By Gregory. J. Chaitin

Chaitin, the inventor of algorithmic info conception, provides during this publication the most powerful attainable model of Gödel's incompleteness theorem, utilizing a knowledge theoretic strategy according to the dimensions of laptop courses. One 1/2 the booklet is worried with learning the halting likelihood of a common computing device if its application is selected via tossing a coin. the opposite part is anxious with encoding the halting likelihood as an algebraic equation in integers, a so-called exponential diophantine equation.

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LISP REGISTER MACHINES 31 of a finite string of characters. Each 8 bits of the base-two numeral for the contents of a register represent a particular character in the LISP alphabet, and the character string is in the register in reverse order. 1 Thus the rightmost 8 bits of a register are the first character in the S-expression, and replacing the contents of a register X by the integer part of the result of dividing it by 256 corresponds to removing the first character of the string. Similarly, if Y is between 1 and 255, replacing X by 256X + Y corresponds to adding the character Y at the beginning of the string X.

X 00 . This is a somewhat special case; the general case of comparison for equality is a little bit harder. B + i Here we use the fact that the absolute value of the difference between two characters cannot exceed 255. As for JUMP’s and GOBACK’s, the corresponding conditions are easily constructed using the above ideas, after introducing a variable ic to represent the instruction counter. Our program for character string reversal does not use JUMP or GOBACK, but if it did, the equation defining the instruction counter vector would be: ic = C (L1) L1 + C (L2) L2 + C (L3) L3 + C (L4) L4 Here C (L1) denotes the non-negative integer that represents the LISP S-expression (L1), etc.

The digit corresponding to q t in ic is a LISP S-expression for the list of the characters in the label of the instruction that is executed at time t. Note that if labels are very long, this may require the base q to be chosen a little larger, to ensure that the list of characters in a label always fits into a single base-q digit. It is amusing to look at the size of the variables in a solution of these exponential diophantine equations. Rough estimates of the size of solutions simultaneously serve to fix in the mind how the equations work, and also to show just how very impractical they are.