Sketch an algorithm to recognize the language, Theory of Computation

First model: Computer has a ?xed number of bits of storage. You will model this by limiting your program to a single ?xed-precision unsigned integer variable, e.g., a single one-byte variable (which, of course, can store only values in the range [0, . . . , 255]), etc. Limityourself, further, to calling input() in just one place in your program. One way of doing this is to call input() in the argument of a multiway branch (e.g., switch) statement. (That statement, of course, will need to be in the scope of some sort of loop, otherwise you would never read more than the ?rst symbol of the input.) The reason for this restriction will become clear in the last part of this question.

(a) Sketch an algorithm to recognize the language: {(ab)i | i ≥ 0} (that is, the set of strings of ‘a's and ‘b's consisting of zero or more repetitions of ab: {ε, ab, abab, ababab, . . .}, where ‘ε' is the empty string, containing no symbols whatsoever).

(b) How many bits do you need for this (how much precision do you need)? Can you do it with a single bit integer?

(c) Sketch an algorithm to recognize the language: {(abbba)i | i ≥ 0} (i.e., {ε, abbba, abbbaabbba, . . .}).

(d) How many bits do you need for this?

(e) Suppose we relax the limitation to calling input() at a single place in the code. Sketch an algorithm for recognizing the language of part (a) using (apparently) no data storage.

[Hint: All you need to do is to verify that the ‘a's and ‘b's occur in the right sequence. If you forget all the restrictions, etc., and just use the simplest program you can think of, you are likely to come up with one that meets these criteria.]

Argue that any algorithm for recognizing this language must store at least one bit of information. Where does your program store it?

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