Computability and Logic

62 ABACUS COMPUTABILITY
5.7 Show that for any k there is a Turing machine that, when started on the leftmost
1 on a tape containing k blocks of 1s separated by single blanks, halts on the
leftmost 1 on a tape that is exactly the same as the starting tape, except that
everything has been moved one square to the right, without the machine in the
course of its operations ever having moved left of the square on which it was
started.
5.8 Review the operations of a Turing machine simulating some give abacus machine
according to the method of this chapter. What is the furthest to the left of
the square on which it is started that such a machine can ever go in the course
of its operations?
5.9 Show that any abacus-computable function is computable by a Turing machine
that never moves left of the square on which it is started.
5.10 Describe a reasonable way of coding abacus machines by natural numbers.
5.11 Given a reasonable way of coding abacus machines by natural numbers, let
d(x) = 1 if the one-place function computed by abacus number x is defined and
has value 0 for argument x, and d(x) = 0 otherwise. Show that this function is
not abacus computable.