## General menu

Each state has its own ancestor matrix; products of several ancestor matrices describe the ancestors of sequences of cells. Consequently, the ancestor menu makes provision for a matrix accumulator which holds the matrix counting the ancestors of all the cells which have been entered, running from left to right. Thus, for a binary automaton,

aplaces the ancestor matrix for state0in the accumulator;bdoes the same for state1. If there are more states, additional letters are used to generate their matrices.Corresponding capital letters multiply the accumulator by the appropriate matrix. Thus

aAABBAwould leave a matrix in the accumulator telling how many ancestors the sequence000110had, classified according to the states in the margins. Summingallthe elements of the matrix gives the total number of ancestors, thetracetells the number of ancestors in a ring of the given circumference, while the element tells how many are embedded in a field of zeroes.A very limited amount of matrix arithmetic is provided; for example

=saves the accumulator andxrecovers it. The operatorXmultiplies the accumulator by the saved matrix;Qsquares the accumulator. The latter is useful for obtaining high powers quickly.The second moment of the ancestor distribution yields the distribution's variance, so the option

tis included to calculate the sum of the tensor squares of the individual de Bruijn matrices. Since its high powers and their traces are of interest (the power corresponds to the length of the chain whose ancestor is being sought), the operatorTsquares the matrix which it encounters in the tensor square position (the first matrix of the sequence must be introduced by the operatort).Ultimately the largest eigenvalue dominates the powers of all these matrices, which are positive matrices to which the analysis of Frobenius and Perron applies. The options

l(de Bruijn matrix in the accumulator) andL(tensor square) estimate these eigenvalues through successive squaring, reporting the result in a little panel of their own.

a- display ancestor matrix for state 0b- display ancestor matrix for state 1h- generate histogramj- classify ancestors by generationl- estimate largest eigenvalue of matrixr- list reversible ruless- summarize the statistics gathered bySt- generate and display second moment matrixu- calculate variance of eigenvector componentsv- calculate variance of column sumsw- print extremals fromSx- recover saved matrixA- multiply by matrixaB- multiply by matrixbQ- square the matrixS- Survey all rulesT- Multiply by tensor squareX- multiply by saved matrix=- save matrix in accumulator/- clear the screen.- refresh panel?- display help panel- CARRIAGE RETURN - return to main menu

Harold V. McIntosh

E-mail:mcintosh@servidor.unam.mx