Given a function and an input for which that function is defined, does the function return an answer? Functions that always return answers are called total, and functions that don’t are called partial.

Note that we are only considering inputs for which the function is defined. It’s easy to define a function on a certain set of inputs (the domain) but forget to cover some cases. This sort of thing happens all the time in both programming (for instance, writing a function to handle lists but forgetting to deal with the empty list) and math (as when a student writing an inductive proof forgets to cover the base case). It’s easy enough to handle. Just add the missing clauses and move on. Alternatively, redefine the domain of the function so that it excludes the missing case. Either way, there will be no “holes” in the modified function.

How could a function fail to return an answer for an input for which it is defined? This problem is essentially retlated to the termination of unbounded while-loops. Functions can be broken down into four classes based on their loop bounds:

1. Primitive computable: no unbounded loops required; all loops bounds can be calculated in advance. (Also known as primitive recurisve.)
2. General computable: unbounded loops required; loop bounds cannot be calculated in advance, but loops can nevertheless be guaranteed to terminate. (Also known as general recurisve.)
3. Uncomputable: unbounded loops required; loops can be guaranteeed not to terminate for some inputs, but which inputs exactly cannot be determined.
4. Unknown: unbounded loops required as far as anyone knows, but this has not been proved.

Let’s take a look at some examples.

# Primitive Computable

Primitive computable functions encompass pretty much everything that is encountered in day-to-day programming and math. A good rule of thumb is that if it can be calculated in Excel, it’s primitive computable. This includes everything from simple arithmetic to solving Go.

Functions in this class can be written using only loops with explicit fixed bounds. Yet it’s common to see real-world code written with unbounded while-loops even in cases where there is no good reason for doing so.

# General Computable

It’s not easy come up with functions that are computable but not primitive. Personally I only know of two examples, and they are both named after their discoverers.

The first of these is the Ackermann function, also known as the Sudan-Ackermann-Peters function. It was devised in the 1920s for the specific purpose of showing that there are computable functions that are not primitive computable. Here’s a definition in Python:

This definition contains an open while-loop, and it cannot be rewritten with a bounded loop. There is no method for estimating the number of loop passes required short of actually calculating it. Still, it can be shown that it will always terminate. On every iteration either `bound` or `total` is decremented. `bound` is increased from time to time, but never past its initial value. The output grows very, very fast with respect to its inputs.

Even faster-growing is the Goodstein function, which has to do with representing a number in terms of sums of powers of some base and then incrementing that base. Like the Ackermann function, the Goodstein function can be proved, despite its staggering growth, to be total. But whereas the totality of Ackermann can be proved without too much difficulty, proving the totality of Goodstein is provably difficult. Specifically, it requires infinitary reasoning and therefore cannot be proved in Peano arithmetic.

# Uncomputable

Uncomputable functions do not return answers for some inputs. The classic example of uncomputability is the halting problem: does a given program halt or not? Any attempt to implement a function to calculate haltingness for arbitrary programs is doomed to be either partial or incorrect.

In practice, a reasonable means of coping with uncomputability is to impose a bound. Rather than asking if a program halts ever, we can ask if it halts within some number of steps. This question in contrast can be answered definitively without any difficulty.

Of course, the Busy Beaver problem is also uncomputable.

# Unknown

Some functions seem to require unbounded loops, but nobody knows if this is actually true. The most famous of these is the Collatz function:

Does this function terminate for all inputs? As far as anybody knows, the answer is yes. Can the open while-loop be rewritten with a bound? As far as anybody knows, the answer is no. This function is awfully simple, but nobody knows how to answer basic questions about its behavior.

The Collatz function belongs to a genre of functions of the following form: apply some transformation to an input until some condition is met. Another example of this genre is the Lychrel function, which adds a number with its digit-reverse until a palindrome turns up:

Does this function terminate for all inputs? The answer here appears to be no: `lychrel(196)` has not been witnessed to terminate. It could be that the sequence does terminate and nobody has checked far enough yet to see it, or it could be that it really doesn’t terminate. If it doesn’t terminate, it would be nice if that could be proved, but nobody knows if that’s possible either.

# Discussion Questions

1. General-purpose programming languages allow users to (attempt to) implement uncomputable functions. Is this a feature or a bug?
2. Suppose your usual programming language was replaced with another language that was exactly the same except unbounded while-loops were prohibited. How long would it take for you to notice? Would you complain?
3. Are there any practical applications for general computable functions?
4. What are some real-world use cases where unbounded while-loops are helpful?
5. What are some real-world use cases where unbounded while-loops are thought to be required but really aren’t?
6. The expression “primitive computable” is, ironically, not very “PC”. What is a better term?