CORREGIMIENTO DE COSTILLA.

In this section the the notion ofn-bisimulation [55] and the appropriaten-round bisimulation game will be introduced. We shall show that for every -concept there is an appropriaten<ωsuch that this concept is invariant undern-bisimulation and conversely thatn-bisimulation can, if the signature is finite, be captured by characteristic -concepts, in the sense that if two interpretations satisfy the same characteristic concept, they aren-bisimilar:

For every natural numbern, then-round bisimulation game G_n(I,d;H,n)is the bisimulation gameG(I,d;H,n)which breaks off aern-rounds.

D 2.1.6 . IIhas a winning strategy inG_n(I,d;H,e)ifIIeither wins the

game withinn-rounds or can play forn-rounds. ♢

Equivalently, one can definen-bisimulation between IandHas a system (or family)(Z_k)_0≤k≤nof relationsZ_k ⊆ ΔI× ΔHsuch that all of the following require- ments are met:

1. for all k ≤ n and all(d₀,e₀) ∈ Z_k and all A ∈ N_C we haved₀ ∈ AI iff e₀∈AH_(ATOM)

2. for allk<n,(d₀,e₀) ∈Z_k+1andr∈N_Rwe have: if there is anr-successor d1ofd0inIthen there is anr-successore1ofe0inHsuch that(d1,e1) ∈Zk (FORTH)

3. for allk<n,(d₀,e₀) ∈Z_k+1andr∈N_Rwe have: if there is anr-successor e₁ofe₀inHthen there is anr-successord₁ofd₀inIsuch that(d₁,e₁) ∈Z_k (BACK)

Proposition 2.1.3 equally holds forn-bisimulations and n-round bisimulation games: Every pair inZ_kgivesIIto a winning strategy for thek-round bisimulation game, henceIIcan playnrounds and conversely, ifIIhas a winning strategy for G_n(I,d;H,e)then for eachk≤nwe obtain a relation

Z_k∶= {(d₀,e₀) ∣IIhas a winning strategy forG_k(I,d₀;H,e₀)} such that(Z_k)_k≤nforms ann-bisimulation betweenIandHwith(d,e) ∈Z_n.

Analogously to bisimulation, we write(I,d) ⟷−− n(H,e)if there is ann-bisimu- lation(Z_k)_k≤nbetweenIandHwith(d,e) ∈Z_n.

O 2.1.7 . For all τ and all n<ω we have

1. ⟷_{−− n}is an equivalence relation on the pointed interpretations 2. If(I,d) ⟷−− n+1 (H,e)then(I,d) ⟷−− n(H,e)

3. If(I,d) ⟷−− (H,e)then(I,d) ⟷−− n(H,e)

Interestingly, the concepts of can be stratified according to the nesting- depth of their quantifications, which we shall call rank. e fragments of - concepts, which are induced by this stratification, are preserved by the appropri- ate notion ofn-bisimulation, wherenis the rank of the concepts.

For allτtherankof anC∈ (τ)is defined as follows:

rankC∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 C∈N_CorC= ⊥ max{rankD, rankE} C=D⊓E rankD C= ¬D

1 + rankD C= ∃r.D, for allr∈N_R

P 2.1.8 . Let C be an -concept of rank≤n as well as(I,d)and(H,e) pointed interpretations. If(I,d) ⟷−− n(H,e)then d∈CI ⟺ e∈CH.

P. e proof is carried out via induction uponn. Forn = 0,Cis w.l.o.g. a concept name. AsIIhas a winning strategy forG0(I,d;H,e), she does not lose

the 0-th round. Hencedandeare atomically equivalent and therefored∈CIiff e∈CH_.

For n+ 1 the casesC = D⊓Eand C = ¬Dcan be reduced to showing the equivalence forDandErespectively. We can therefore w.o.l.g. assume thatC = ∃r.Dfor somer∈N_R.

Letd∈ (∃r.D)I_{, thus there is anr-successord}

0ofdinIsuch thatd0 ∈DI. As

IIhas a winning strategy forG_n+1(I,d;H,e), there is anr-successore₀ ofeinH

such thatIIhas a winning strategy forG_n(I,d₀;H,e₀). e induction hypothesis yields nowe0∈DHand hencee∈ (∃r.D)H. e only-if direction follows the same

rationale. 2

C 2.1.9 . If(I,d) ⟷_{−− (}H,e)then for every -concept C we have d∈CI iff e∈CH_.

Although the corollary is obvious, it reveals the interesting fact that no prop- erty is expressible by which could distinguish the tree-unravelling from its original structure. Furthermore each conceptCis local:

LetIℓ_d be defined as the restriction of the tree-unravellingI_dto path-elements

up to lengthℓ:

ΔIℓd ∶= {dr

0d1r1⋅ ⋅ ⋅rkdk∣k< ℓ} AIℓ_{d ∶=A}I_{d ∩ Δ}Iℓ_d

rIℓ_{d ∶=r}I_{d ∩ Δ}Iℓ_{d × Δ}Iℓ_d

L 2.1.10 . For allℓ <ω we have(I,d) ⟷−− ℓ (Iℓd,d).

P. Consider the gameG_ℓ(I_d,d;Iℓ_d,d). As the game starts uniformly ond in both interpretations, the path-elements in each configuration get successively longer with each round, but do not exceed lengthℓ. By simply copying every move ofI,IIhas a winning strategy inG_ℓ(I_d,d;Iℓ_d,d).

As(I,d) is in particularℓ-bisimilar to(I_d,d) and⟷_{−− ℓ} is transitive we have

(I,d) ⟷−− ℓ(Iℓd,d). 2

D 2.1.11 . An -conceptCisℓ-localif for every interpretationIand

everyd∈ ΔIwe haved∈CIiffd∈CIℓd. _♢

it appears in the context of Gaifman-Graphs [54] for which the notion of being ℓ-local is defined in much greater generality. It is not unusual though to restrict locality in the context of standard modal logics and therefore also in our context to forward reachable elements [103].

C 2.1.12 . Let C be an -concept of rankℓ. en C isℓ-local.

is result shows that -concepts can only express properties whose truth can be checked within finite depth from distinguished point. -concepts are also oblivious to copies of successors, i.e. the number of successors, which are bisimilar to each other, nor can distinguish elements that have predecessors from those that do not.

-concepts are not affected by disjoint unions with other structures either: LetIandHbe two structures such thatΔIandΔHare disjoint. enI⊎His

called thedisjoint union of IandH, has the carrier set ΔI⊎H ∶= ΔI⊎ ΔH and AI⊎H_∶=AI_⊎AH_{for allA∈N}

CandrI⊎H∶=rI⊎rHfor allr∈NR.

Since the tree-structure retains only those elements accessible on the connected component of its root, we haveI_d= (I⊎H)_d, so locality of -concepts entails for alld∈Iand every interpretationHthat(I,d) ⊨C ⟺ (I⊎H,d) ⊨C.

e notion of beingℓ-local for -concepts or modal logic formulae however plays a central role in the characterisation of standard modal logics: de Rijke [113] used locality as characteristic property to defineabstract modal logics(abstract in the sense of generic) which he namedfinite rank, alluding to the statement Co- rollary 2.1.12. Clearly, locality is also the reason why we can decide for an - concept, whether or not it is satisfiable. For anℓ-local conceptC, we only need to build every tree of depthℓinvolving only the (finitely many) signature symbols of C. e result goes back to [124] where the decidability of the 2-variable fragment ofFOwithout equality is proven. Later, the result is extended in [97] allowing equality, though using completely different techniques. A complexity theoretical investigation is conducted in [86, 68] showing PSPACE-completeness for the sat- isfiability problem of standard modal logic formulae. Fourteen years aer [86] an -specific version showing PSPACE-completeness for concept satisfiability is presented along with the logic itself in [122]. How a naïve search for a model could be accomplished will be clearer when looking at characteristic concepts.