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Table 1.
Experimental results.
Benchmark
Classical method
New method
Ratio
Name
n
Time
Box
Time
Box
Box
Brown
4
1
1 018
0
76
13
Brown
5
2
12 322
0
94
131
Brown
6
43
183 427
0
67
2 737
Brown
7
1 012
3 519 097
0
88
39 989
Wood
4
1
3 688
0
325
11
Wood
8
256
452 590
3
4 747
95
Wood
12
?
?
42
45 751
?
Transistor
9
1 061
236 833
130
23 887
9
Product
3
1
52
0
31
1
Product
5
459
5 698 714
0
37
154 019
Product
7
?
?
0
40
?
Extended Product
3
1
3 217
0
16
201
Extended Product
4
1 346
19 315 438
0
19
101 660
Extended Product
5
?
?
0
19
?
Estimation
11
6
11 581
3
5 467
2
4.3
On Strategies
A critical component of the algorithm is the strategy for combining terms. The
first approach is to tune the method according to the problem structure, typically
exponential sums. This allows one to develop so-called global constraints, namely
complex constraints associated with specific and ecient algorithms.
The second approach is to tackle the general problem. In our implemen-
tation, only consecutive constraints are combined in order to introduce a rea-
sonable number of new constraints and patterns are determined wrt. specific
tree-representations of constraints. These limitations have to be relaxed with
the aim of controling the combinatorial explosion.
5Con lu on
In this paper, we have introduced a general framework for improving consis-
tency techniques using redundant constraints. Redundancies are obtained by
combination of terms and simplification according to the precision of interval
computations. In particular, the well-known problem of processing exponential
sums arising in mathematical modeling of dynamic systems has been eciently
handled. In this case the simplification process follows from the property of the
exponential function
e
a
e
b
=
e
a
+
b
.
Thefirstissueistodesignanecientcombination strategy for the general
problem. For this purpose, the study of term rewriting engines will be useful
[4]. The second issue is to develop new global constraints, e.g., for geometric
problems modeled by trigonometric functions.
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