15–212: Principles of Programming Basic Computability Theory

Computers can be programmed to perform an impressive variety of tasks, ranging from numerical computations to natural language processing. The computer is surely one of the most versatile tools ever devised, playing a critical role in solving all manner of “real world” problems. Some would argue that computers can solve any problem that a human can solve; some would argue the opposite; and some regard the question as irrelevant. Whatever view one adopts, it is still interesting to consider whether there are any limits to what can be solved by a computer. Any given computer has only a finite memory capacity, so certainly there are problems that are too large for it to solve. We abstract away from these limitations, and consider the question of what can be solved by an ideal computing device, one which is not limited in its memory capacity (but is still required to produce an answer in a finite amount of time). Computability theory is concerned with exploring the limitations of such idealized computing devices. Complexity theory (which we shall not study here) is concerned with calibrating the resources (both time and space) required to solve a problem. Our treatment of computability theory is based on problems pertaining to ML programs. We begin by considering questions about ML functions such as “does the function f yield a value when applied to an input x?” or “are functions f and g equal for all inputs?”. It would be handy to build a debugging package that included ML programs to answer these (and related) questions for us. Can such a package be built? You may well suspect that it cannot, but how does one prove that this suspicion is well-founded? We then go on to consider questions about ML programs (presented as values of a datatype representing the abstract syntax of ML). The difference lies in the fact that in ML functions are “black boxes” — we can apply them to arguments, but we can’t look inside the box. “Of course,” you might think, “one can’t test convergence of a function on a given input, but what if it were possible to look at the code of the function? What then? Maybe then one can decide convergence on a given input.” Unfortunately (or fortunately, depending on your point of view), the problem remains undecidable. ∗Modified from a draft by Robert Harper, 1997.