### Table 1: Test for necessary and sufficient conditions for various classes of codes for PCRC.

708

"... In PAGE 24: ...Discussion and Simulation Results The results of our necessary and sufficient conditions (16), (46) and (47) as well as the sufficient condition in [18], evaluated for various classes of codes for PCRC are shown in Table 1. As can be seen from the last column of Table1 , the sufficient condition in [18] identifies only COD2 (Alamouti) and CUW4 as SSDs for PCRC. However, our conditions (16, (46) and (47) identify CIOD4, RR8, and CODs from RODs, in addition to COD2 and CUW4, as SSDs for PCRC (4th column of Table 1).... In PAGE 24: ... As can be seen from the last column of Table 1, the sufficient condition in [18] identifies only COD2 (Alamouti) and CUW4 as SSDs for PCRC. However, our conditions (16, (46) and (47) identify CIOD4, RR8, and CODs from RODs, in addition to COD2 and CUW4, as SSDs for PCRC (4th column of Table1 ). It is noted that, CIOD4 being a construction by using G = COD2 in (50) and coordinate interleaving, it is SSD for PCRC from Lemma 2 and Theorem 4.... ..."

### TABLE I VERIFYING SATISFIABILITY OF SUFFICIENT CONDITION OF THEOREM 1

in Finite Bisimulation of Reactive Untimed Infinite State Systems Modeled as Automata with Variables

### Table 1: Sufficient conditions for the existence of various Nash equilibria.

### Table III. Rules for Merging Conditions Comparing Multiple Variables into a Single Sufficient Condition

### Table 9. A set of sufficient conditions on ai for the differential path given in Table 8.

2005

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### Table 8. Sufficient conditions on ai for 21-95 rounds of the differential path in Table 5.

"... In PAGE 8: .... We compute M1 from eq.(1). 3. If the last row of Table8 is also satisfied, then we have done. Finding M0: 1.... In PAGE 9: ... Next suppose that the above M0 is given. Then we can find M1 if the last row of Table8 is satisfied. Thereore, the success probability of finding M1 is given by Pr[a92,32 = 1] = 1/2.... ..."

### Table 1: The necessary (but not always sufficient) conditions for inclusion in a region based on the distances along seven plane normal vectors.

"... In PAGE 3: ... In that way, the computed dis- tance from the plane varies from zero to one along the valid length of the edge. Table1 summarizes the requisite condi- Figure 4: Illustration of the seven voxelization regions around a triangle. Each affected voxel is either closer to the triangle face, an edge, or a corner.... In PAGE 6: ...ng, but p is typically small (i.e., four or eight). After the seven plane distances are calculated, the val- ues flow down the pipeline where tests are done in the next pipeline stage to determine in which region the current voxel resides. Only seven comparators are needed to decide the out- come of the truth table (see Table1 ), due to the mutual exclu- sion of some cases. For instance, in Figure 5 if you are on the negative (lower) side of plane b, then it is not necessary to test the distances from plane f or g depending on the value of the distance from plane e.... ..."

### Table 2: Necessary and sufficient validity conditions and (local) change in score for each operator in E-space

1996

"... In PAGE 18: ... PDAG Space DAG Space Current State Result of Operator (2) Apply operator to the consistent extension (1) Create a consistent extension of the current state (3) Show that the result is a consistent extension of the PDAG that results from the operator Figure 9: Approach taken to prove each of the operator results. In Table2 we summarize the results of the six theorems and the corresponding six corollaries. In particular we provide, for each of the E-space operators, both necessary and sufficient conditions for that operator to be valid, and the increase in score that results from applying that operator.... In PAGE 28: ...n Section A.4 through Section A.9, we use the results from Section A.2 and Section A.3 to prove, for each operator, that the (necessary and sufficient) validity conditions and increases in score given in Table2 are correct. Finally, in Section A.... ..."

Cited by 82

### Table 1: Necessary and sufficient validity conditions and (local) change in score for each operator Operator

2002

"... In PAGE 21: ... We use NAY;X to denote the set of nodes that are neighbors of node Y and are adjacent to node X in the current state. The proofs of these results, which are summarized in Table1 , are given in Appendix B. Theorem 15 Let Pc be any completed PDAG, and let Pc0 denote the result of applying an Insert(X;Y;T) operator to Pc.... In PAGE 22: ... There are a number of tricks we can apply to generate more efficiently the candidate operators corresponding to a pair of nodes. Consider the first validity condition for the Insert operator given in Table1 : namely, that the set NAY;X [ T must be a clique. If this test fails for some set T,thenit will also fail for any T0 that contains T.... In PAGE 23: ... In all of the experiments we have performed, however, including those presented in the next section, we have yet to encounter a domain for which GES encounters a state that has too many neighbors. As is evident from the simplicity of the validity conditions from Table1 , there are a number of ways to efficiently update (i.... In PAGE 23: ... Suppose that all the operators have been generated and scored at a given step of (the first phase of) GES, and we want to know whether these operators remain valid and have the same score after applying some operator. From Table1 , we see that if the neighbors of Y have not changed, the first validity condition must still hold for all previously-valid operators; because we are adding edges in this phase, any clique must remain a clique. Furthermore, if the parents of node Y have not changed, we need only check the second validity condition (assuming the first holds) if the score of the operator is higher than the best score seen so far; otherwise, we know that regardless of whether the operator is valid or not, it will not be chosen in the next step.... In PAGE 37: ... Appendix B: Operator Proofs In this appendix, we provide proofs for the main results in Section 5. We show that the conditions given in Table1 are necessary and sufficient for an Insert and Delete operator to be valid for the... In PAGE 39: ... But from Corollary 39, A and C are not adjacent, and thus A ! B C is a v-structure, yielding a contradiction. The conditions from Table1 include checking that some set of neighbors of a node in a com- pleted PDAG are a clique. It follows immediately from Lemma 34 that if any set of neighbors is a clique, then that set of neighbors is a clique of undirected edges.... In PAGE 39: ... B.2 The Insert Operator In this section, we show that the conditions in Table1 are necessary and sufficient for determining whether an Insert operator is valid during the first phase of GES. In particular, we show in Theorem 15 that the conditions hold if and only if we can extract a consistent extension G of the completed PDAG Pc to which adding a single directed edge results in a consistent extension G0 of the com- pleted PDAG Pc0 that results from applying the operator.... In PAGE 41: ... B.3 The Delete Operator In this section, we show that the conditions in Table1 are necessary and sufficient for determin- ing whether a Delete operator is valid during the second phase of GES. In particular, we show in Theorem 17 that the conditions hold if and only if we can extract a consistent extension G of the completed PDAG Pc to which deleting a single directed edge results in a consistent extension G0 of the completed PDAG Pc0 that results from applying the operator.... ..."

Cited by 66