By Edith Spaan

This can be a doctoral dissertation of Edith Spaan below the supervision of prof. Johan van Benthem.

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**Example text**

The satisfiability problem for this class of frames is in NP. e. pO(^ )w }|

\). We start by describing the form of the frames in T \ and T 2. 3 Let T \ and T 2 be two classes of uni-modal frames closed under disjoint union. If we are not in case I and A through F, then for some a £ {1,2}, we are in one of the following three cases: not skeleton subframe of T a not skeleton subframe of Ta • CY ct G •A • ; •— •> \• ; •— • • H • \ , * ** \ ) • • CY^^ ^_^0—^0 ^ 00 ^0 ^ ^ ^CY , CY } 9 ^CY J In the proof of the first requirement, we have seen that / \ m , and 9 + 2 „ are not skeleton subframes of T \ nor of T 2.

4 can be applied in cases A through F follows from the previous section. 4 with cr = abc, and F the following frame: c £)a 48 CHAPTER 3. 2. Let 0! be the set of indices a such that ^ is a skeleton subframe of T a. Since we are not in case I, it follows that |H'| > 2. 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.

It follows that we need DSPACE(s(n2) + (m — l)s(rc2)) = DSPACE(ms(n2)). On the other hand, the time that is needed is proportional to the number of queries multiplied by the time per query. It follows that we need DTIME(£(n2)£(n2)m_1) = DTIME(*(n2)m). For nondeterministic time classes, this proof does not go through, since in general these classes are not known to be closed under complementation. However, a slight variation of the construction above gives the wanted transfer result. Instead of computing whether a query belongs to 7Za, we guess the set of queries in lZa and verify that all these queries are indeed in 7Z^.