By Edith Spaan

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

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**Example text**

First suppose that 0! consists of two elements, say a and 6. 6, © aGn T a satisfiability is polynomial time reducible to T a ® satisfiability. By the previous section, T a ® T\> satisfiability is in NP, and thus © a€n T a satisfiability is in NP as well. Finally, suppose that 0! contains at least three elements. Since we are not in case N, it follows that for all a £ fi, is not a skeleton subframe of T a. Furthermore, since we are not in case A, we know that / \ is not a skeleton subframe of T a.

Then satisfiability is PSP ACE-hard. 1. We will prove the analog for the general join of uni-modal logics. There is no reason why there shouldn’t exist such a theorem for multi modal logics. However, there are very many cases which force PSPACE hardness, and we haven’t managed to obtain a complete characterization. 1. To see why the situation is more complex if we consider the join of multi-modal logics, look at the following example: CHAPTER 3. THE COMPLEXITY OF THE JOIN 46 Q = {{a, a'}, {&, 6'}}, and .

E. for a £ {1,2}, either all three worlds are a reflexive, or all three worlds are a irreflexive. Next, we want the constructed tree to be an T \ 0 T 2 subframe. This is not true for every frame F: Suppose for instance that wqR\W i , and that T \ consists of the closure under disjoint union of the frame •—~ . Then the tree constructed from F is not an T \ 0 T 2 subframe, as identification of w0 and wi worlds leads to arbitrarily long Ri paths. This problem can be avoided by requiring that Wo has no non-reflexive Ri edges, and wt and wr have no non-reflexive R 2 edges or vice versa.