Extensions of K5: Proof Theory and Uniform Lyndon Interpolation

We introduce a Gentzen-style framework, called layered sequent calculi, for modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to investigate the uniform Lyndon interpolation property (ULIP), which implies both the uniform interpolation property and the Lyndon interpolation property. We obtain complexity-optimal decision procedures for all logics and present a constructive proof of the ULIP for K5, which to the best of our knowledge, is the first such syntactic proof. To prove that the interpolant is correct, we use model-theoretic methods, especially bisimulation modulo literals.


Introduction
The uniform interpolation property (UIP) is an important property of a logic.It strengthens the Craig interpolation property (CIP) by making interpolants depend on only one formula of an implication, either the premise or conclusion.A lot of work has gone into proving the UIP, and it is shown to be useful in various areas of computer science, including knowledge representation [16] and description logics [24].Early results on the UIP in modal logic include positive results proved semantically for logics GL and K (independently in [9,31,34]) and negative results for logics S4 [10] and K4 [5].A proof-theoretic method to prove the UIP was first proposed in [29] for intuitionistic propositional logic and later adapted to modal logics, such as K and T in [5].A general proof-theoretic method of proving the UIP for many classical and intuitionistic (non-)normal modal logics and substructural (modal) logics based on the form of their sequent-calculi rules was developed in the series of papers [2,3,15].
Apart from the UIP, we are also interested in the uniform Lyndon interpolation property (ULIP) that is a strengthening of the UIP in the sense that interpolants must respect the polarities of the propositional variables involved.Kurahashi [17] first introduced this property and proved it for several normal modal logics, by employing a semantic method using layered bisimulations.A sequent-based proof-theoretic method was used in [1] to show the ULIP for several non-normal modal logics and conditional logics.
Our long-term goal is to provide a general proof-theoretic method to (re)prove the UIP for modal logics via multisequent calculi (i.e., nested sequents, hypersequents, labelled hypersequents, etc.).Unlike many other ways of proving interpolation, the proof-theoretic treatment is constructive in that it additionally yields an algorithm for constructing uniform interpolants.Towards this goal, we build on the modular treatment of multicomponent calculi to prove the CIP for modal and intermediate logics in [8,18,20,22,23].First steps have been made by reproving the UIP for modal logics K, D, and T via nested sequents [12] and for S5 via hypersequents [11,13], the first time this is proved proof-theoretically for S5.
Towards a modular proof-theoretic treatment, we introduce a new form of multisequent calculi for these logics that we call layered sequent calculi, the structure of which is inspired by the structure of the Kripke frames for the concerned logics from [26].For S5, this results in standard hypersequents [4,25,30].For K5 and KD5, the presented calculi are similar to grafted hypersequent calculi in [21] but without explicit weakening.Other, less related, proof systems include analytic cut-free sequent systems for K5 and KD5 [33], cut-free sequent calculi for K45 and KD45 [32], and nested sequent calculi for modal logics [7].
The layered sequent calculi introduced in this paper adopt a strong version of termination that only relies on a local loop-check based on saturation.For all concerned logics, this yields a decision procedure that runs in co-NP time, which is, therefore, optimal [14].We provide a semantic completeness proof via a countermodel construction from failed proof search.
Finally, layered sequents are used to provide the first proof-theoretic proof of the ULIP for K5.The method is adapted from [11,13] in which the UIP is proved for S5 based on hypersequents.We provide an algorithm to construct uniform Lyndon interpolants purely by syntactic means using the termination strategy of the proof search.To show the correctness of the constructed interpolants, we use model-theoretic techniques inspired by bisimulation quantification in the setting of uniform Lyndon interpolation [17].

Preliminaries
The language of modal logics consists of a set Pr of countably many (propositional ) atoms p, q, . .., their negations p, q, . .., propositional connectives ∧ and ∨, boolean constants ⊤ and ⊥, and modal operators and ♦.A literal ℓ is either an atom or its negation, and the set of all literals is denoted by Lit.We define modal formulas in the usual way and denote them by lowercase Greek letters Table 1.Modal axioms and their corresponding frame conditions.
Throughout the paper we employ the semantics of Kripke frames and models.

Definition 2 (Kripke semantics).
A Kripke frame is a pair (W, R) where W is a nonempty set of worlds and R ⊆ W × W a binary relation.A Kripke model is a triple (W, R, V ) where (W, R) is a Kripke frame and V : Pr → P(W ) is a valuation function.A formula ϕ is defined to be true at a world w in a model M = (W, R, V ), denoted M, w ϕ, as follows: M, w ⊤, M, w ⊥ and , the relation R is total on C. We write wRC iff wRv for all v ∈ C.
We work with specific classes of Kripke models sound and complete w.r.t. the logics.The respective frame conditions for the logic L, called L-frames, are defined Semantics for extensions of K5 (see [26,28]).Everywhere not ρRρ for the root ρ, set C is a finite cluster, and ⊔ denotes disjoint union.
More precisely, we consider rooted frames and completeness w.r.t. the root, i.e., ⊢ L ϕ iff for all L-models M with root ρ, M, ρ ϕ (we often denote the ifcondition as L ϕ).For each logic, this follows from easy bisimulation arguments.

Theorem 3 ( [26]
).Any normal modal logic containing K5 is sound and complete w.r.t. a class of finite Euclidean Kripke frames (W, R) of one of the following forms: (a) W = {ρ} consists of a singleton root and R = ∅, (b) the whole W is a cluster (any world can be considered its root), or (c) W \{ρ} is a cluster for a (unique) root ρ ∈ W such that ρRw for some w ∈ W \{ρ} while not ρRρ.
Theorem 5.If a logic L has the ULIP, then it also has the UIP.
Proof.We define a uniform interpolant of ϕ w.r.
Table 3. Layered sequent calculi L.L: in addition to explicitly stated rules, all L.L have axioms id P and id ⊤ and rules ∨, ∧, ♦c, and t (see Fig. 1).Note that the rules of system L.L may only be applied to L-sequents.

Layered Sequents
Definition 6 (Layered sequents).A layered sequent is a generalized onesided sequent of the form where Γ i , Σ i , Π i are finite multisets of formulas, n, m, k ≥ 0, and if k ≥ 1, then m ≥ 1.A layered sequent is an L-sequent iff it satisfies the conditions in the rightmost column of Table 3.Each Σ i , each Π i , and i Γ i is called a sequent component of G.The formula interpretation of a layered sequent G above is: Layered sequents are denoted by G and H.The structure of a layered sequent can be viewed as at most two layers of hypersequents ([ ]-components Σ i and [[ ]]-components Π i forming the first and second layer respectively) possibly nested on top of the sequent component i Γ i as the root.Following the arboreal terminology from [21], the root is called the trunk while [ ]-and [[ ]]-components form the crown.Analogously to nested sequents representing tree-like Kripke models, the structure of L-sequents is in line with the structure of L-models introduced in Sect. 2. We view sequents components as freely permutable, e.g., Remark 7. The layered calculi presented here generalize grafted hypersequents of [21] and, hence, similarly combine features of hypersequents and nested sequents.In particular, layered sequents are generally neither pure hypersequents (except for the case of S5) nor bounded-depth nested sequents.The latter is due to the fact that the defining property of nested sequents is the tree structure of the sequent components, whereas the crown components of a layered sequent form a cluster.Although formally grafted hypersequents are defined with one layer only, this syntactic choice is more of a syntactic sugar than a real distinction.Indeed, the close relationship of one-layer grafted hypersequents for K5 and KD5 in [21] to the two-layer layered sequents presented here clearly manifests itself when translating grafted hypersequents into the prefixed-tableau format (see grafted tableau system for K5 [21, Sect.6]).There prefixes for the crown are separated into two types, limbs and twigs, which match the separation into [ ]-and [[ ]]-components.
We sometimes use unary contexts, i.e., layered sequents with exactly one hole, denoted { }.Such contexts are denoted by G{ }.The insertion G{Γ } of a finite multiset Γ into G{ } is obtained by replacing { } with Γ .The hole { } in a component σ can also be labeled G{ } σ .We use the notations and to refer to either of [ ] or [[ ]].
Using Fig. 1 and the middle column of Table 3, we define layered sequent calculi L.K5, L.KD5, L.K45, L.KD45, L.KB5, and L.S5, where L.L is the calculus for the logic L. Following the terminology from [21], we split all modal rules into trunk rules (subscript t) and crown rules (subscript c) depending on the position of the principal formula.We write Definition 9 (Saturation).Labeled formula σ : ϕ ∈ G is saturated for L.L iff -ϕ equals p or p for an atom p, or equals ⊥, or equals ⊤; -ϕ = ϕ 1 ∧ ϕ 2 and σ : ϕ i ∈ G for some i; -ϕ = ϕ 1 ∨ ϕ 2 and both σ : ϕ 1 ∈ G and σ : ϕ 2 ∈ G; -ϕ = ϕ ′ , the unique rule applicable to σ : ϕ ′ in L.L is either t or c (i.e., a rule creating a [ ]-component), and •i : ϕ ′ ∈ G for some i; -ϕ = ϕ ′ , the unique rule applicable to σ : ϕ ′ in L.L is c ′ (i.e., a rule creating a [[ ]]-component), and i : ϕ ′ ∈ G for some i.In addition, we define for any label σ and formula ϕ: and d) G is not of the from H{⊤} or H{q, q} for some q ∈ Pr.
Theorem 10.Proof search in L.L modulo saturation terminates and provides an optimal-complexity decision algorithm, i.e., runs in co-NP time.
Proof.Given a proof search of layered sequent G, for each layered sequent H in this proof search, consider its labeled formulas as a set F H = {σ : ϕ | σ : ϕ ∈ H}.Let s be the number of subformulas occurring in G and N be the number of sequent components in G. Since we only apply rules (that do not equal id P or id ⊤ ) to non-saturated sequents, sets F H will grow for each premise.Going bottom-up in the proof search, at most s labels of the form •i and at most s labels of the form i can be created, and each label can have at most s formulas.Therefore, the cardinality of sets F H are bounded by s(N +s+s), which is polynomial in the size of F G .Hence, the proof search terminates modulo saturation.Moreover, since each added labeled formula is linear in the size F G and the non-deterministic branching in the proof search is bounded by (N + s + s)s(N + s + s), again a polynomial in the size of F G , this algorithm is co-NP, i.e., provides an optimal decision procedure for the logic.
⊓ ⊔ → W such that the following conditions apply whenever the respective type of labels exists in G: ) for all labels of the form •i and j in Lab(G); 4.Not I(•)R I(j) for any label of the form j ∈ Lab(G).
Definition 12 (Sequent semantics).For any given interpretation G is valid in L, denoted L G, iff M, I G for all L-models M and interpretations I of G into M.We omit L and M when clear from the context.
The proof of the following theorem is based on a countermodel construction (with more standard parts of the proof relegated to the Appendix): Theorem 13 (Soundness and completeness).For any L-sequent G, We show a cycle of implications.The left-to-middle implication, i.e., that ⊢ L.L G =⇒ L ι(G), can be proved by induction on the L.L-derivation of G.
For the middle-to-right implication, i.e., L ι(G) =⇒ L G, let G be a sequent of form (1). We prove that M, I G implies Finally, we prove the right-to-left implication by contraposition using a countermodel construction: from a failed proof search of G, construct an L-model refuting G from (1).In a failed proof-search tree (Theorem 10), since L.L G, at least one saturated leaf defines multiformulas, where σ : ϕ is a labeled formula.Lab(℧) denotes the set of labels of ℧.
Definition 16 (Bisimilarity).Let M = (W, R, V ) and M ′ = (W ′ , R ′ , V ′ ) be models and ℓ ∈ Lit.We say M ′ is ℓ-bisimilar to M, denoted M ′ ≤ ℓ M iff there is a nonempty binary relation Z ⊆ W × W ′ , called an ℓ-bisimulation between M and M ′ , such that the following hold for every w ∈ W and w ′ ∈ W ′ : literals ℓ .if wZw ′ , then a) M, w q iff M ′ , w ′ q for all atoms q / ∈ {ℓ, ℓ} and b) if M ′ , w ′ ℓ, then M, w ℓ; forth.if wZw ′ and wRv, then there exists v ′ ∈ W ′ such that vZv ′ and w ′ R ′ v ′ ; back.if wZw ′ and w ′ R ′ v ′ , then there exists v ∈ W such that vZv ′ and wRv.M and M ′ are bisimilar, denoted M ∼ M ′ , iff there is a relation Z = ∅ satisfying forth and back, as well as part a) of literals ℓ for any p ∈ Pr, in which case Z is called a bisimulation.We write (similarly for ∼ instead of ≤ ℓ ): - Note that ≤ ℓ is a preorder and we have Definition 18 (BLUIP).Logic L is said to have the bisimulation layeredsequent uniform interpolation property (BLUIP) iff for every literal ℓ and every L-sequent G, there is a multiformula A ℓ (G), called BLU interpolant, such that: Lemma 19.The BLUIP for L implies the ULIP for L.
⊓ ⊔ To show that calculus L.K5 enjoys the BLUIP for K5, we need two important ingredients: some model modifications that are closed under bisimulation and an algorithm to compute uniform Lyndon interpolants.
Lemma 21.Let model N be obtained by copying a world w from a K5-model M (away from the root).Let I : X → M and I ′ : X → N be interpretations such that for each x ∈ X, either I(x) = I ′ (x) or I(x) = w while I ′ (x) = w c .Then, N is a K5-model and (M, I) ∼ (N , I ′ ).
In the construction of interpolants, we use the following rules d ′ t and dd and sets G c and ♦G c of formulas from the crown of G: shows similarities with rule d t from logics KD5 and KD45, but is only applied in the absence of the crown.Rule d ′ t is sound for K5 because it can be viewed as a composition of an (admissible) cut on ⊥ and ♦⊤ in the trunk, followed by t in the left premise on ⊥ that creates the first crown component (though ⊥ is dropped from it), which is populated using several ♦ t -rules for ♦ψ ∈ Γ .The label of this crown component is always •1. Rule dd provides extra information in the calculation of the uniform interpolant and is needed primarily for technical reasons.We highlight the two new sequent components created by the last instance of dd using special placeholder labels •d and d for the respective brackets.These labels are purely for readability purposes and revert to the standard •j and k labels after the next instance of dd.
To compute a uniform Lyndon interpolant ∀ℓξ for a formula ξ, we first compute a BLU interpolant A ℓ (0, ∅; ξ • ) by using the recursive function A ℓ (t, Σ c ; G) with three parameters we present below.The main parameter is a K5-sequent G, while the other two parameters are auxiliary: t ∈ {0, 1} is a boolean variable such that t = 1 guarantees that rule dd has been applied at least once for the case when G contains diamond formulas; Σ c ⊆ ♦G c is a set of modal formulas that provides a bookkeeping strategy to prevent redundant applications of rule dd.
To calculate A ℓ (t, Σ c ; G) our algorithm makes a choice of which row from Table 4 to apply by trying each of the following steps in the specified order: where j is the smallest integer such that •j / ∈ G and the SCNF of where j is the smallest integer such that j / ∈ G and the SCNF of where (b) else, if G = Γ consists of the trunk only, apply rule d ′ t as follows: where the SDNF of where SDNF of The computation of the algorithm can be seen as a proof search tree (extended with rules d ′ t and dd).In this proof search, call A ℓ (t, Σ c ; G) is sufficient (to be a BLU interpolant for G) if each branch going up from it either stops in Steps 1 or 4a or continues via Steps 4b or 4d.Otherwise, it is insufficient, if one of the branches stops in Step 4c, say, calculating A ℓ (1, Σ c ; H).In this case, A ℓ (1, Σ c ; H) is not generally a BLU interpolant for H, but these leaves provide enough information to find a BLU interpolant from some sequent down the proof search tree.

Now, by
Step 4d, and converting into a new SDNF, we get A p (0, ∅; Γ, Further applications of ∨ and t keep this interpolant intact.Note that the application of d ′ t does not require to continue proof search for the right Instead, Step 4b prescribes that A p (0, ∅; ϕ, p, ♦♦(p ∨ q)) ≡ • : p Simplifying, we finally obtain To check that p ∨ ♦♦q is a uniform Lyndon interpolant for ϕ w.r.t.literal p, it is sufficient to verify that ( 8) is a BLU interpolant for G by checking the conditions in Def.18.We only check BLUIP(iii) as the least trivial.If M, I • : (p ∨ ♦♦q) for an interpretation I into a K5-model M = (W, R, V ), then, by Defs.14 and 11, M, ρ p ∨ ♦♦q for the root ρ of M. For ℓ = p, we have an ℓ-bisimulation (M ′ , I) ≤ ℓ (M, I) for M ′ = (W, R, V ′ ) with V ′ (p) = {ρ} and V ′ (r) = V (r) for r = p since literals p allows to turn p from true to false.It is easy to see that M ′ , ρ p ∨ ♦♦(p ∨ q).Thus, M ′ , I • : ϕ.
The following properties of the algorithm are proved in the Appendix.Lemma 23.All recursive calls A ℓ (t, Σ c ; G) in a proof search tree of A ℓ (0, ∅; ϕ) have the following properties: 1.The algorithm is terminating.

When
Step 4b is applied, t = 0 and every branch going up from it consists of Steps 2-3 followed by either final Step 1 or continuation via Step 4d. 3.After Step 4d is applied, every branch going up from it consists of Steps 2 followed by a call Proof.It is sufficient to prove that, once the algorithm starts on A ℓ (0, ∅; ϕ), then every sufficient call A ℓ (t, Σ c ; G) in the proof search returns a BLU interpolant for a K5-sequent G.Because the induction on the proof-search is quite technical and involves multiple cases, we demonstrate only a few representative cases, relegating most to the Appendix and omitting simple ones, e.g., BLUIP(i), altogether.
BLUIP(ii) We show that M, I A ℓ (t, Σ c ; G) implies M, I G for any interpretation I of G into any K5-model M = (W, R, V ).The hardest among Steps 1-3 is Step 3 using row 5 in Table 4. Let G = G ′ , ϕ and M, I i.e., for each 1 ≤ i ≤ h either M, ρ δ i or M, I(τ ) γ i,τ for some τ ∈ G.For an arbitrary v such that ρRv and the the smallest j such that •j / ∈ G, clearly By IH, The only other case we consider (here) is Step 4d.Let M, I A ℓ (t, Σ c ; G) for A ℓ (t, Σ c ; G) from (5), i.e., for some 1 ≤ i ≤ h we have M, ρ ♦δ i , and M, I(•1) ♦δ ′ i , and M, I(τ ) γ i,τ for all τ ∈ G.In particular, M, v δ i for some ρRv and M, u δ ′ i for some I(•1)Ru.Let M ′ be obtained by copying u into u ′ away from the root in M and let G. Since we have (M, I) ∼ (M ′ , J ) by Lemma 21, we have M, I G by Lemma 17(2) in all cases.
BLUIP(iii) We show the following statement by induction restricted to sufficient calls: if Here we only consider Step 4 as the other steps are sufficiently similar to K and S5 covered in [12,13].Among the four subcases, Step 4a is tedious but conceptually transparent.Step 4c is trivial because the induction statement is only for sufficient calls while Step 4c calls are insufficient by Lemma 23.Out of remaining two steps we only have space for Step 4d, which is conceptually the most interesting because its recursive call may be insufficient, precluding the use of IH for it.Let M, I A ℓ (t, Σ c ; G) for A ℓ (t, Σ c ; G) from (5).We first modify M and I to obtain an injective interpretation ) is not empty and partitioned into pairs (v, u) with I ′ (•)Rv and not I ′ (•)Ru.To this end we employ copying as per Def.20, constructing a sequence of interpretations I i from G into models N i = (W i , R i , V i ) starting from N 0 = M and I 0 = I as follows: 1.If I i (τ 1 ) = I i (τ 2 ) for τ 1 = τ 2 , obtain N i+1 by copying I i (τ 2 ) to a new world w and redirect τ 2 to this new world, i.e., and obtain N ′ by copying: for each y ∈ Y , copy I K (•1) away from the root to a new world y 2 ; -for each z ∈ Z, copy for all y ∈ Y , and not I ′ (•)R ′ y 2 for all y ∈ Y , and I ′ (•)R ′ z 1 for all z ∈ Z, and not I ′ (•)R ′ z for all z ∈ Z.Thus, we obtain the requisite partition Ordinarily, here we would use IH, but this is only possible for sufficient calls, which, alas, is not guaranteed for (6).What is known by Lemma 23(3) is that every branch going up from (6) leads to a call of the form where Θ j ⊇ Θ and Φ j ⊇ Φ, that returns multiformula ℧ j and is either sufficient or insufficient but saturated.Let Ξ denote the multiset of these multiformulas ℧ j returned by all these calls.Since Step 2 is the only one used between that call and all the calls comprising (11), it is clear that (6) is their conjunction, i.e., Collecting all this together, we conclude that for each pair (v, u) ∈ P there is some We distinguish between two cases.First, suppose for at least one pair (v, u) ∈ P there is a sufficient (12).By IH for this ℧ v,u there is an interpretation Finally, by restricting to labels of G, we can see that Otherwise, (12) does not hold for any pair (v, u) ∈ P and any sufficient ℧ v,u ∈ Ξ.

Conclusion
We presented layered sequent calculi for several extensions of modal logic K5: namely, K5 itself, KD5, K45, KD45, KB5, and S5.By leveraging the simplicity of Kripke models for these logics, we were able to formulate these calculi in a modular way and obtain optimal complexity upper bounds for proof search.We used the calculus for K5 to obtain the first syntactic (and, hence, constructive) proof of the uniform Lyndon interpolation property for K5.Due to the proof being technically involved, space considerations prevented us from extending the syntactic proof of ULIP to KD5, K45, KD45, KB5, and S5.For S5, layered sequents coincide with hypersequents, and we plan to upgrade the hypersequent-based syntactic proof of UIP from [11] to ULIP (see also [13]).As for KD5, K45, KD45, and KB5, the idea is to modify the method presented here for K5 by using the layered sequent calculus for the respective logic and making other necessary modifications, e.g., to rule dd, to fit the specific structure of the layers.We conjecture that the proof for K45, KD45, and KB5 would be similar to that for S5, whereas KD5 would more closely resemble K5.

we apply
Step 4c to it which makes it insufficient.
Proof.In addition to the cases provided in the proof in the main text, here we show additional representative cases: BLUIP(ii), some of the cases from Steps 1-3.row 2 of Table 4: G = G ′ {q, q} σ .Then either M, I(σ) q or M, I(σ) q, hence M, I G. row 4 of Table 4:

BLUIP(ii), remaining cases from Step 4.
Steps 4a and 4c.We can treat them simultaneously because both are calculated according to (2).This case is easy as the truth of the interpolant implies that some literal in G is true.
Step 4b.Suppose M, I A ℓ (0, Σ c ; Γ ) for the multiformula presented in (3) Obviously, either M, ρ ⊥ or M, ρ ♦⊤.In the latter case, it follows from the right conjunct of ( 19) that some literal in Γ is true, resulting in M, I Γ .In the former case, based on the left conjunct of ( 19), there is an 1 and M, I Γ .Otherwise, M, J Γ , so again M, I Γ .BLUIP(iii), some of the cases from Steps 1-3.Let us start with the induction steps in which G is not saturated and A ℓ (t, Σ c ; G) is calculated according to a row in Table 4 (one of Steps 1-3 in the algorithm): row 2 of Table 4: G cannot be of the form G ′ {q, q} σ since A ℓ (t, Σ c ; G) = σ : ⊤ and we assumed M, I A ℓ (t, Σ c ; G). row 4 of Table 4: The cases are symmetric, so let us only treat the former.By assumption, G is a sufficient layered sequent and thus so is G ′ {ϕ ∧ ψ, ϕ} σ by definition.So we can apply the IH and get a model M ′ and interpretation •j as defined in (10).From M, I A ℓ (t, Σ c ; G) it follows that for some 1 ≤ i ≤ h, both M, I(•) δ i and also for all τ ∈ G, M, I(τ ) γ i,τ .From the former it follows that there is v such that I(•)Rv and M, v δ i .Define a new interpretation J = I ⊔ {(•j, v)}.By inspection of (10) we can easily see that M, J A ℓ t, Σ c ; G ′ , ϕ, [ϕ] •j .Note that G ′ , ϕ, [ϕ] •j is a sufficient layered sequent, and by IH there are M ′ and J ′ such that (M ′ , J ′ ) ≤ ℓ (M, J ) and M ′ , J ′ G ′ , ϕ, [ϕ] •j .Now define I ′ = J ′ ↾ Dom(I).Clearly, (M ′ , I ′ ) ≤ ℓ (M, I) and M, I G. row 9 of Table 4: We construct an ℓ-bisimilar model M ′ with interpretation I ′ such that each occurrence of ℓ in G is still falsified in the corresponding world in M ′ according to I ′ .We proceed in two steps.Informally speaking, we first copy worlds to make an injective interpretation I ′ and after that we modify the valuation of p ∈ {ℓ, ℓ}, as desired.1. Divide the domain of I into equivalence classes such that σ and τ belong to the same class if and only if I(σ) = I(τ ).One could say that each class is represented by a world in M. For each equivalence class with n > 1 elements, make n − 1 copies of the corresponding world and call this model N = (W ′ , R ′ , V N ).Define I ′ to be the interpretation assigning different labels in an equivalence class to copies of the worlds in such a way that I ′ is injective and, by Lemma 21, (N , I ′ ) ∼ (M, I).
2. Define M ′ = (W ′ , R ′ , V ′ ) to be the same as N except for valuations of p: We prove that M ′ , I ′ (σ) ϕ whenever σ : ϕ ∈ G by induction on the structure of ϕ.Recall that G does not contain formulas of the form ♦ψ.
-If σ : ℓ ′ ∈ G with ℓ ′ ∈ Lit \ {ℓ, ℓ}, then M, I(σ) ℓ ′ by ( 20 In the latter case, M is a one-world (irreflexive) model.We only need to define a new valuation on M to define M ′ as in Step 2 and the proof proceeds the same way and is left to the reader.In particular, note that Γ has no boxed formulas as it only consists of the trunk.For the former case, M consists of more than one world.In particular, for each 1 ≤ i ≤ h, either M, I(•) γ i or for all worlds v such that I(•)Rv we have M, v δ i .For all such v, consider the interpretation J v : {•, •1} → M defined by mapping label •1 to world v.By inspection of (4), observe that for all such v we have M, J v A ℓ (0, Σ c ; Γ, [{ψ | ♦ψ ∈ Γ }] •1 ).It will be sufficient to fix such a v. Since A ℓ (0, Σ c ; Γ, [{ψ | ♦ψ ∈ Γ }] •1 ) is sufficient by Lemma 23(2), we can apply the induction hypothesis to obtain (M ′ , J ′ ) ≤ ℓ (M, J v ) such that M ′ , J ′ Γ, [{ψ | ♦ψ ∈ Γ }] •1 .Now let I ′ be J ′ restricted to the domain {•}.Hence, M ′ , I ′ G.

Example 8 .
G = ϕ, ψ, [χ], [ξ], [[θ]] is a layered sequent with the trunk and three crown components: two [ ]-components and one [[ ]]-component.Since it has both the trunk and a [[ ]]-component, it can only be a K5-or KD5-sequent.A corresponding labeled sequent is . Taking I of G into M as the identity function (or I(•) = 1 in case of KB5), we have M, I G as desired.⊓ ⊔ 4 Uniform Lyndon Interpolation Definition 14 (Multiformulas).The grammar

( c )
else, if t = 1 and ♦G c ⊆ Σ c , stop and return A ℓ (t, Σ c ; G) = LitDis ℓ (G).(d) else, apply the rule dd as follows (where w.l.o.g.•1 ∈ G): one of the following types: (a) sufficient and final when calculated via Step 1; (b) sufficient and propositionally saturated when calculated via Step 3, with every branch going up from there consisting of more Steps 2-3, followed by either final Step 1 or continuation via Step 4d; (c) insufficient and saturated when calculated via Step 4c.Theorem 24.Logic K5 has the BLUIP and, hence, the ULIP.