### TABLE 4 PROPOSITION TESTING

2007

### Table 4: Distribution of Propositional

### Table 5.1: Probabilistic Domain Description Language (propositions proposition-1 proposition-2 ... ;list of propositions (proposition-i proposition-j ... ) ;mutually exclusive propositions ...)

### Table 2: Comparison of the average objective values of Maxregret and its improved version Maxregret 2 To exploit the structure of the coe cients even more we propose two oth- er heuristics. The heuristic Simple 3 is based on the Algorithm Simple and works as follows: Given three sequences A, B and C then Simple 3 executes Simple (A; B; C), Simple (B; C; A) and Simple (C; A; B) | the three possibil- ities of applying Proposition 2.1 twice | and reports the best solution value of these three di erent constructed thee-dimensional assignments. To illustrate Simple 3 consider the following example: Let three sequences, say A = (1; 2; 3; 4), B = (1; 2; 4; 5) and C = (2; 3; 3; 6), be given. Then all three di erent solutions generated by Simple 3 are given by: 0

1996

"... In PAGE 14: ...1 are set to in nity and then Maxregret is applied to the modi ed cost array. Table2 shows the di erences in the optimal objective values of applying Maxregret to the cost array with and without irrelevant cost coe cients, i.... ..."

Cited by 15

### Table 2: Extended Typing Rules time and has only patterns of the forms x; (P j Q) and ]z . On the other hand, by proposition 2.2, we can state the relation between two derivations ? . M: C and Decon(?) . M: C in the following way: Lemma 2.1 For any type-checking derivation D that ends with ? . M: C there is type-checking derivation of height at most that of D, which ends with Decon(?) . M: C. Proof. By induction on the height of the derivation ? . M : C, then by cases according to the last rule used in the derivation, using the proposition 2.2 below. End of proof. Proposition 2.2 (Commutation) Withing type derivations, the rules ( left); (layered) and (wildcard) com- mute with all the other rules. Proof. The proof is by case-analysis and is quite straightforward, we only show three cases to illustrate how it works. ( left) commutes with ( right)

1993

Cited by 47

### Table 7. Now the relations R and S are equal, and they have the following six congruence classes f;, fag, : : :, fc; dgg, ffa; b; cgg, ffa; b; dgg, ffa; c; dgg, ffb; c; dgg, and fUg. Similarly, the relations R and S are identical and they have six congruence classes f;g, ffagg, ffbgg, ffcgg, ffdgg, and ffa; bg, : : :, Ug. It can be easily seen that also R and S are the same. They have 11 equivalence classes. By Proposition 6.2.2,

1999

### Table 2: Operational semantics (symmetric versions of (Sum), (Par) and (Com) omitted)

1999

"... In PAGE 16: ...?!, i.e.: if P Q and P ??! P 0 then there exists Q0 such that Q ??! Q0 and P 0 Q0 (the proof goes by inspection of the rules; see also [16]). The key to soundness is the following proposition, that relates equivalence on environ- ments ( ) to the (conventional) operational semantics of Table2 (its proof can be found in Appendix B). Proposition 4.... ..."

Cited by 54

### Table 2: Results for Cluster Deletion: (1) Enumerating all size-s graphs containing a P3; (2) Expansion scheme utilizing Proposition 2 size time isom concat graphs maxbn avgbn bvmax bvmed maxlen bvset (1) 4 lt; 1 sec 12% 12% 5 1.77 1.65 4 2 5 4

2004

"... In PAGE 23: ...Since this problem is a special case of Cluster Editing, where only edge deletions are allowed, all problem-speci c rules devised for Cluster Editing can also be used for this problem without any modi cation. However, the rst implementation with all rules for Cluster Editing showed that (as shown in the rst half of Table2 ) the resulting worst-case branching number 1:62 is determined by the (1; 2)-branching of Proposition 1 which is used in Rule 3. In order to achieve a better branching rule for Cluster Deletion, we improved the branching in Proposition 1 as follows: Proposition 2.... In PAGE 24: ... Results. See Table2 . The measured values are de ned in the beginning of Sect.... In PAGE 24: ... 4. Incorporating Proposition 2 into Rule 3, we obtained the results shown in the second half of Table2 . Here, the (1; 3)-branching in Proposi- tion 2, which corresponds to a branching number of 1:47, is not the worst case any more.... ..."

Cited by 9

### Table 2: Results for Cluster Deletion: (1) Enumerating all size-s graphs containing a P3; (2) Expansion scheme utilizing Proposition 2 size time isom concat graphs maxbn avgbn bvmax bvmed maxlen bvset (1) 4 lt; 1 sec 12% 12% 5 1.77 1.65 4 2 5 4

2004

"... In PAGE 23: ... Since this problem is a special case of Cluster Editing, where only edge deletions are allowed, all problem-speci c rules devised for Cluster Editing can also be used for this problem without any modi cation. However, the rst implementation with all rules for Cluster Editing showed that (as shown in the rst half of Table2 ) the resulting worst-case branching number 1:62 is determined by the (1; 2)-branching of Proposition 1 which is used in Rule 3. In order to achieve a better branching rule for Cluster Deletion, we improved the branching in Proposition 1 as follows: Proposition 2.... In PAGE 24: ... Results. See Table2 . The measured values are de ned in the beginning of Sect.... In PAGE 24: ... 4. Incorporating Proposition 2 into Rule 3, we obtained the results shown in the second half of Table2 . Here, the (1; 3)-branching in Proposi- tion 2, which corresponds to a branching number of 1:47, is not the worst case any more.... ..."

Cited by 9