### Table 1: Deterministic sampling using aBDD (static and dynamic)

1999

"... In PAGE 5: ...Experiment 1 ( Table1 , and Figure 2): First, we use the order computed by sampling to build the BDD statically. Except for slightly inferior orderings on c499 and c1355 (both circuits are functionally equiva- lent) we find that our methods always produce better variable orderings than those produced by DFS search based static techniques (Table 1).... In PAGE 5: ...Experiment 1 (Table 1, and Figure 2): First, we use the order computed by sampling to build the BDD statically. Except for slightly inferior orderings on c499 and c1355 (both circuits are functionally equiva- lent) we find that our methods always produce better variable orderings than those produced by DFS search based static techniques ( Table1 ). For many industrial examples we find that DFS-MIN cannot even process the circuits.... In PAGE 5: ... It is easy to see that window based sampling gives much better results than cube based methods. Interestingly, for EX3 and EX6, aBDD based methods can create a small BDD for the output function, but cube based sampling fails for some of the runs! Experiment 2 ( Table1 and Figure 3) show the utility of window based sampling in a dynamic vari- able ordering scheme. That is, we show how dynamic reordering techniques can be significantly improved if they are supplied with an initial variable ordering generated using a window based sampling technique.... In PAGE 5: ... That is, we show how dynamic reordering techniques can be significantly improved if they are supplied with an initial variable ordering generated using a window based sampling technique. In Table1 , we find that we can produce far smaller graphs than the traditional dynamic reordering meth- ods (sift, sift-convergence). Also, for most of the large circuits we take less time.... ..."

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### Table 1: Computational results on the static graph: average values

2008

"... In PAGE 8: ... The original network has 7778913 junctions and 17154042 road segments; the number of nodes and arcs in each level is as follows. level 0 1 2 3 4 5 6 7 8 9 nodes 7778913 1517291 433286 182474 91888 53376 34116 23338 16445 11790 arcs 17154042 3461385 1283000 583380 308249 183659 119524 81170 57235 41092 We show the results for queries on the full graph without dynamic travelling times in Table1 ; in this case, all paths computed with the HH algorithm are fastest paths. In Table 2, instead, we record our results on a graph with dynamic travelling times; we also report the relative distance of the solution found with our heuristic version of the HH algorithm and the fastest path computed with Dijkstra, and, for comparative reasons, the results of the naive approach which consists in computing the traffic-free optimal solution with the HH algorithm (i.... In PAGE 8: ... All tables report average values over 5000 queries. All computational results in Table1 and 2 have been obtained on a multiprocessor Intel Xeon 2.6 GHz with 8GB RAM running Microsoft Windows Server 2003, compiling with Miscrosoft Visual Studio 2005 and optimization level 2.... ..."

### Table 1. State graphs of the dynamic control

"... In PAGE 11: ... 3.4 Complexity Table1 gives an overview of a subset of the state graphs we have generated using di erent reduction techniques and allows to compare their sizes. Execution Time With respect to execution time, the following observation can be made: execution times are roughly proportional to the size of the generated graphs, which means that the di erent reduction methods do not introduce any signi cant overhead.... In PAGE 11: ... For partial order reduction it is the case be- cause we use a simple static dependency relation. Table1 shows only minimization results for relatively small graphs (ap4a and mt4a) so that minimization time is small anyway. Nev- ertheless, it can be seen that minimization for observational equivalence is more expensive than for safety equivalence, as the computation of the transitive closure transition relation \ a quot; is required (where represents a non-observable and a an observable transition).... In PAGE 13: ... This lead to a number of error traces which we considered to be \probably because of too loose assumptions on the environment quot; and we added corresponding restrictions for subsequent veri cations. The graphs mentioned in Table1 have been obtained using the most restrictive environment. Property: Association Establishment Req1.... ..."

### Table 1. State graphs of the dynamic control

2001

"... In PAGE 11: ... 3.4 Complexity Table1 gives an overview of a subset of the state graphs we have generated using di erent reduction techniques and allows to compare their sizes. Execution Time With respect to execution time, the following observation can be made: execution times are roughly proportional to the size of the generated graphs, which means that the di erent reduction methods do not introduce any signi cant overhead.... In PAGE 11: ... For partial order reduction it is the case be- cause we use a simple static dependency relation. Table1 shows only minimization results for relatively small graphs (ap4a and mt4a) so that minimization time is small anyway. Nev- ertheless, it can be seen that minimization for observational equivalence is more expensive than for safety equivalence, as the computation of the transitive closure transition relation \ a quot; is required (where represents a non-observable and a an observable transition).... In PAGE 11: ... For the considered system, we get similar reductions when slicing according to the 4 main sub-protocols (1 to 2 addi- tional orders of magnitude), where connection opening is slightly more complicated than the others (it involves more signal exchanges than the others), and thus we get a bit less reduction. It was impossible to generate the state graph of the global system as a whole, thus we started to consider ap and mt in isolation (see rst two parts of the Table1 ). Finally, we were able... In PAGE 13: ... This lead to a number of error traces which we considered to be \probably because of too loose assumptions on the environment quot; and we added corresponding restrictions for subsequent veri cations. The state graphs mentioned in Table1 have been obtained using the most restrictive environment. Property: Association Establishment Req1.... ..."

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### Table 1. Forward computation of slices. dynamic dynamic static

2004

"... In PAGE 3: ... Algorithm 1 Updating Slicing Information Procedure Update(a33 ) 1: a34a36a35a37a33a39a38a41a40 = a42a36a33a19a43 ; 2: a44a19a33a39a45a46a40a48a47 a33a50a49 = a44a19a33a39a45a46a40a26a34a51a44a53a52a54a45a56a55 ++; 3: for (each use a57 in a58a59a34a60a40a11a47 a33a50a49 ) do 4: a34a36a35a37a33a39a38a41a40 = a34a36a35a37a33a39a38a41a40a62a61a64a63a48a40a66a65 a34a60a35a67a33a21a38a51a40a11a47 a57a68a49 ; 5: end for 6: a63a11a38a51a63 = the statement a34 in a69a59a70a71a47 a33a39a49 which has the maximum a44a19a33a39a45a46a40a48a47 a34a51a49 value; 7: a34a36a35a37a33a39a38a41a40 = a34a36a35a37a33a39a38a41a40a72a61a46a34a36a35a37a33a39a38a41a40a11a47 a63a11a38a51a63a73a49a21a47 a35a74a40a36a33a75a47 a63a11a38a76a63a73a49a77a49 ; 8: a34a36a35a37a33a39a38a41a40a11a47 a33a50a49a78a47 a35a74a40a36a33a75a47 a33a50a49a79a49 = a34a36a35a37a33a39a38a41a40 ; 9: a35a74a40a36a33a75a47 a33a39a49 ++; 10: for (each definition a63 in a70a14a40a66a65a80a47 a33a50a49 ) do 11: a63a48a40a66a65 a34a36a35a37a33a39a38a41a40a11a47 a63a73a49 = a34a36a35a37a33a39a38a51a40 ; 12: end for Forward computation of dynamic slices for example in Fig. 1 is shown in Table1 . In the execution step of a20a34a17a81a2 , which is a0a82a30a83a10 a20 a85a84a39a86 a11 a76a87 a11 a76a0a3a88 , a0 is defined, a4 a86 a11 a76a87 a11 a76a0a89a5 are the variables that are used.... ..."

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### Table 1. Forward computation of slices. dynamic dynamic static

2004

"... In PAGE 3: ... Algorithm 1 Updating Slicing Information Procedure Update(a33 ) 1: a34a36a35a37a33a39a38a41a40 = a42a36a33a19a43 ; 2: a44a19a33a39a45a46a40a48a47 a33a50a49 = a44a19a33a39a45a46a40a26a34a51a44a53a52a54a45a56a55 ++; 3: for (each use a57 in a58a59a34a60a40a11a47 a33a50a49 ) do 4: a34a36a35a37a33a39a38a41a40 = a34a36a35a37a33a39a38a41a40a62a61a64a63a48a40a66a65 a34a60a35a67a33a21a38a51a40a11a47 a57a68a49 ; 5: end for 6: a63a11a38a51a63 = the statement a34 in a69a59a70a71a47 a33a39a49 which has the maximum a44a19a33a39a45a46a40a48a47 a34a51a49 value; 7: a34a36a35a37a33a39a38a41a40 = a34a36a35a37a33a39a38a41a40a72a61a46a34a36a35a37a33a39a38a41a40a11a47 a63a11a38a51a63a73a49a21a47 a35a74a40a36a33a75a47 a63a11a38a76a63a73a49a77a49 ; 8: a34a36a35a37a33a39a38a41a40a11a47 a33a50a49a78a47 a35a74a40a36a33a75a47 a33a50a49a79a49 = a34a36a35a37a33a39a38a41a40 ; 9: a35a74a40a36a33a75a47 a33a39a49 ++; 10: for (each definition a63 in a70a14a40a66a65a80a47 a33a50a49 ) do 11: a63a48a40a66a65 a34a36a35a37a33a39a38a41a40a11a47 a63a73a49 = a34a36a35a37a33a39a38a51a40 ; 12: end for Forward computation of dynamic slices for example in Fig. 1 is shown in Table1 . In the execution step of a20a34a17a81a2 , which is a0a82a30a83a10 a20 a85a84a39a86 a11 a76a87 a11 a76a0a3a88 , a0 is defined, a4 a86 a11a76a87 a11a76a0a89a5 are the variables that are used.... ..."

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### Table 1. Forward computation of slices. dynamic dynamic static

2004

"... In PAGE 3: ... Algorithm 1 Updating Slicing Information Procedure Update(a33 ) 1: a34a36a35a37a33a39a38a41a40 = a42a36a33a19a43 ; 2: a44a19a33a39a45a46a40a48a47 a33a50a49 = a44a19a33a39a45a46a40a26a34a51a44a53a52a54a45a56a55 ++; 3: for (each use a57 in a58a59a34a60a40a11a47 a33a50a49 ) do 4: a34a36a35a37a33a39a38a41a40 = a34a36a35a37a33a39a38a41a40a62a61a64a63a48a40a66a65 a34a60a35a67a33a21a38a51a40a11a47 a57a68a49 ; 5: end for 6: a63a11a38a51a63 = the statement a34 in a69a59a70a71a47 a33a39a49 which has the maximum a44a19a33a39a45a46a40a48a47 a34a51a49 value; 7: a34a36a35a37a33a39a38a41a40 = a34a36a35a37a33a39a38a41a40a72a61a46a34a36a35a37a33a39a38a41a40a11a47 a63a11a38a51a63a73a49a21a47 a35a74a40a36a33a75a47 a63a11a38a76a63a73a49a77a49 ; 8: a34a36a35a37a33a39a38a41a40a11a47 a33a50a49a78a47 a35a74a40a36a33a75a47 a33a50a49a79a49 = a34a36a35a37a33a39a38a41a40 ; 9: a35a74a40a36a33a75a47 a33a39a49 ++; 10: for (each definition a63 in a70a14a40a66a65a80a47 a33a50a49 ) do 11: a63a48a40a66a65 a34a36a35a37a33a39a38a41a40a11a47 a63a73a49 = a34a36a35a37a33a39a38a51a40 ; 12: end for Forward computation of dynamic slices for example in Fig. 1 is shown in Table1 . In the execution step of a20a34a17a81a2 , which is a0a82a30a83a10 a20 a85a84a39a86 a11 a76a87 a11 a76a0a3a88 , a0 is defined, a4 a86 a11 a76a87 a11 a76a0a89a5 are the variables that are used.... ..."

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### Table 2. Scalability over the number of the computers

"... In PAGE 21: ... Practical running time of these two approaches can differ, but it depends on the particular graphs. We have accomplished yet another set of tests (see Table2 ) in order to validate the scalability of the DSP-R algorithm. The table shows how the number of computers in uences the computation time.... ..."

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### Table 2. Scalability over the number of the computers

"... In PAGE 21: ... Practical running time of these two approaches can differ, but it depends on the particular graphs. We have accomplished yet another set of tests (see Table2 ) in order to validate the scalability of the DSP-R algorithm. The table shows how the number of computers influences the computation time.... ..."

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### Table 4. Runtime for different priority and multiple propagator approaches. No priority Static priority Dynamic priority

2004

"... In PAGE 13: ... Table 3 shows the number of propagation steps required to solve each benchmark, relative to the base solver. Table4 shows the relative execu- tion times. Important additional benchmarks are color-1 and color-2 imple- menting graph coloring on large graphs (50 nodes) with large cliques (for each clique a domain-consistent alldifferent propagator is used).... ..."

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