DYNAMIC CANTOR DERIVATIVE LOGIC

. Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d -logics. Unlike logics based on the topological closure operator, d -logics have not previously been studied in the framework of dynamical systems, which are pairs ( X, f ) consisting of a topological space X equipped with a continuous function f : X → X . We introduce the logics wK4C , K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d -semantics in this dynamical setting. In particular, we prove that wK4C is the d -logic of all dynamic topological systems, K4C is the d -logic of all T D dynamic topological systems, and GLC is the d -logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H , K4H and GLH . The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d -logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation – something known to be impossible over the class of all spaces.


Introduction
Dynamic (topological) systems are mathematical models of processes that may be iterated indefinitely.Formally, they are defined as pairs ⟨X, f ⟩ consisting of a topological space X = ⟨X, τ ⟩ and a continuous function f : X → X; the intuition is that points in the space X 'move' along their orbit, x, f (x), f 2 (x), . . .which usually simulates changes in time.Dynamic topological logic (DTL) combines modal logic and its topological semantics with linear temporal logic (see Pnueli [Pnu77]) in order to reason about dynamical systems in a decidable framework.
Due to their rather broad definition, dynamical systems are routinely used in many pure and applied sciences, including computer science.To cite some recent examples, in data-driven dynamical systems, data-related problems may be solved through data-oriented research in dynamical systems as suggested by Brunton and Kutz [BK19].Weinan [Wei17] proposes a dynamic theoretic approach to machine learning where dynamical systems are used to model nonlinear functions employed in machine learning.Lin and Antsaklis's [LA14] points.Further, we consider dynamical systems where f is a homeomorphism, i.e.where f −1 is also a continuous function.Such dynamical systems are called invertible.
The basic dynamic d-logic we consider is wK4C, which consists of wK4 and the temporal axioms for the continuous function f .In addition, we investigate two extensions of wK4C: K4C and GLC.As we will see, K4C is the d-logic of all T D dynamical systems, and GLC is the d-logic of all dynamical systems based on a scattered space.Unlike the generic logic of the trimodal topo-temporal language L • * ♢ , we conjecture that a complete finite axiomatisation for GLC, extended with axioms for the 'henceforth' operator, will not require changes to the trimodal language.This logic is of special interest to us as it would allow for the first finite axiomatisation and completeness results for a logic based on the trimodal topo-temporal language.
Outline.This paper is structured as follows: in Section 2 we give the required definitions and notations necessary to understand the paper.In Section 3 we provide some background on prior work on the topic of dynamic topological logics.Moreover, we motivate our interest in GLC, the most unusual logic we work with.
In Section 4 we present the canonical model, and in Section 5 we construct a 'finitary' accessibility relation on it.Both are then used in Section 6 in order to develop a proof technique that, given the right modifications, would work for many d-logics above wK4C.
In particular, we use it to prove the finite model property, soundness and completeness for the d-logics wK4C, K4C and GLC, with respect to the appropriate classes of Kripke models.
In Section 7 we prove topological d-completeness of K4C, wK4C and GLC with respect to the appropriate classes of dynamical systems.In Section 8 we present logics for systems with homeomorphisms and provide a general completeness result which, in particular, applies to the d-logics wK4H, K4H and GLH.Finally, in Section 9 we provide some concluding remarks.

Preliminaries
In this section we review some basic notions required for understanding this paper.We work with the general setting of derivative spaces, in order to unify the topological and Kripke semantics of our logics.
Definition 2.1 (topological space).A topological space is a pair X = ⟨X, τ ⟩, where X is a set and τ is a subset of ℘(X) that satisfies the following conditions: The elements of τ are called open sets, and we say that τ forms a topology on X.A complement of an open set is called a closed set.
The main operation on topological spaces we are interested in is the Cantor derivative.Definition 2.2 (Cantor derivative).Let X = ⟨X, τ ⟩ be a topological space.Given S ⊆ X, the Cantor derivative of S is the set d(S) of all limit points of S, i.e. x ∈ d(S) ⇐⇒ ∀U ∈ τ s.t.x ∈ U, (U ∩ S)\{x} ̸ = ∅.We may write d(S) or dS indistinctly.
Given subsets A, B ⊆ X, it is not difficult to verify that d satisfies the following properties: (1) In fact, these conditions lead to a more general notion of derivative spaces:1 Definition 2.3 (derivative space).A derivative space is a pair ⟨X, ρ⟩, where X is a set and ρ : ℘(X) → ℘(X) is a map satisfying properties 1-3 (with ρ in place of d).
When working with more than one topological space, we will often denote the map ρ on the topological space ⟨X, τ ⟩ by ρ τ .The intended derivative spaces discussed in this paper are of the form ⟨X, d τ ⟩.However, there are other examples of derivative spaces.The standard topological closure may be defined by c(A) = A ∪ d(A).Then, ⟨X, c⟩ is also a derivative space, which satisfies the additional property A ⊆ c(A) (and, a fortiori, cc(A) = c(A)), which together form Kuratowski's axioms; we call such derivative spaces closure spaces.Similarly to the Cantor derivative, we will often denote the closure of the topological space Note that if ρ = d τ , then the topology τ is uniquely determined, but not ever derivative operator is of the form d τ .In particular, weakly transitive Kripke frames provide examples of derivative spaces which do not necessarily arise from a topology.For the sake of succinctness, we call these frames derivative frames.Definition 2.4 (derivative frame).A derivative frame is a pair F = ⟨W, ⟩ where W is a non-empty set and is a weakly transitive relation on W , meaning that w v u implies that w ⊑ u, where ⊑ is the reflexive closure of .
Below and throughout the text, we write ∃x y φ instead of ∃x(y x ∧ φ), and adopt a similar convention for the universal quantifier and other relational symbols.We chose the notation because it is suggestive of a transitive relation, but remains ambiguous regarding reflexivity, as there may be irreflexive and reflexive points.Note that is weakly transitive iff ⊑ is transitive.Given A ⊆ W , we define ↓ as a map ↓ : ℘(W ) → ℘(W ) such that

Similarly, we define
The following is then readily verified: Lemma 2.5.If ⟨W, ⟩ is a derivative frame, then ⟨W, ↓ ⟩ is a derivative space.
There is a connection between derivative frames and topological spaces.Given a derivative frame ⟨W, ⟩, we define a topology τ on W such that U ∈ τ iff U is upwards closed under , in the sense that if w ∈ U and v w then v ∈ U .Topologies of this form are Aleksandrov topologies.The following is well-known and easily verified.
Lemma 2.6.Let ⟨W, ⟩ be a derivative frame and τ = τ .Then, d τ = ↓ iff is irreflexive and c τ = ↓ iff is reflexive.Dynamical systems consist of a topological space equipped with a continuous function.Recall that if ⟨X, τ ⟩ and ⟨Y, υ⟩ are topological spaces and f : continuous, open and bijective.It is well-known (and not hard to check; see e.g.[Mun00]) that f is continuous iff c τ f −1 (A) ⊆ f −1 c υ (A) for all A ⊆ Y .By unfolding the definition of the closure operator, this becomes We thus arrive at the following definition.
Definition 2.7.Let ⟨X, ρ X ⟩ and ⟨Y, ρ Y ⟩ be derivative spaces.We say that f : It is worth checking that these definitions coincide with their standard topological counterparts.
Lemma 2.8.If ⟨X, τ ⟩ and ⟨Y, υ⟩ are topological spaces with Cantor derivatives d τ and d υ respectively, and f : X → Y , then (1) f is continuous as a function between topological spaces if and only if it is continuous as a function between derivative spaces, and (2) f is a homeomorphism as a function between topological spaces if and only if it is a homeomorphism as a function between derivative spaces.
Proof.We prove the first claim and leave the second to the reader.Suppose that f : X → Y is continuous in the topological sense and let , so we may assume otherwise.Let U be any neighbourhood of f (x); note that Since by assumption f (x) / ∈ A, we obtain y ̸ = f (x), and since U was arbitrary,

continuous as a function between derivative spaces and let
We are particularly interested in the case where X = Y , which leads to the notion of dynamic derivative system.Definition 2.9.A dynamic derivative system is a triple S = ⟨X, ρ, f ⟩, where ⟨X, ρ⟩ is a derivative space and f : X → X is continuous.If f is a homeomorphism, we say that S is invertible.
for some topology τ , we say that S is a dynamic topological system and identify it with the triple ⟨X, τ, f ⟩.If ρ = ↓ for some weakly transitive relation , we say that S is a dynamic Kripke frame and identify it with the triple ⟨X, , f ⟩.
It will be convenient to characterise dynamic Kripke frames in terms of the relation .
, and weakly monotonic if The function f is persistent if it is a bijection and for all w, v ∈ W , w v if and only if f (w) f (v).We say that a Kripke frame is invertible if it is equipped with a persistent function.
Lemma 2.11.If ⟨W, ⟩ is a derivative frame and f : W → W , then (1) if f is weakly monotonic if and only if it is continuous with respect to ↓ , and (2) if f is persistent if and only if it is a homeomorphism with respect to ↓ .
Our goal is to reason about various classes of dynamic derivative systems using the logical framework defined in the next section.

Dynamic Topological Logics
In this section we discuss dynamic topological logic in the general setting of dynamic derivative systems.Given a non-empty set PV of propositional variables, the language L • ♢ is defined recursively as follows: where p ∈ PV.It consists of the Boolean connectives ∧ and ¬, the temporal modality , and the modality ♢ for the derivative operator with its dual □ := ¬♢¬.The interior modality may be defined as φ := φ ∧ □φ.Definition 3.1 (semantics).A dynamic derivative model (DDM) is a quadruple M = ⟨X, ρ, f, ν⟩ where ⟨X, ρ, f ⟩ is a dynamic derivative system and ν : PV → ℘(X) is a valuation function assigning a subset of X to each propositional letter in PV.Given φ ∈ L • ♢ , we define the truth set ∥φ∥ ⊆ X of φ inductively as follows: We write M, x |= φ if x ∈ ∥φ∥, and M |= φ if ∥φ∥ = X.We may write ∥ • ∥ M or ∥ • ∥ ν instead of ∥ • ∥ when working with more than one model or valuation.
We define other connectives (e.g.∨, →) as abbreviations in the usual way.The fragment of L • ♢ that includes only ♢ will be denoted by L ♢ .Since our definition of the semantics applies to any derivative space and a general operator ρ, we need not differentiate in our results between d-logics, logics based on closure semantics and logics based on relational semantics.Instead, we indicate the specific class of derivative spaces to which the result applies.
In order to keep with the familiar axioms of modal logic, it is convenient to discuss the semantics of □.Accordingly, we define the dual of the derivative, called the co-derivative.Definition 3.2 (co-derivative).Let ⟨X, ρ⟩ be a derivative space.For each S ⊆ X we define ρ(S) := X\ρ(X\S) to be the co-derivative of S.
The co-derivative satisfies the following properties, where A, B ⊆ X: It can readily be checked that for each dynamic derivative model ⟨X, ρ, f, ν⟩ and all formulas φ, ∥□φ∥ = ρ(∥φ∥).The co-derivative can be used to define the standard interior of a set, given by i(A) = A ∩ ρ(A) for each A ⊆ X.This implies that U ⊆ ρ(U ) for each open set U , but not necessarily ρ(U ) ⊆ U .Next, we discuss the systems of axioms that are of interest to us.Let us list the axiom schemes and rules that we will consider in this paper: The 'base modal logic' over L ♢ is given by but we are mostly interested in proper extensions of K. Let Λ and Λ ′ be logics over languages L and L ′ respectively.We say that Λ extends Λ ′ if L ′ ⊆ L and all the axioms and rules of Λ ′ are derivable in Λ.A logic over L ♢ is normal if it extends K.If Λ is a logic and φ is a formula, Λ + φ is the least extension of Λ which contains every substitution instance of φ as an axiom.
We then define wK4 := K + w4, K4 := K + 4, S4 := K4 + T and GL := K4 + L. These logics are well-known and characterise certain classes of topological spaces and Kripke frames which we review below.In addition, for a logic Λ over L ♢ , ΛF is the logic over L • ♢ given by 2 ΛF := Λ + Next ¬ + Next ∧ + Nec .This simply adds axioms of linear temporal logic to Λ, which hold whenever is interpreted using a function.Finally, we define ΛC := ΛF + C and ΛH := ΛF + H, which as we will see correspond to derivative spaces with a continuous function or a homeomorphism respectively.The purely topological fragments of these logics have been well-studied, starting with the following well-known result dating back to McKinsey and Tarski [MT44].
Theorem 3.3.S4 is the logic of all topological closure spaces, the logic of all transitive, reflexive derivative frames, and the logic of the real line with the standard closure.
Analogously, Esakia demonstrated that wK4 is the logic of topological derivative spaces [Esa01].
Theorem 3.4.wK4 is the logic of all topological derivative spaces, as well as the logic of all weakly-transitive derivative frames.
The logic K4 includes the axiom □p → □□p, which is not valid over the class of all topological spaces.The class of spaces satisfying this axiom is denoted by T D , defined as the class of spaces in which every singleton is the result of an intersection between an open set and a closed set.Moreover, Esakia showed that this is the logic of transitive derivative frames [Esa04].
2 Logics of the form ΛF correspond to dynamical systems with a possibly discontinuous function.We will not discuss discontinuous systems in this paper; see [ADN97] for more information.

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Theorem 3.5.K4 is the logic of all T D topological derivative spaces, as well as the logic of all transitive derivative frames.
Many familiar topological spaces, including Euclidean spaces, satisfy the T D property, making K4 central in the study of topological modal logic.A somewhat more unusual class of spaces, which is nevertheless of particular interest to us, is the class of scattered spaces.Definition 3.6 (scattered space).A topological space ⟨X, τ ⟩ is scattered if for every S ⊆ X, S ⊆ d(S) implies S = ∅.This is equivalent to the more common definition of a scattered space where a topological space is called scattered if every non-empty subset has an isolated point.Scattered spaces are closely related to converse well-founded relations.Below, recall that ⟨W, ⟩ is converse well-founded if there is no infinite sequence w 0 w 1 . . . of elements in W .
Proof.First suppose that ⟨W, τ ⟩ is scattered and let A ⊆ W be non-empty.Let x be an isolated point of A. Then, x is isolated in A, and U := {w} ∪ ↑ ({w}) is the least neighbourhood of w so we must have that U ∩ A = {w}, i.e. w is a -maximal point of A.
Conversely, if ⟨W, ⟩ is converse well-founded and A ⊆ W is non-empty, it is readily checked that any -maximal point of A is isolated in A.
Theorem 3.8 (Simmons [Sim75] and Esakia [Esa81]).GL is the logic of all scattered topological derivative spaces, as well as the logic of all converse well-founded derivative frames and the logic of all finite, transitive, irreflexive derivative frames.
Aside from its topological interpretation, the logic GL is of particular interest as it is also the logic of provability in Peano arithmetic, as was shown by Boolos [Boo84].Meanwhile, logics with the C and H axioms correspond to classes of dynamical systems.Lemma 3.9.
(1) If Λ is sound for a class of derivative spaces Ω, then ΛC is sound for the class of dynamic derivative systems ⟨X, ρ, f ⟩, where ⟨X, ρ⟩ ∈ Ω and f is continuous.
(2) If Λ is sound for a class of derivative spaces Ω, then ΛH is sound for the class of dynamic derivative systems ⟨X, ρ, f ⟩, where ⟨X, ρ⟩ ∈ Ω and f is a homeomorphism.
The above lemma is easy to verify from the definitions of continuous functions and homeomorphisms in the context of derivative spaces (Definition 2.7).

Prior Work.
The study of dynamic topological logic originates with Artemov, Davoren and Nerode, who observed that it is possible to reason about dynamical systems within modal logic.They introduced the logic S4C and proved that it is decidable, as well as sound and complete for the class of all dynamic closure systems (i.e.dynamic derivative systems based on a closure space).Kremer and Mints [KM05] considered the logic S4H and showed that it is sound and complete for the class of dynamic closure systems where f is a homeomorphism.
The latter also suggested adding the 'henceforth' operator, * , from Pnueli's linear temporal logic (LTL) [Pnu77], leading to the language we denote by L • * ♢ .The resulting trimodal system was named dynamic topological logic (DTL).Kremer and Mints offered an axiomatisation for DTL, but Fernández-Duque proved that it is incomplete; in fact, DTL is not finitely axiomatisable [Fer14].Fernández-Duque also showed that DTL has a natural axiomatisation when extended with the tangled closure [FD12].In contrast, Konev et al. established that DTL over the class of dynamical systems with a homeomorphism is non-axiomatisable [KKWZ06b].
3.2.The tangled closure on scattered spaces.Our interest in considering the class of scattered spaces within dynamic topological logic is motivated by results of Fernández-Duque [FD12].He showed that the set of valid formulas of L • * ♢ over the class of all dynamic closure systems is not finitely axiomatisable.Nevertheless, he found a natural (yet infinite) axiomatisation by introducing the tangled closure and adding it to the language of DTL [Fer11].Here, we use the more general tangled derivative, as defined by Goldblatt and Hodkinson [GH17].
Definition 3.10 (tangled derivative).Let ⟨X, ρ⟩ be a derivative space and let S ⊆ ℘(X).Given A ⊆ X, we say that S is tangled in A if for all S ∈ S, A ⊆ ρ(S ∩ A).We define the tangled derivative of S as The tangled closure is then the special case of the tangled derivative where ρ is a closure operator, i.e. ρ = c.Fernández-Duque's axiomatisation is based on the extended language L • * ♢ * .This language is obtained by extending L • * ♢ with the following operation.Definition 3.11 (tangled language).We define L The logic DGL is an extension of GLC that includes the temporal operator * .Unlike the complete axiomatisation of DTL that requires the tangled operator, in the case of DGL, we should be able to avoid this and use the original spatial operator ♢ alone.This is due to the following: Theorem 3.12.Let X = ⟨X, τ ⟩ be a scattered space and {φ 1 , . . ., φ n } a set of formulas.Then ♢ * {φ 1 , . . ., φ n } ≡ ⊥.
This leads to the conjecture that the axiomatic system of Kremer and Mints [KM05], combined with GL, will lead to a finite axiomatisation for DGL.While such a result requires techniques beyond the scope of the present work, the completeness proof we present here for GLC is an important first step.Before proving topological completeness for this and the other logics we have mentioned, we show that they are complete and have the finite model property for their respective classes of dynamic derivative frames.

The Canonical Model
The first step in our Kripke completeness proof will be a fairly standard canonical model construction.For a given logic Λ, a maximal Λ-consistent set (Λ-MCS) w is a set of formulas that is Λ-consistent, i.e. w ⊬ Λ ⊥, and every set of formulas that properly contains it is Λ-inconsistent.
It follows that g is weakly monotonic and is weakly transitive and hence M Λ c is a wK4C model.
Suppose that Λ extends K4C, then weak monotonicity holds as before since K4C extends wK4C.Therefore, we only need to prove that is transitive.Suppose w v u.We consider an arbitrary □ψ ∈ w.Since □p → □□p ∈ wK4C, then □□ψ ∈ w.Then, w v u implies ψ ∈ u and since □ψ is arbitrary, w u, as required.
It follows that g is weakly monotonic and is transitive and thus It is a well-known fact that the transitivity axiom □p → □□p is derivable in GL (see [Smo85]).Therefore, GLC extends the system K4C.The proofs of the following two lemmas are standard and can be found for example in [BdRV01].
Lemma 4.2 (existence lemma).Let Λ be a normal modal logic and let M Λ c = ⟨W c , c , g c , ν c ⟩.Then, for every w ∈ W c and every formula φ in Λ, if ♢φ ∈ w then there exists a point v ∈ W c such that w c v and φ ∈ v. Lemma 4.3 (truth lemma).Let Λ be a normal modal logic.For every w ∈ W c and every formula φ in Λ, M Λ c , w |= φ iff φ ∈ w.Corollary 4.4.The logic wK4C is sound and complete with respect to the class of all weakly monotonic dynamic derivative frames, and K4C is sound and complete with respect to the class of all weakly monotonic, transitive dynamic derivative frames.

A finitary accessibility relation
One key ingredient in our finite model property proof will be the construction of a 'finitary' accessibility relation Φ on the canonical model.This accessibility relation will have the property that each point has finitely many successors, yet the existence lemma will hold for formulas in a prescribed finite set Φ.This is a kind of selective filtration (see [CZ97]).
We define the c -cluster C(w) for each point w ∈ W c as Definition 5.1 (φ-final set).A set w is said to be a φ-final set (or point) if w is an MCS, φ ∈ w, and whenever w c v and φ ∈ v, it follows that v ∈ C(w).
Let us write ⊑ c for the reflexive closure of c .It will be convenient to characterise ⊑ c in the canonical model syntactically.Recall that φ := φ ∧ □φ.
Lemma 5.2.If Λ extends wK4C and w, v ∈ W c , then w ⊑ c v if and only if whenever φ ∈ w, it follows that φ ∈ v.
Proof.First assume that w ⊑ c v. Obviously if w = v then φ ∈ w implies φ ∈ v.If φ ∈ w then from □φ ∈ w we obtain φ ∈ v and by w4 we have □□φ ∈ w and so □φ ∈ v.
The following version of Zorn's lemma will be used to prove an important existence property.
Lemma 5.3 (Zorn's Lemma).Let (A, ≤) be a preordered set where A is non-empty.A chain is a subset C ⊆ A whose elements are totally ordered by ≤.Suppose that every chain C has an upper bound in A. Then, A has a ≤-maximal element.
The following proves a sufficient condition for the existence of a final point accessible from a given point.
Lemma 5.4.If ♢φ ∈ w, then there is φ-final point v such that w c v.
Proof.Suppose that ♢φ ∈ w and let A := c (w) ∩ ∥φ∥ (where c (w) = {v ∈ W c : w c v}).Note that A is non-empty by Lemma 4.2, so in order to apply Zorn's lemma, consider a ⊑ c -chain C in A. We show that there is an upper bound of C that belongs to A.
Choose a formula θ as follows: If w is an irreflexive world (i.e.w ̸ c w), then choose θ so that □θ ∧ ¬θ ∈ w; such a θ exists since otherwise the definition of c would yield w c w. Otherwise, w is reflexive and we simply set θ = ⊤.Let Γ be the set Suppose ψ 1 , . . ., ψ n ∈ Γ.For each i there is w i ∈ C such that ψ i ∈ w i .Since C is a chain, for some j we have that w j = max n i=1 w i .Then, using Lemma 5.2 we see that φ, θ, ψ 1 , . . ., ψ n ∈ w j , hence {φ, θ, ψ 1 , . . ., ψ n } is consistent.Since { ψ 1 , . . ., ψ n } is an arbitrary finite subset of Γ, we have that Γ ∪ {φ, θ} is consistent.
By the Lindenbaum lemma we can extend Γ ∪ {φ, θ} to an MCS, which we denote by w * .Suppose ψ ∈ v ∈ C, then by construction ψ ∈ w * .It follows by Lemma 5.2 that v ⊑ c w * .Hence w * is an upper bound for C. Note that the assumption that ♢φ ∈ w implies that there is some v c w such that φ ∈ v, that is, v ∈ A. Hence w c v ⊑ c w * , which yields w ⊑ c w * .To see that indeed w c w * , if w is reflexive there is nothing to prove, otherwise since θ ∈ w * we have that w ̸ = w * .Moreover φ ∈ w * by construction, so w * ∈ A.
Thus, by Lemma 5.3 we conclude that A contains a maximal element, and this maximal element is clearly a φ-final point above w.
We are now ready to prove the main result of this section regarding the existence of the finitary relation Φ .The idea is that Φ is a sub-relation of c that only chooses enough points to provide witnesses for any formula ♢φ ∈ Φ.We also want w Φ v to depend only on the cluster of w, except possibly in the case that w = v; in other words, ⊑ Φ will be cluster-invariant.This will allow us to pick finite submodels of the canonical model, by including only the Φ -successors of each world rather than all of the c -successors (which are typically infinitely many).
Lemma 5.5.Let Λ extend wK4C and Φ be a finite set of formulas closed under subformulas.There is an auxiliary relation Φ on the canonical model of Λ such that: (i) Φ is a subset of c ; (ii) For each w ∈ W , the set Φ (w) is finite; (iii) If ♢φ ∈ w ∩ Φ, then there exists v ∈ W with w Φ v and φ ∈ v; (iv) If w c v c w then Φ (w) ⊆ ⊑ Φ (v); (v) Φ is weakly transitive.Moreover, if Λ extends K4C then Φ is transitive, and if Λ extends GLC then Φ is irreflexive.
Proof.Let C be any cluster of points in W and define We construct the weakly transitive relation Φ as follows: Using Lemma 5.4 we use the axiom of choice to choose a function that for each formula φ and each cluster C such that ♢φ ∈ C, assigns a φ-final point w(φ, C) such that w(φ, C) ∈ c (C).We choose a second point w ′ (φ, C), possibly equal to w(φ, C), such that Set u 0 Φ v iff u c v and there exists ψ ∈ Φ such that ♢ψ ∈ u and v ∈ {w(ψ, C(u)), w ′ (ψ, C(u))}.
Let Φ be the weakly transitive closure of 0 Φ .It is clear that (i), (iii) and (iv) follow directly from the construction: (i) follows from the fact that Φ is the transitive closure of 0 Φ ; (iii) follows from Lemma 5.4; (iv) follows from the fact that v ∈ C(w) and by assuming that v ⊑ Φ u and unraveling the definition of ⊑ Φ .
We continue to verify conditions (ii) and (v).For condition (ii), first observe that 0 Φ (u) is finite by construction, as it contains at most two elements for each φ ∈ Φ.Now, if u Φ v then there is a sequence By taking a minimal such sequence, we may assume that it is injective, i.e. each element appears only once in the sequence.Consider the tree consisting of all such sequences (ordered by the initial segment relation).This is a finitely-branching tree, as 0 Φ (x) is always finite.Moreover, if Φ (u) is infinite, then this tree is infinite.By König's lemma, there is an infinite sequence Since Φ is finite, there is some θ ∈ Φ such that v i is θ-final for infinitely many values of i.Let i 0 be the least such value.If i > i 0 is any other such value, ) is finite, contradicting that the chain is infinite and injective.
Finally, we verify condition (v).The relation Φ is weakly transitive by definition, and if Λ extends K4C, we have that x Φ y Φ z implies x ⊑ Φ z, which in the case that x = z implies that x c x and also that x, y are in the same cluster.This together with y Φ z implies that x Φ z, as Φ is cluster-invariant.Now let Λ extend GLC, and suppose that w is φ-final.We claim that ♢φ ̸ ∈ w, which immediately yields w ̸ Φ w, as needed.By the contrapositive of L, we have that ♢φ → ♢(φ ∧ □¬φ) ∈ w.If ♢φ ∈ w, then ♢(φ ∧ □¬φ) ∈ w.It follows that there is v Φ w with φ ∧ □¬φ ∈ v.But □¬φ ∈ v implies that v ̸ Φ w, contradicting the φ-finality of w.So ♢φ / ∈ w and w is irreflexive, as required.

Stories and Φ-morphisms
In this section we show that the logics wK4C, K4C and GLC have the finite model property by constructing finite models and truth preserving maps from these models to the canonical model.If is a weakly transitive relation on A, ⟨A, ⟩ is tree-like if whenever a ⊑ c and b ⊑ c, it follows that a ⊑ b or b ⊑ a.We will use labelled tree-like structures called moments to record the 'static' information at a point; that is, the structure involving , but not f .Definition 6.1 (moment).A Λ-moment is a structure m = ⟨|m|, m , ν m , r m ⟩, where ⟨|m|, m ⟩ is a finite tree-like Λ frame with a root r m , and ν m is a valuation on |m|.
In order to also record 'dynamic' information, i.e. information involving the transition function, we will stack up several moments together to form a 'story'.Below, denotes a disjoint union.Definition 6.2 (story).A story (with duration I) is a structure S = ⟨|S|, S , f S , ν S , r S ⟩ such that there are I < ω, Λ-moments S i = ⟨|S i |, i , ν i , r i ⟩ for each i ≤ I, and functions (f i ) i<I such that: (1) |S| = i≤I |S i |; (2) S = i≤I i ; (3) ν S (p) = i≤I ν i (p) for each variable p; (4) r S = r 0 ; (5) f S = Id I ∪ i<I f i with f i : |S i | → |S i+1 | being a weakly monotonic map such that f i (r i ) = r i+1 for all i < I (we say that f i is root preserving), and Id I is the identity on We often omit the subindices m or S when this does not lead to confusion.We may also assign different notations to the components of a moment, so that if we write m = ⟨W, , ν, x⟩, it is understood that W = |m|, = m , etc.
Recall that a p-morphism between Kripke models is a type of map that preserves validity.It can be defined in the context of dynamic derivative frames as follows: Definition 6.3 (dynamic p-morphism).Let M = ⟨W M , M , g M ⟩ and N = ⟨W N , N , g N ⟩ be dynamic derivative frames.Let π : W M → W N .We say that π is a dynamic p-morphism if w M v implies that π(w) N π(v), π(w) N u implies that there is v M w with π(v) = u, and π • g M = g N • π.
It is then standard that if π : W M → W N is a surjective, dynamic p-morphism, then any formula valid on M is also valid on N.However, our relation Φ will allow us to weaken these conditions and still obtain maps that preserve the truth of (some) formulas.If we drop condition 2, we say that π is a Φ-morphism; the latter notion will mostly be applied to moments, viewed as stories of duration one.
We now show that a dynamic Φ-morphism π preserves the truth of formulas of suitable -depth, where the latter is defined as usual in terms of nested occurrences of in a formula φ.
Lemma 4.1.If Λ extends wK4C, then the canonical model for Λ is a wK4C model.If Λ extends K4C, then the canonical model of Λ is a K4C model.

Figure 1 .
Figure 1.An example of a GL-story.The squiggly arrows represent the relation S while the straight arrows represent the function f S .Each vertical slice represents a GL-moment.In the case of other types of stories, we may also have clusters besides singletons.