### TABLE II PROBABILITIES FOR N-STATE MODEL

### Table 1 Results of the analysis of the Lehmann-Rabin algorithm. n states ts vs transitions ett it

"... In PAGE 9: ... The mean number of philosophers who are simultaneously eating is instead obtained by summing up the steady state probabilities of any state of the underlying Markov reward chain multiplied by the number of philosophers eating in that state, which is derived by yield reward 1 assigned to action having type eati in term P00 i . In Table1 we summarize the results we have obtained by varying the num- ber n of philosophers from 3 to 6, where i = 4 and i = 0:75 for any 0 i n ? 1. The columns show, respectively, the number of philosophers, the number of states of the integrated interleaving semantic model, the num- ber of tangible states (which coincides with the number of states of the Markov reward chain), the number of vanishing states, the number of transitions, the number of exponentially timed transitions, the number of immediate transi- tions, the absence of adjacent philosophers simultaneously eating (veri ed by means of CWB-NC), the absence of deadlock (veri ed by means of CWB- NC), and the mean number of philosophers who are simultaneously eating... ..."

### Table 2 The probability that a randomly generated n-state IPD

"... In PAGE 10: ... 8. 0 For later purposes, Table2 computes the ex- pected number of self-cooperative n-state IPD machines in a randomly-chosen population of size 30. To recap, Lemma 1 counts the total number of IPD machines with n states.... In PAGE 19: ... On the other hand, if a single cooperator can outscore an otherwise de- fecting initial population, then cooperators will eventually have offspring with whom they can cooperate, do even better, and take over. Interestingly, for populations of size 30 it fol- lows from Table2 that initial populations whose players have 16-state IPD machine representa- tions include, on average, only 1.47 players whose self-play string is purely cooperative, whereas populations whose players have one-state IPD machine representations have a much greater ex- pected number of self-play cooperators - 7.... ..."

### Table 1. Verification runs for the dining philosophers problem Test case N State space Fired trans Exec time Result GT (wrong) 5 235 405 0.12 dead

"... In PAGE 32: ... In a traditional graph transformation system, the first solution would typically supercede the other in perfor- mance as the application of a more complex rule takes much more time than formalizing the problem with a set of rules of smaller complexity (by complexity we mean the number of nodes in the a170a13a171a15a172 and a173a41a174a42a175 graphs since the performance of pattern matching is the critical phase in rule application). However, after all the preprocessing (collecting potential matchings) performed at compile time, it will turn out that having a small number of relatively complex rules yield a better performance for verification than a larger number of rules with relatively low complexity (see Table1 for a comparison later in Sec.... In PAGE 36: ...etamodel (in Fig. 11), and not by the classes of the model (Fig. 7). Even though for meta-level encodings the verification managed to terminate only for few number of philoso- phers (as shown in Table1 ), the verification process automatically detected a non-trivial error in the our graph transformation semantics of UML statecharts which leads our system into a state where our safety criterion is violated. The problem relies in the fact that testing whether the fork is not held by anyone, and actually acquiring the fork is not an atomic operation.... In PAGE 37: ...Table1 ), for each verification run at most 100 Megabytes of system memory was allocated to store the state space, and Mura194 was running on a 550 MHz Pentium III machine. Table 1.... ..."

### Table 5.1, page 60, lists the number of different regular languages associated with UDFAs with s final states for n = 7 and n = 8. When generating UDFAs with n = 7 or n = 8 states, we want to simulate these frequencies. We can use a uniform random number generator to simulate the desired frequency of final states as described in Example 34, on page 60. Example 34 Using a uniform random number generator which generates deviates between zero and one, we are able to simulate the desired frequency of final states. This is done by calculating the number of regular languages accepted by n-state UDFAs with s final states as a proportion of the total number of regular languages associated with n-state UDFAs. As shown in Table 5.2, for n = 7, a random deviate, Ui, in the interval [0, 1 738) results in the generation of a UDFA with 0 final states. The probability of Ui falling in this interval is consistent with the probability of a regular language being associated with an n-state UDFA with zero final states. This method of choosing the number of final states according to the interval of Ui will result in the desired frequency of final states.

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### Table 5.10 shows that, for three state UNFAs, the maximum possible number of regular lan- guages can only be obtained with one start state. As soon as a start state set containing more than one element occurs, the number of regular languages which could be associated with the UNFA decreases. Therefore, this table leads us to expect that generating n-state UNFAs with varied number of elements in the start state set will have bad results over the domain of the regular languages. Experiment 20, page 76, confirms this. In the rest of our investigation of randomly

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### Table 2.1: Rain Attenuation Models [10] Variable Function Frequency Range

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### Table 4. Results of Two-Deep Cross-Validation Averaged over 100 Splitsa

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"... In PAGE 8: ... This was done to make the calculation more similar to a normal test set q2. The entire two-deep cross-validation is carried out 100 times, and the mean values are reported in Table4 . The 3.... ..."