1. A leaping Turing machine is similar to an ordinary Turing machine except that it can skip over a finite number of cells in a single transition. It can skip both while moving its tape head to the left and to the right. The number of cells to be skipped is shown in parentheses in a transition. For instance, a transition a → b, R for an ordinary Turing machine specifies that when the symbol under the tape head is a, the machine should replace it by b and move the tape head one cell to the right. A transition a → R (4)'' for a leaping Turing machine specifies that when the symbol under the tape head is a, the machine should replace it by b and move the tape head 4 cells to the right. When a leaping move would take the Turing machine to the left of the leftmost tape cell, the machine stays at the first tape cell just like an attempt to move left on the leftmost tape position in an ordinary Turing machine.
Part a. Show the domain and the range of the transition function for leaping Turing machines.
Part b. Show that this type of Turing machine recognizes the class of Turing-recognizable languages, i.e., this variant Turing machine model is equivalent to the ordinary Turing machine model. Remember that equivalency means implications in both directions.
2. (Exercise 4.2) Consider the problem of testing whether a DFA and a regular expression are equivalent. Express this problem as a language and show that it is decidable.
3. (Exercise 4.3) Let ALLDFA = {<A> | A is a
DFA that recognizes Σ*}. Show that ALLDFA is decidable.