### Table 4: Data structures and notation for the SLCN arc consistency algorithm.

1995

"... In PAGE 33: ... Note that SLCN arc consistency reduces to CN arc consistency when the number of sentences is one. The SLCN arc consistency algorithm builds and maintains several data structures, described in Table4 . Figure 24 shows the code for initializing the data structures, and Figure 25 contains the algorithm for eliminating inconsistent role values from the domains.... ..."

Cited by 19

### Table 2. Number of constraint checks using Backtracking ltered with Arc- Consistency.

"... In PAGE 9: ... We considered all constraints as global constraints, that is, all con- straints had maximum arity. Thus, Table2 sets three of the parameters and varies the other one in order to evaluate the algorithm performance when this parameter increases. We evaluated 100 test cases for each type of problem and each value of the variable parameter.... In PAGE 9: ...The number of constraint checks using BT ltered by arc-consistency (as a preprocessing) (BT-AC) and BT-AC using our model (HASCOH+BT-AC) is presented in Table2... ..."

### Table 2. Number of constraint checks using Backtracking filtered with Arc- Consistency.

"... In PAGE 9: ... We considered all constraints as global constraints, that is, all con- straints had maximum arity. Thus, Table2 sets three of the parameters and varies the other one in order to evaluate the algorithm performance when this parameter increases. We evaluated 100 test cases for each type of problem and each value of the variable parameter.... In PAGE 9: ...The number of constraint checks using BT filtered by arc-consistency (as a preprocessing) (BT-AC) and BT-AC using our model (HASCOH+BT-AC) is presented in Table2... ..."

### Table 1: A Descriptions Data Base; intensional descriptions and extensional assertions

1996

"... In PAGE 4: ... The overall structure of a typical DL knowledge base is given in Figure 2. Table1 presents a particu- lar descriptions KB. The Tbox includes 8 primitive concepts, 3 primitive roles, and 2 de ned concepts.... In PAGE 6: ... For example, the query: Is a plant, necessarily located at a place? is rephrased as the subsumption query:?plant v all(located at; place) The answer is true, since the range of the role located at is place, and a plant is located at somewhere. Similarly, the descriptions data base of Table1 implies the subsumption risky place v place , since the domain of the role inv(buried at) is place. It also implies pr 2 mechanical product, and dp 2 risky place.... In PAGE 7: ... The integration of rules and descriptions is expected to extend their separate inferential capabilities. For example, consider the following integration: Descriptions data base: Table1 without the assertion (a1). Rules: Table 2.... ..."

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### Table 1. Establishing Arc Consistency on Domino instances

2006

"... In PAGE 14: ... Reversely, if activity A becomes invalid then all activities C such that (C ) A) 2 Dep are made invalid (see rule /1/ below). Keeping the exclusions and dependencies explicitly has the advantage of stronger ltering ( Table1 ). In particular, if exclusion fA; Bg is to be added to Ex and there is a dependency (A ) B) 2 Dep then we can make activ- ity A invalid (and the exclusion is resolved so it does not need to be kept in Ex).... In PAGE 14: ... Keeping explicit dependencies and exclusions will also help us later to deduce a better estimate of the number of valid/invalid activities that is used in cost-based ltering (see rule /3/ below). Table1 . Reasoning on exclusions and dependencies.... In PAGE 33: ... We propagate the Lex constraints either using the specialised algorithm GACLex given in [8] or the Slide encoding described in Section 7. Table1 shows the results on some large instances described as v; b; r; k; . As both propagators maintain GAC, we report the runtime results.... In PAGE 34: ...97 71.81 Table1 . BIBD generation.... In PAGE 38: ... The results that we obtain when running AC3, AC2001, AC3.2 and the new algorithm AC3rm on some instances of this problem are depicted in Table1 . We do not consider AC3r as it is always outperformed by AC2001 (even if, on these very special instances, they have the same behaviour).... In PAGE 61: ... In all the experiments, the search variables were the 0/1 decision variables. In Table1 , we report the cost of the best solution found within the CPU time limit, and a entry of signifies that no solution was found in the time limit. What we see is that Bound-SAC can indeed be beneficial, allowing us to frequently find better solutions than just using MAC on its own.... In PAGE 62: ...Lecoutre and Prosser Maintaining Instance MAC B-SACdn B-SACst B-SAC la01 666 666 666 666 la02 655 655 655 655 la03 653 597 603 603 la04 628 598 590 590 la05 593 665 665 665 la06 1245 1146 1233 1237 la07 1214 897 1336 1359 la08 1161 1084 1400 1393 la09 1498 1049 1527 1520 la10 1658 972 1192 1259 la11 1453 1787 la12 1467 1504 la13 2899 2310 la14 1970 1784 la15 2368 2200 Table1 . Cost of best solution found for Lawrence scheduling instances, given 10 minutes CPU.... In PAGE 77: ...2 Results The algorithms SAC, LSAC, PSAC, and PLSAC were applied to a variety of known prob- lems. Table1 shows the results obtained on some representative instances of variety of known problems: (1) average of 100 satisfiable instances of balanced Quasigroup with Holes problems bqwh15 106 and bqwh18 141 (2) three attacking prime queen in- stances qa-6, qa-7 and qa-8, (3) two queen-knights instances K25 Q8 and K25 Q8, (4) RLFAP instances scene5 and scene11, (5) modified RLFAP instance scene11 f6, (6) GRAPH instances graph10 and graph14, (7) two job-shop instances enddr1-10-by-5-10 and enddr2-10-by-5-2, which are called js-1 and js-2 in [9] respectively, and (8) two sets of composed random instances composed-25-10-20 and composed-75-1-80. In Table 1 #rem denotes the number of removed values.... In PAGE 77: ... Table 1 shows the results obtained on some representative instances of variety of known problems: (1) average of 100 satisfiable instances of balanced Quasigroup with Holes problems bqwh15 106 and bqwh18 141 (2) three attacking prime queen in- stances qa-6, qa-7 and qa-8, (3) two queen-knights instances K25 Q8 and K25 Q8, (4) RLFAP instances scene5 and scene11, (5) modified RLFAP instance scene11 f6, (6) GRAPH instances graph10 and graph14, (7) two job-shop instances enddr1-10-by-5-10 and enddr2-10-by-5-2, which are called js-1 and js-2 in [9] respectively, and (8) two sets of composed random instances composed-25-10-20 and composed-75-1-80. In Table1 #rem denotes the number of removed values. The intention is not to test if the preprocessing by SAC has any effect in the global cost of solving the problem, but to see if the same results can be achieved by doing considerably less computation by using probabilistic support inference.... In PAGE 78: ... The number of ineffective revisions in the worst- case increases to O(n2d2(ed nd)) for a SAC algorithm such as SAC-1, when its under- lying arc consistency algorithm is a coarse-grained algorithm, since in the worst-case SAC-1 can call its underlying arc consistency algorithm n2d2 times. One can observe in Table1 that probabilistic versions of SAC algorithm perform revisions far fewer than their original versions. This clearly shows that PRC is good in saving many ineffective revisions.... In PAGE 79: ...73 Table1 . Comparison between SAC, LSAC, PSAC and PLSAC problem SAC LSAC PSAC PLSAC #chks 274,675 186,172 32,394 25,671 bqwh15 106 #time 0.... In PAGE 110: ... Fur- thermore, the ILL-consistency is obtained at step 2 for all graphs having more than 800 vertices. jV j jL j T k jLlabelj Tlabel 200 199,64 0 3,40 191,92 0,01 400 400,00 0 2,88 399,87 0,07 600 600,00 0 2,14 600,00 0,19 800 800,00 0,01 2,01 800,00 0,36 Table1 . Results for the little graphs having a density of 1%.... ..."

### Table 1. The five problems related to genaralized arc consistency

2004

"... In PAGE 2: ... U denotes the set of all constraint types. Table1 contains the five problems. The first problem we consider is GACSUPPORT.... ..."

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### Table 1. The five problems related to genaralized arc consistency

2004

"... In PAGE 2: ... CD denotes the set of all constraint types. Table1 contains the five problems. The first problem we consider is GACSUPPORT.... ..."

Cited by 7

### Table 1. The five problems related to genaralized arc consistency

2004

"... In PAGE 2: ... a14 denotes the set of all constraint types. Table1 contains the five problems. The first problem we consider is GACSUPPORT.... ..."

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### Table 1: 100th and 90th percentiles in backtracks re- quired to complete a quasigroup of order 10. ACg ? ACg corresponds to generating and solving using gen- eral arc consistency, BC ? ACg corresponds to gen- erating using Gomes and Selman apos;s backward checking method and solving using general arc consistency, and BC ? ACb corresponds to generating preassignments with the backward checking method and solving using arc consistency on binary constraints. A means that the instance was abandoned after 10000 leaf nodes had been visited. 100 problems were solved at each data point.

1998

"... In PAGE 3: ... Experiments with Gomes and Selman apos;s random pre- assignment method show that enforcing general arc consistency on n?ary all-di erent constraints instead of arc consistency on binary constraints reduces the cost of solving the problem drastically. This is demon- strated in Table1 , giving the percentiles in backtracks required to complete a quasigroup of order 10 with p% of its entries preassigned. Figures 3 and 4 show how the cost of quasigroup completion scales when using general arc consistency to perform both preassignments and completion.... ..."

Cited by 20

### Table 4. Runtime and propagation steps for extensional propagation

1993

"... In PAGE 10: ... Table4 compares runtime (left in a table cell) and number of propagator executions (right in a table cell) for different extensional propagators. base is the GAC-2001 propagator, cheap is a GAC-Schema propagator where the prop- agator searches for new supports, and expensive is a GAC-Schema propagator where advisors search for new supports.... In PAGE 11: ...n2-ary extensional constraints on 0/1 variables. Table4 clarifies that using an incremental approach to propagate extensional constraints reduces the number of propagator executions. Using advisors to re- move supports also reduces runtime.... ..."

Cited by 1