or (3). Consider then b 0 ∩ b 1 ∩ · · · ∩ b n for n ≥ 1. If one of the b i is of the form I, then it follows immediately that R T (b 0 ∩ b 1 ∩ · · · ∩ b n ) ⊆ R T I = ∅. If one of them, without loss of generality b 0 , is of the By the uniqueness of predecessors, this is equivalent to having x ∈ b and x ∈ R T (b 1 ∩ · · · ∩ b n ). As the latter was admissible by the induction hypothesis, it follows that Finally, if none of the b i are of the form I or R T [b ] for some b ∈ A T , then they must all be of the form b i = π −1 [a i ], from which it follows that: Now towards the main result of the paper, it is useful to find out which properties of the general frame are transferred to its unravelling cover.

REMARK 4.4 For all a ∈ A T and n ∈ N, the set (R T ) n [a] is admissible.
In order to obtain the van Benthem result for descriptive models, we will show that properties defining descriptive frames are preserved under this construction. PROOF. Let x = (x i ) i≤n , y = (y j ) j≤m ∈ W T be distinct paths. If n > m, then y / ∈ (R T ) n [W T ] x. So assume n = m. Then their distinction must mean there is some k ≤ n such that x k = y k . Then by differentiatedness of g there is an a ∈ A such that y k / ∈ a x k . But then If n := l( y) = l( x) + 1, then there must be a k ≤ l( x) such that x k = y k . By assumption, g is differentiated, so there exists an a ∈ A with x k / ∈ a y k . This implies that y ∈ ( Although surprising, it may be understandable that tightness is not required, as the structure of the forest itself already separates unconnected points automatically. PROPOSITION 4.7 Let g = (W , R, A) be a general frame and T(g) its unravelling cover. Take x ∈ W T . Then π R T [ x] : ] is a homeomorphism between the subspace topologies.
PROOF. It is obviously a continuous bijection as π −1 [a] ∈ A T for all a ∈ A, which is the basis of the topology on g. To prove continuity of the inverse, it is sufficient to show that all basis elements in the subspace topology of By induction on the recursion schema for A T . It is obvious for the restriction of π −1 [a] and I. If b ∈ A T , then by using again that distinct points have disjoint R T -successor sets. Finally, π −1 distributes over intersection and union, completing the induction.

The descriptive unravelling
The unravelling cover of a compact frame is not necessarily compact, as the following example demonstrates. This shows that while differentiatedness of g implies T(g) is differentiated and tight, compactness may not be preserved. In fact, no collection of admissible sets can be constructed with which an unravelling forest of a descriptive frame with arbitrarily long paths is descriptive. Like for the proof on finite models from [27], the unravelling must be modified to become descriptive. In principle, this could be done in the same way as in [27]. Putting the original frame at a sufficiently long distance from I would likely suffice for a reproduction of the argument. However, for descriptive frames, there exists an alternative construction that will be used in this paper: the descriptive completion using Jónsson-Tarski duality discussed in Definition 2.6. 9 DEFINITION 4.12 Let g be a general frame. Then let its descriptive unravelling be the descriptive completion (Definition 2.6) of the unravelling cover (Definition 4.2) of g. Write g := ((T(g)) * ) * to abbreviate.
This will turn out to be a very well-behaved construction for descriptive frames and will be key to the theorem.
PROOF. For the implication from right to left, assume that y ∈ R [x]. To prove that F x R * F y , let a ∈ F y . Then y ∈ a. From xRy it follows that x ∈ R a, yielding R a ∈ F x . Since a was arbitrary, this holds for all a ∈ F y , so that F x R * F y .
For the implication from left to right, assume F x R * F. By definition, if a ∈ F then R a ∈ F x , implying x ∈ R a. Therefore, there exists an x ∈ a such that xRx .
So for every a ∈ F, we have a ∩ R[x] = ∅. Since F is closed under finite intersections, we obtain that {a ∩ R[x] : a ∈ F} has the finite intersection property. By compactness of such that for all a ∈ F, we have y ∈ a. So F = F y for some y ∈ R[x], because it is an ultrafilter. REMARK 4.14 One might have expected tightness to show up in the proposition above to prove that F x R * F y implies xRy, but tightness as an immediate consequence of image-compactness and differentiability, so it may be reasonable to expect that image-compactness is strong enough to prove something not quite as strong as tightness. PROPOSITION 4.15 Let g be a differentiated and image-compact frame. Then g (g * ) * is a generated subframe 10 through a topological embedding ι g : x → F x . That is, g is homeomorphic to its image under ι g .
PROOF. Lemma 4.13 gives immediately that it is a bounded morphism. From the fact that g is differentiated, it follows that ι g injective, making g a generated subframe. To see that it is a homeomorphism on its image, let a be an admissible set on g. Then so that ι g and ι −1 g preserve the basis elements of the topology, ensuring continuity for both it and its inverse restricted to the image. 9 Actually, the construction is quite natural in that it admits several potential equivalent definitions, including an explicit construction and a topological definition [18, Definition 2.86, Definition 5.14] as well as a definition through a universal property, but these will not be needed in this paper. 10 Generated subframes are given by injective bounded morphisms.

THEOREM 4.16
Let g be a descriptive frame. Then ι = ι T(g) : T(g) g is a continuous embedding.
PROOF. Note that descriptive frames are in particular, image-compact and differentiated, so Corollary 4.8 and Proposition 4.5 gives that T(g) is image-compact and differentiated. Proposition 4.15 then give the theorem.
In fact, an even stronger claim is true.

THEOREM 4.17
Let g be a descriptive frame. Then g # = T(g) # L for some unspecified frame of limit points L, 11 where the #-operation takes the underlying Kripke frame of a general frame.
PROOF. Let g = (W , R, A). From Theorem 4.16, it is sufficient to show that two points in g can only be (R T ) * -related if they are both inside or both outside T(g) # . Theorem 4.16 implies that if w is in the image of ι : T(g) g, then the (R T ) * -successor set of w is, too. To complete the theorem, the predecessor set has to be, as well. This means that if F(R T ) * F x for some ultrafilter F and the ultrafilter F x generated by x, Towards contraposition, assume that F = F y for any y ∈ (R T ) −1 [x]. In T(g), each point has at most one predecessor, so (R T ) −1 [x] ⊆ {y} for some y. In particular, this means there exists some , either because it has no predecessor or because y / ∈ a. By As such, there is an isomorphic copy of T(g) # in g # . The next step is to upgrade the descriptive unravelling to a descriptive model. PROOF. Both π and ι are bounded morphisms, so they satisfy the back and forth conditions by construction. The propositional requirement is satisfied because The final remark is then given by [6, Theorem 5.2].

Preservation under finite bisimulations
The previous section provides a tool with which to show invariance under bisimulation implies invariance under some finite bisimulation. This tool will now be used to achieve this through Ehrenfeucht-Fraïssé methods. To this end, there is a final combinatorial construction that will prove useful: a duplication process. It will be useful to copy points in a manner that preserves compactness. DEFINITION 4.19 Let A and B be fields of sets over universes X and Y . Write A ⊗ B for the field of sets over the universe X × Y generated by {a × b | a ∈ A, b ∈ B}. PROPOSITION 4.20 Let A and B be fields of sets generating topological spaces X and Y as bases. Then A ⊗ B is a basis for the product space X × Y.
PROOF. To see that A ⊗ B generates topology that is at least as fine, note that the basis of X × Y consists of all U × V where U and V are open subsets of X and Y, respectively. This means that U = η∈I a η and V = ξ ∈J b ξ for a η ∈ A and b ξ ∈ B and index sets I and J . But then it is immediate that To PROOF. This follows immediately from Proposition 4.20 and the fact that the product of two compact spaces is compact and the product of totally separated spaces is totally separated.

DEFINITION 4.22
Let g = (W , R, A) be a general frame, and let F be a field of sets over a universe X . Define the F-multiplier of g by which will be called the F-multiplier of m.

REMARK 4.23
There is an obvious surjective continuous bounded morphism π 0 : F ⊗X F given by projection on the first coordinate.

LEMMA 4.24
Let g = (W , R, A) be a descriptive frame, and let F over X be a compact and differentiated field of sets. Then g ⊗F is a descriptive frame.
PROOF. Compactness and differentiatedness follow immediately from Corollary 4.21. To see tightness, let ((w, x 1 ), (v, x 2 )) / ∈ R ⊗ X . Then (w, v) / ∈ R. From tightness of g follows the existence of a ∈ A such that v ∈ a but w / ∈ R a. Then in particular, (v, x 2 ) ∈ a × X , but (w, x 1 ) / ∈ ( R a) × X . It is a general frame in the first place because completing the proof.
As mentioned before, this construction will be used to apply Ehrenfeucht-Fraïssé method. Multiple constructions are conceivable, but the approach taken here is adopted for its convenience. It will be inspired by Hanf's lemma [15,Lemma 2.3]. Like Hanf's lemma, it relies on the notion of the Gaifman neighbourhood.
. That is, the -neighbourhood of S is the set of points that can be reached in steps along or against R. The letter N is used for the set, and N is used for the subframe and the submodel.
That is, the -neighbourhood of a set S is the collection of sets that can be reached in at most steps forwards or backwards along R from S.
PROOF. The proof uses the Ehrenfeucht-Fraïssé method. The reader is referred to [14] for an exposition of the method.
By induction on the number of rounds played so far, it will be shown that duplicator can counter any move by spoiler. To this end, write (k) := 3 n−k .
More precisely, let the induction hypothesis denote that for any (a i ) i<k and (b i ) i<k such that A move a k can be countered with a move b k such that the above condition holds with k replaced by k + 1. In particular, there will be a local isomorphism between the elements chosen. From symmetry, the response of an a k to a b k can be given similarly. Inductively performing this until k = n then gives victory for duplicator. Suppose that there have been k turns and the inductive hypothesis holds. When spoiler makes a move a k , there are two cases to consider: In the former case, let θ : be the isomorphism assumed in the induction hypothesis, and let b k = θ(a k ). Observe that 2 · 3 n−k−1 + 3 n−k−1 = 3 n−k , so and the same for b. It follows immediately that the restriction of θ is again an isomorphism between these two smaller neighbourhoods.
In the second case, remark that N (k) ({b i } i<k ) can only intersect at most k connected components. Because J was infinite per assumption, there is a ρ ∈ J such that N (k) ({b i } i<k ) ∩ C ρ = ∅. By assumption, there is a π such that for some v. Therefore, there is a b k in C ρ (namely the image of v under the isomorphism directly above) such that N (0) (a k ) ∼ = N (0) (b k ). Choosing it, one obtains where denotes disjoint union of models and the isomorphism on the different disjoint components is preserved, so that it is still the identity on X and sends the a i to the b i .
Symmetry of the models M and N guarantees that the exact same argument provides a response a k to a move b k by spoiler.
At the end, this gives duplicator an isomorphism which, using (n) = 1, can be restricted to a local isomorphism winning the game for duplicator. The implication (1) ⇒ (2) is a well-known property of Kripke bisimulations and modal logic [8,Theorem 2.20]. Modal formulae are bisimulation-invariant, so any first-order formula equivalent to one is also bisimulation-invariant.
The only interesting implication is the implication towards (1). Towards the implication (2) ⇒ (1), it will turn out to be sufficient that Kripke-bisimulation-invariance implies finite-bisimulationinvariance.
Let α(x) be a bisimulation-invariant formula in one free variable with quantifier depth q(α). Write = 2 · 3 q(α) and assume that m, w and n, v are pointed descriptive models with respective universes W and W such that • m, w ↔ n, v over the propositional variables corresponding to the predicates occurring in α.
Let κ be an infinite cardinal greater than those of the universes of m and n. Then the order topology is compact and totally separated on the ordinal number κ + 1. Write K for the field of clopens of this topology. Remark 4.23 gives that m ⊗K , (w, 0) ↔ n ⊗K , (v, 0) and Corollary 4.18 implies ,0)) .
Write w := F ((w,0)) and v := F ((v,0) where the equalities marked by * follow from the fact that two paths can only be related in the tree if one extends the other, and two paths with different starting points cannot extend one another. Recall from Definition 4.2 that T x (M) is the submodel of paths starting at x. The submodels L and Λ are unspecified frames of limit points. For any η, ξ ∈ κ + 1, through the map of switching the initial point. Moreover, and B η = C η := w∈W K-duplicates. The same reasoning applies to f i (x), so because they agree on each ♦ϕ per assumption, for each theory x and f i (x) either both have κ many successors with that theory or none. It follows that for every ϕ ∈ Σ, there exists a bijection g x ϕ between the successors of x satisfying ϕ and the successors of f i (x) satisfying ϕ. Choosing one such g x ϕ for each ϕ allows the construction of this preserves all new relations and predicates. Moreover, y ∈ R[x] with x ∈ R i [ w] has modal theory characterized by ϕ ∈ Σ and therefore by construction y −i−1 g x ϕ (y). Since this also held for f i by induction hypothesis, the inductive condition is satisfied again.

Conclusions and future work
The main result of this paper is the validity of the van Benthem characterization theorem on the class of models based on descriptive frames. We also showed the failures of the compactness theorem, the upward Löwenheim-Skolem theorem, the Beth definability theorem and the Craig interpolation theorem on this class of models.
We will now brief ly discuss potential directions for future work. As pointed out in the introduction, many generalizations and adaptations exist of the classical van Benthem characterization theorem. Deserving special mention is the Janin-Walukiewicz theorem [19]. This theorem is an analogue of the van Benthem characterization theorem for the modal μ-calculus. It states that the modal μ-calculus is the bisimulation-invariant fragment of monadic first-order logic. The proof of this theorem also relies centrally on unravelling trees and the convenient properties of trees for gametheoretic semantics. In [7], a subclass of descriptive frames is designed to allow interpretation of the modal μ-calculus. We leave it as an open problem whether an analogue of the Janin-Walukiewicz theorem can be proved for these descriptive μ-frames. We only note that for this task, one could try to work with an appropriate version of descriptive unravellings defined in this paper.
There are also other important language expansions of modal logic such as hybrid logic [3] and guarded fragment [1]. The analogues of van Benthem's theorem have been proved for these languages [2,5]. For the guarded fragment, the theorem also holds for finite models [26]. It is natural to ask whether the analogue of these results hold for descriptive models of these logics. These will also provide interesting new venues for the proof strategy via descriptive unravellings given in this paper. However, to formulate such results, descriptive frames and models for hybrid logic and guarded fragment need to be defined first. For hybrid logics, one might draw inspiration from the algebraic treatment in [23]. Therefore, we leave it as an open problem to develop appropriate version of descriptive models for these languages and to prove an analogue of van Benthem characterization theorem for these models.
The coalgebraic generalizations of the van Benthem characterization theorem for finite frames are inspired by the view of Kripke frames as coalgebras for the powerset functor on the category of sets. Similarly, the finite Kripke frames can be viewed as coalgebras for the powerset functor in the category of finite sets. This then leads to a strategy based on the pseudotrees as in [27] to obtain the van Benthem characterization theorem for finite supported coalgebras. As was shown in this paper, descriptive frames are model-theoretically very similar to finite frames. As descriptive frames are coalgebras for the Vietoris functor on the category of Stone spaces, one could consider combining the constructions from this paper with the approach used in [29]. There again, a type of pseudotree is introduced to apply arguments from finite model theory to obtain the result. Replacing the pseudotree with the descriptive unravelling could give way to a similar result for alternative Vietoris-like coalgebras on Stone spaces.
In [24,25], the van Benthem characterization theorem is treated for intuitionistic frames. Descriptive intuitionistic frames are known as Esakia spaces [13] and with the techniques from this paper, one could pursue modal characterization theorems on these interesting classes.
Finally, neighbourhood structures have been given the structure of Stone coalgebras in [17]. Thus, the constructions presented here may be combined with the approach from [16] to achieve a modal characterization result on these neighbourhood structures over Stone spaces.