AREAS DE MANEJO ESPECIAL Plano de manejo ambiental.

e notion of bisimulation was introduced to modal logics by van Benthem in [129] called p-relation and later zigzag-relation [118]. Bisimulation and its vari- ants will be central throughout all chapters of this thesis. It will be adapted for every logic that is to be characterised. It is a model-theoretic notion, in that it relates interpretations that satisfy the same -concepts. Hence all these inter- pretations are models of the same -theory on concept-level.

D 2.1.1 . LetIandHbe interpretations of the same signatureτ. en

the relationZ⊆ ΔI×ΔHis calledbisimulationbetweenIandHif for all(d,e) ∈Z the following holds:

2. for allr ∈ N_R we have if there isd₀ ∈ ΔIwith(d,d₀) ∈ rI_{then there is}

e0∈ ΔHwith(e,e0) ∈rHand(d0,e0) ∈Z. (FORTH)

3. for allr ∈ N_R we have if there ise₀ ∈ ΔHwith(e,e₀) ∈ rHthen there is d₀∈ ΔIwith(d,d₀) ∈rI_and(d

0,e0) ∈Z. (BACK)

We write(I,d) ⟷−− (H,e)and say(I,d)isbisimilar to(H,e)if there is a bisimu-

lationZbetweenIandHsuch that(d,e) ∈Z. ♢

It is sometimes more convenient to show that two pointed structures are bisim- ilar, if the bisimulation is described by a pebble game between two players I andII, where IIhas a winning strategy. Bisimulation games are derived from Ehrenfeucht-Fraïssé games [48]. e laer correspond to potential isomorphisms which are the analogous notion [42] of bisimulation onFOlevel and can be found in standard textbooks like [47]. Bisimulation games are of particular interest in characterisations of model logics over finite models [115, 104]. Although we shall not endeavour into finite model theory, we shall use them because their descript- ive and illustrative nature facilitates our arguments in proofs to come.

In thebisimulation game G(I,d;H,e), the interpretations(I,d)and(H,e) are considered as separate graphs. For each interpretation there is one pebble which can be placed on an element of the interpretation, and whenever a pebble is on some elementd₀ ∈ ΔIand there isd₁ ∈ ΔIwith(d₀,d₁) ∈ rI then the pebble can bemoved along an r-edgetod₁. e same applies forH.

e position of both pebbles in each interpretation is summarised asconfigur- ation(I,d₀,H,e₀)whenever the pebble inIis placed ond₀and the pebble inHis

placed one₀.

e gameG(I,d;H,e)is played according to the following rules: e 0-th round starts in configuration(I,d;H,e), where for allA∈N_C:d ∈AIiffe∈AHmust hold, orIIhas lost the game in the 0-th round. For any configuration(I,d₀;H,e₁) the following rules apply:

1. In each roundIpicks one of the two interpretations,Isay, and moves the pebble along some edge, say anr-edge emerging from the pebble’s position to some element d₁. If there is no such edge along which playerI could move,IIhas won the game.

2. IIhas to respond to playerI’s move in the other interpretation, hereH, by moving the pebble frome₀along some edge with the same label, in our case anr-edge, to some elemente₁such that for allA∈N_C:d₁∈AI_iffe

In case playerIIcannot move accordingly,IIhas lost the game; otherwise a new round begins.

Clearly, these rules mirror the ATOM-property as well as the FORTH- and BACK-property, respectively, depending on which interpretationIchooses. D 2.1.2 . IIhas awinning strategy in the game G(I,d;H,e)if she can respond to every move ofIsuch that she either wins the game or can play forever. ♢

P 2.1.3 . IIhas a winning strategy in the game G(I,d;H,e)iff(I,d) ⟷−− (H,e).

P. For the if-direction, we define the following relation, which is obviously a bisimulation:Z∶= {(d,e) ∈ ΔI_×ΔH_{∣IIhas a winning strategy forG(I,d;H,e)}.}

For the only-if direction, letIIplay such that each new configuration(I,d₀;H,e₀) has a pair(d₀,e₀) in Z. To see that this is possible, let the game have reached (I,d0;H,e0)such that (d0,e0) ∈ Z. LetImove inI, say, fromd0along somer-

edge tod₁. en, by definition ofZ, there must be a pair(d₁,e₁) ∈Zsuch thate₁ is anr-successor ofe₀inHandd₁is atomically equivalent toe₁. HenceIImoves toe1. us,IIeither wins the game or can play forever. 2

For each signatureτthe bisimulation relation is an equivalence relation on the pointedτ-interpretations. We therefore talk, w.r.t. to a signatureτ, about bisim- ulation types, which are the classes formed by this equivalence relation.

e bisimulation type links interesting structures together: Tree-unravellings are interpretations which, viewed as a graph, form a tree whose elements cor- respond to all finite path sequences. According to [18, Notes to Chapter 2] and [20] the method of tree unravelling first appeared 1959 in [46] but became pop- ular due to [117]. In particular, every pointed interpretation is bisimilar to its tree-unravelling:

D 2.1.4 . Let (I,d) be a pointed τ interpretation. We call I_d its tree- unravelling in dwhereΔId is recursively defined by

1. d∈ ΔId

2. Let _d̄ _{∶= d r}

1d1r2…rndn ∈ ΔId. For allr ∈ NR: if (dn,dn+1) ∈ rI then

̄

d⋅r⋅d_n+1 ∈ ΔId, whered̄_⋅r_⋅d

For allA∈N_Candr∈N_Rwe define

rI_{d ∶= {( ̄d, ̄drd) ∣ ̄d, ̄drd ∈ Δ}I_d}

AI_{d ∶= { ̄d∈ Δ}I_{d ∣ ̄d=d r}

1d1r2…rndnanddn ∈AI}

♢ Hence each element_d̄_{=d r}

1d1r2…rndninΔId represents a path of finite length beginning indand noting all traversed nodes and the edges by which they were accessed.I_dis a proper tree-interpretation:dcannot be reached via any edge and every other element inI_dhas exactly one predecessor and exactly one edge with which it is connected to its predecessor.

By the recursive definition of a tree-unravelling, the depth of such a tree-structure is at most countably infinite.

O 2.1.5 . (I,d) ⟷−− (Id,d)

P. By maintaining configurations(I,d;I,d r₁d₁r₂⋅ ⋅ ⋅r_nd_nr_n+1d)IIhas a winning strategy in the gameG(I,d;I_d,d). 2