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Recursion Theory and Undecidability

In the second half of the class, we will explore limits on computation. These are questions of the form “What types of things can be computed at all?” “What types of things can be computed efficiently?” “How fast can problem XYZ be solved?” We already saw some positive examples in class: problems we could solve, and solve efficiently. For instance, we saw that we can sort n numbers in time O(n log n). Or compute the edit distance of two strings in time O(nm). But we might wonder if that’s the best possible. Maybe, there are still faster algorithms out there. Could we prove that these problems cannot be solved faster? In order to answer these questions, we first need to be precise about what exactly we mean by “a problem” and by “to compute”. In the case of an algorithm, we can go with “we know it when we see it”. We can look at an algorithm, and agree that it only uses “reasonable” types of statements. When we try to prove lower bounds, or prove that a problem cannot be solved at all, we need to be more precise: after all, an algorithm could contain the statement “Miraculously and instantaneously guess the solution to the problem.” We would probably all agree that that’s cheating. But how about something like “Instantaneously find all the prime factors of the number n”? Would that be legal? This writeup focuses only on the question of what problems can be solved at all. Running time lower bounds and the theory of NP-completeness are covered in the textbook, and we will cover them in class next. The material covered here is necessarily only a small excerpt of what’s known in the field. For more information on these topics, check out Part II of the book by Sipser [6], the book by Hopcroft, Motwani, and Ullman [3], or the book by Kfoury, Moll, and Arbib [4].