"... In PAGE 7: ... For other plausible applications of the subgroup membership problem, the reader is also referred to [12] in which the DDH assumption is applied to the cryptographic schemes which only known method to construct is to base on the QR assumption. We summarize the examples above in Table2 , however, the table is not exhaustive at all. Table 2.... ..."

"... In PAGE 2: ...nown for these problems, see e.g. [8]. Membership problems for tree automata were investigated in [13] for ranked trees (see Table1 for the results of [13]) and [14] for un- ranked trees from the perspective of computational complexity. Here we extend this line of research by investigating the computational complexity of membership problems for various classes of tree automata on compressed trees (dags and SL cf tree grammars).... In PAGE 2: ... For all cases, we present upper and lower bounds which vary from NL (nondeterministic logspace) to PSPACE (polynomial space). Our results are collected in Table1 . We also briefly consider the parameterized complexity [15] of membership problems for tree automata.... In PAGE 5: ... class of SL cf tree grammars (e.g., the class of all dags). The membership problem for the fixed TA A and the class G is the following decision problem: INPUT: G 2 G QUESTION: Does eval(G) 2 T(A) hold? For a class C of tree automata, the uniform membership problem for C and the class G is the following decision problem: INPUT: G 2 G and A 2 C QUESTION: Does eval(G) 2 T(A) hold? The upper part of Table1 collects the complexity results that were obtained in [13] for uncompressed trees. The statement that for instance the membership problem for TA is NC1-complete means that for every fixed TA the membership problem is in NC1 and that there exists a fixed TA for which the membership problem is NC1-hard.... In PAGE 7: ... Theorem 4 and 5) we obtain the complexity results for linear SL cf tree grammars with a fixed number of parameters (resp. unrestricted SL cf tree grammars) in Table1 , see lin.... ..."

"... In PAGE 4: ... In column 7 the time for the transformation (from uIMC to uCTMDP) is shown. In Table2 we collect statistics about the implementation of the timed reachability algorithm... ..."

"... In PAGE 2: ... This is the first algorithm that breaks the O(n2) update barrier for all graphs with o(n2) edges. The new algorithm and some of the previously existing algorithms for the dynamic reachability problem are compared in Table1 . Our algorithm has the fastest update time, among all algorithms that work on all graphs, but alas the slowest query time.... ..."

"... In PAGE 2: ... This is the first algorithm that breaks the O(n2) update barrier for all graphs with o(n2) edges. The new algorithm and some of the previously existing algorithms for the dynamic reachability problem are compared in Table1 . Our algorithm has the fastest update time, among all algorithms that work on all graphs, but alas the slowest query time.... ..."

"... In PAGE 4: ... 4.1 Reachability Analysis Results for reachability analysis on the benchmark circuits are shown in Table1 . The name of the circuit appears in Column 1, and number of latches in Column 2 #28marked #23L#29.... In PAGE 5: ...63. To demonstrate this e#0Bect of ICR, we present details of the experimental results for the prolog circuit in Table 2, where labels of the Columns are similar to Table1 . Note that use of ICR results in less number of BDD leaves #28sub-problems#29 in most image computation steps of the complete traversal.... ..."