### Table 6: Tableaux expansion rule for universal constraints

1997

"... In PAGE 12: ...constraint: 8x:x : :Ulcer t 8hasLocation::Stomach t 9hasLocation:StomachLining This constraint states that every individual in a valid model is either not an Ulcer or not located in the Stomach or is located in the StomachLining. The expansion rule for universal constraints, given in Table6 on this page, ensures that when S contains the universal constraint 8x:x : C, every variable y in S will be subject... In PAGE 20: .... Perform all deterministic expansions; e.g. apply the u, 9, 8 and 8x rules from Table 5 on page 9 and Table6 on page 10. 2.... ..."

Cited by 10

### Table 3. Universal Constraint Descriptions (UCD)

"... In PAGE 5: ... performs better in the case of UCD descriptions ( Table3 ). The majority of the descriptions can be compressed with a compression factor between approximately 8 and 12.... ..."

### Table 1: How constraints are transformed. The rst column is the constraint currently being transformed. The second column is the set of constraints resulting from the transformation. The third column is the constraint that will be considered if this point in the execution is reached again via backtracking.

1997

"... In PAGE 10: ... The induction steps of constraint transformation handle constraints containing boolean con- nectives and universal and existential quanti ers. These transformations are summarized in Table1 . Conjuncts can be considered separately as both must hold for the constraint to be satis ed.... ..."

### Table 1: How constraints are transformed. The #0Crst column is the constraint currently being

1998

"... In PAGE 10: ... The induction steps of constraint transformation handle constraints containing boolean con- nectives and universal and existential quanti#0Cers. These transformations are summarized in Table1 . Conjuncts can be considered separately as both must hold for the constrainttobe satis#0Ced.... ..."

### Table 1: CCEL Class Member Functions 2. A set of declarations for universally quanti ed CCEL variables. Such variables take as their values components of C++ programs. Each CCEL variable has a type; this type is one of the CCEL classes shown in Figure 1. 3. An assertion that comprises the essence of the constraint. Assertions may use universally quanti ed CCEL variables and may declare and use existentially quanti ed CCEL variables. 4. A scope speci cation that determines the region of applicability of the constraint in relation to the C++ source being checked. By default, constraints are globally applicable, but they may be restricted to a single le, function, or class. 5. A message to be issued when a violation of the constraint is detected. Of these ve parts, only the constraint identi er and the assertion are required. If omitted, the set of CCEL variables is empty, the scope of applicability is global, and constraint violations are indicated by a message in a default format. As an example of a CCEL constraint, consider Meyers apos; admonition [16] that every base class in C++ should declare a virtual destructor:

### Table 1. Comparison between the University of Melbourne current software (UM) and our hybrid algorithm. The best timetable was deemed to be the one with the fewest clashes, with ties resolved by choosing the lowest objective score. The corresponding score for the best timetable can exceed the average. Note that it was not possible to compare directly to the UM program for the mel01s1 data set, and consequently the hybrid method was only compared to the timetable produced by the university for that semester. This timetable had been manipulated afterwards to satisfy constraints not known when the timetable was produced, and was not minimizing the same objective.

2003

"... In PAGE 15: ... (Table 3 in [10]) and Caramia et al. ( Table1 in [7]). The results of the comparison can be seen in Table 3.... In PAGE 18: ... [10], Table 3 in Caramia et al. [7], Table1 in Di Gaspero and Schaerf [13], and Table 9 in White and Xie [19]). Note that for all these data sets, the hybrid algorithm produced a clash-free timetable within the maximum allowed number of sessions, without recourse to the fourth greedy heuristic stage.... ..."

Cited by 3

### Table 4: Residual and extrapolation based estimation of the modelling error. the modelling error accurately as the model order increases. References [1] Actis, R.L., Hierarchic Models for Laminated Plates, Doctoral Dissertation Washington University, St. Louis, Mo 1991. [2] Antman, S.S. and R.S. Marlow, Material Constraints, Lagrange Multipliers, and Compati- bility. Applications to Rod and Shell Theories, Arch. Rat. Mech. Anal. 116 1991, 257-299.

"... In PAGE 35: ... Finally, we compare the residual modelling error estimates according to Theorem 4.1 with the ones obtained by extrapolation in Table4 where we used that 0 lt; 2(G(un) ? G(u)) = E(u) ? E(un) = kuk2 ? kunk2 = kun ? uk2 = kenk2: We see that the residual based estimators are guaranteed upper estimators which follow... ..."

### Table 2: Percentage of connections violating timing constraints after detailed routing completion. 5 Conclusions We have presented a timing-driven router for FPGAs with segments of various lengths. The router is based on the hierarchical strategy and suited for the special properties of FPGA routing architectures. Experimental results show that our router is very e ective in reducing the number of connections violating timing constraints. Acknowledgments The authors would like to thank Steve Brown and Baharam Fallah of University of Toronto for providing us with the benchmark circuits, Nick Haruyama for helpful discussions, and Cherng-Shiuan Wang for implementing the detailed router.

"... In PAGE 10: ...nd loose cases for timing constraints. If the delay bound B(ti) was less than dmin1, then set B(ti) = dmin. Each circuit was routed by the algorithm and the percentage of source-sink pairs violating the delay bounds was computed. The results are shown in the column \Timing-driven quot; of Table2 . For the purpose of comparison, we also routed the circuits by the same routing algorithm, with the cost function for the linear assignment to minimize the wire length and the delay bound distributions/redistributions being turned o , i.... In PAGE 10: ...o minimize the wire length and the delay bound distributions/redistributions being turned o , i.e., C(3) ij in Equation (4) was set according to the cost function illustrated in Figure 8(b); this leads to a non-timing-driven routing approach [19, 25, 30]. The results are given in the column \Non-timing-driven quot; of Table2 . For all the circuits, the timing-driven routing algorithm substantially reduced the percentage of connections violating the delay bounds.... ..."

### Table 1. De nitions of Four Topological Relations in C0 Relation Description Set Equation Model Constraint

"... In PAGE 9: ... In the former case the interpretation is that, in the situation being described, the denotation of the set-term equals the entire universe. Table1 shows how four spatial relations can be characterised with model constraints stated in terms of the classical propositional calculus. Table 1.... ..."