By Edith Spaan

It is a doctoral dissertation of Edith Spaan lower than the supervision of prof. Johan van Benthem.

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R \r 18 an alternating □ C ase J Suppose / \ , M and , „ are not skeleton subframes of T \ , and / \ , cy_^5 • >• and . ^ a r e not skeleton subframes of T 2. As we have seen in the proof of the first criterion, any frame in Rt(T\) is of the form: ... , ... :.. ^ >m ... or a skeleton subframe of It is easy to see that any frame in R t(T2) is a singleton, or the frame •—~ . Let F = (W, R \,R 2) be a frame in T \ 0 T 2, and let w0 G W. 2 For v a proper successor of w0, let Wv consist of all the worlds in W that are reachable from v by a path that does not contain w0.

Or 4. A skeleton subframe of the frame: We’d like to use the following algorithm for T satisfiability: guess a model M = (W, R, tt) of size at most |0| and a world w0, verify that M, w0 f= 0 and that there exists a frame F € T such that F^wlwoRpath^)1^ } = (W, R). Unfortunately, verifying the last condition can be of arbitrary complexity. For suppose A is a subset of the natural numbers. Define T in such a way that a cyclic frame F is a member of T iff the length of the cycle is in A. Then A is reducible to verification of the last condition for T .

VF, Sa) = F a ® ((W ,S a) \ ( W \ W a)). Proof. Let F[ = (W[, R[) consist of the disjoint union of F\ and 2\W2\ — 1 frames isomorphic to F\, and let F'2 = consist of the disjoint union of F2 and 2 \W i \ — 1 frames isomorphic to F2. Since T a is closed under disjoint union, F'a £ T a. Define F = (W, S i,S 2) in such a way that there exist isomorphisms f a from Fl to (VF, Sa) and for all w £ Wa : f a{w) = It is obvious that if F can be constructed, then F satisfies the requirements of the lemma.

### Complexity of Modal Logics [PhD Thesis] by Edith Spaan

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