Modeling of Artificial Intelligence

3-21

Authors: Eric Grigoryan

The n-Queens problem is considered. A description of the regularities in a sequential list of all solutions, both complete and short, is given. Determined that: 1. The fraction of total solutions in the general list of all solutions decreases, with increasing value of n. 2. Complete solutions are distributed in a sequential list of all solutions in such a way that the most likely solutions are complete solutions located in the list close to each other. 3. There is a symmetry in the order of the location of the complete solutions in the general list of all solutions. If the solution is complete in the i-th position from the beginning of the list, then the symmetric solution from the end of the list, located in the position n-i + 1, is also complete (rule of symmetry of solutions). 4. Any pair of solutions, both short and full, arranged symmetrically in the list of all solutions, are complementary – the Queen position indices sum of the corresponding rows is a constant and is equal to n + 1 (the rule of complementarity of solutions). This suggests that only the first half of the list of all complete solutions is "unique", any complete solution from the second half of the list can be obtained on the basis of the complementarity rule. The consequence of this rule is the fact that for any value of n, the number of complete solutions will always be an even number. For an arbitrary matrix of a solution of size n x n, it is established that: 5.The probability of completion to a full solution an arbitrary composition of k queens, gradually decreases with increasing value of k to a certain minimum, and then increases, with a further increase in the value of k. 6. There is some minimum value of the size of the composition k0, such that any composition whose size is less than or equal to k0 can always be completed to a complete solution. As the value of n increases, the value of k0 also increases. 7. The activity of row cells in solution matrix is symmetric with respect to the horizontal axis passing through the middle of this matrix. This means that the cells activity in the i-th row always coincides with the cells activity in the row n- i + 1. By activity is meant the frequency with which the cell index occurs in the corresponding row of the list of complete solutions. Similarly, the activity of the cells of the columns of the solution matrix is symmetrical about the vertical axis dividing the matrix into two equal parts 8. For any n, in the sequential search for all solutions, the first complete solution appears only after some sequence of short solutions. The size of the initial sequence of short solutions increases with increasing n. The length of the list of short solutions until the first complete solution for even values of n appears is much larger than for the nearest odd values. 9. The row in the solution matrix, on which difficulties begin to move forward, and the first short solution is formed, divides the matrix according to the rule of the golden section. For small values of n, such a conclusion is approximate, but with an increase in the value of n, the accuracy of such an output asymptotically increases to the level of the standard rule.

29-37

Authors: Sardarkhоdja K. Kurganov

In this paper, the limiting behavior of trajectories ω(x^0 )={x^0,Vx^0,V^2 x^0,…} of the evolution operator for STR loci (D8S1179, D21S11, D7S820, CSF1PO, D3S1358, TH01, D13S317, D16S539, D2S1338, D19S433, vWA, TPOX, D5S818, D18S51 and FGA) alleles without calculating the occurrence of mutations for the population.