### Table 1. The detection of symmetry problem with 3 inputs.

2005

"... In PAGE 4: ... The problem is to detect the binary activity levels of a one-dimensional array of in put neurons are symmetrical about the centre point. For example, the input-output mapping in the case of 3-inputs is shown in Table1 . Output is 1 means that the input is symmetric, and 0 asymmetric.... ..."

### Table 2. The detection of symmetry problem with 3 inputs.

2005

### Table 1: Simulation results of the symmetry problem in di erent dimensions

in References

"... In PAGE 3: ... of Train. Vec. # of Test Vec. Success Rate 2?D 3 1 1=1 3?D 2 6 2=6 3?D 3 5 5=5 4?D 10 6 4=6 4?D 12 4 4=4 5?D 16 16 7=16 5?D 25 7 7=7 6?D 20 44 30=44 6?D 32 32 32=32 Table 2: Generalization of the symmetry problem in di erent dimensions Table1 . In 5?D the model has discovered two di erent polynomials depending on di erent random initial coe cients.... ..."

### Table 2. The size of the solution symmetry group and number of solutions for the n-queens problem with constraints to eliminate the constraint symmetry.

2005

"... In PAGE 19: ...resulting CSP by finding all the solutions and finding the automorphisms of the graph that has an n-ary hyperedge for every solution. Table2 shows the size of the solution symmetry group. Table 2.... ..."

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### Table 2: Solution of the gripper problems with satisfiability algorithms. Symmetry-breaking for all pairs of operators.

### Table 3: Generalization ability of neural networks obtained for the mirror symmetry problem. Each result is averaged over 30 trials.

### Table 3. Search e ort and running time to nd all optimal solutions to the armies of queens problem, with ECLiPSe. No symmetry breaking With SBDS

2002

"... In PAGE 4: ... It would be possible to achieve some of the speed-up without SBDS, by adding constraints to the model, for instance that the top half of the board contains more white queens than the bottom, but simple constraints of this kind cannot remove all the symmetry. Table3 compares nding all solutions with and without symme- try breaking using SBDS. It proved impracticable to nd all solutions for the 8 8 board without any symmetry breaking: there are evidently hundreds of... ..."

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### Table 5.5: Generalization of symmetry problem in di erent dimensions. Dim. # of Train. Vec. # of Test Vec. Success Rate

### Table 2: Number of nonnull weights contained in the neural networks generated for the mirror symmetry problem. Each result is averaged over 30 trials.

"... In PAGE 43: ... Every test set is formed by 4000 samples whereas the number s of patterns in the training set assumes the values s = 100; 200; 400; 600. Table2 and 3 show respectively the average number of nonnull weights and the percentage of the test set correctly classified. The average CPU time of each training algorithm is reported in Tab.... ..."

### Table 2. Behaviour of the algorithm of symmetries detection

2002

"... In PAGE 11: ... 7.2 The Behaviour of the Symmetry Detection Table2 shows the behaviour of symmetry detection on three different problems when using SEM (with LNH) as a finite model generator. We can see for the Abelian groups that 82% of symmetry calls (i.... ..."

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