►A Data-Centric Introduction to Computing
▼VII Advanced Topics
►22 Partial Domains
On this page:
22.1 A Non-Solution
22.2 Exceptions
22.3 The Option Type
22.4 Total Domains, Dynamically
22.5 Total Domains, Statically
22.6 Summary
22.7 A Note on Notation
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22 Partial Domains

    22.1 A Non-Solution

    22.2 Exceptions

    22.3 The Option Type

    22.4 Total Domains, Dynamically

    22.5 Total Domains, Statically

    22.6 Summary

    22.7 A Note on Notation

Sometimes, we cannot precisely capture the domain of a function with the precision we would like. In mathematics, if a function cannot accept all values in its domain, it is called partial. This is a problem we encounter more often than we might like in programming, so we need to know how to handle it. There are actually several programming strategies that we can use, with different benefits and weaknesses. Here, we will examine some of them.

Consider some functions on lists of numbers, such as computing the median or the average. In both cases, these functions don’t work on all lists of numbers: there is no median for the empty list, and we can’t compute its average either, because there are no elements (so trying to compute the average would result in a divison-by-zero error). Thus, while it is a convenient fiction to write 

average :: List<Number> -> Number

it is just that: a (bit of a) fiction. The function is only defined on non-empty lists.

We will now see how to handle this from a software engineering perspective. We’ll specifically work through average because the function is simple enough that we can focus on the software structure without getting lost in the solution details. There are at least four solutions, and one non-solution.

22.1 A Non-Solution

We will start with a strategy that has often been used by programmers in the past, but that we reject as a non-solution. This strategy is to make the above contract absolutely correct by returning a value in the erroneous case; this value is often called a sentinel. For instance, the sentinel might be 0. Here is the full program: 

type LoN = List<Number>

fun sum(l :: LoN) -> Number:
  fold({(a, b): a + b}, 0, l)
end

avg0 :: LoN -> Number

fun avg0(l):
  cases (List) l:
    | empty => 0
    | link(_, _) =>
      s = sum(l)
      c = l.length()
      s / c
  end
end

and here are a few tests: 

check:
  avg0([list: 1]) is 1
  avg0([list: 1, 2, 3]) is 2
  avg0([list: 1, 2, 3, 10]) is 4
end

Is there a test missing here? Yes, for the empty list! Should we add it? 

check:
  avg0(empty) is 0
end

The question is, should we be happy with this “solution”? There are two problems with it.

First, every single use of avg0 needs to check for whether it got back 0 or not. If it did not, then the answer is legitimate, and it can proceed with the computation. But if it did, then it has to assume that the input may have been illegitimate, and cannot use the answer.

Second, even that’s not quite true. To understand why, we need to write a few more tests: 

check:
  avg0([list: -1, 0, 1]) is 0
  avg0([list: -5, 4, 1]) is 0
end

So the problem is that when avg0 returns 0, we don’t know whether that’s a legitimate answer or a “fake” answer that stands for “this is not a valid input”. So even our strategy of “check everywhere” fails!

Ah, but maybe the problem is the use of 0! Perhaps we could use a different number that would work. How about 1? Or -1? The question is: Is there any number that reasonably can’t be the average of an actual input? (And in general, for all problems, can you be sure of this?) Well, of course not.

That’s why this is a non-solution. It has created several problems:

  • We can’t tell from the output whether the input was invalid.

  • That means every caller needs to check.

  • A caller that forgets to check may compute with nonsense.

  • Compositionality is ruined: any function passed average needs to know to check the output (and there is nothing in the contract to warn it!).

Indeed, decades of experience tells us that some of the world’s most sophisticated programmers have not been able to handle this issue even when it matters most, resulting in numerous, pernicious security problems. Therefore, we should now regard this as a flawed approach to software construction, and never do it ourselves.

Let’s instead look at four actual solutions.

22.2 Exceptions

One technique that many languages, including Pyret, provide is called the exception. An exception is a special programming construct that effectively halts the computation because the program cannot figure out how to continue computing with the data it has. There are more sophisticated forms of exceptions in some languages, but here we focus simply on using them as a strategy for handling partiality.

Here is the average program written using an exception (we reuse sum from before): 

avg1 :: LoN -> Number

fun avg1(l):
  cases (List) l:
    | empty => raise("no average for empty list")
    | link(_, _) =>
      s = sum(l)
      c = l.length()
      s / c
  end
end

check:
  avg1([list: 1]) is 1
  avg1([list: 1, 2, 3]) is 2
  avg1([list: 1, 2, 3, 10]) is 4
end

The way raise works is that it terminates everything that is waiting to happen. For instance, if we were to write 

1 + avg1(empty)

the 1 + … part never happens: the whole computation ends. raise creates exceptions.

Again, we’re missing a test. How do we write it? check: avg1(empty) raises "no average for empty list" end The raises form takes a string that it matches against that provided to raise. In act, for convenience, any sub-string of the original string is permitted: we can, for instance, also write check: avg1(empty) raises "no average" avg1(empty) raises "empty list" end

In many programming languages, the use of exceptions is the standard way of dealing with partiality. It is certainly a pragmatic solution. Observe that we got to reuse sum from earlier; the contract looks clean; and we only needed to use raise at the spot where we didn’t know what to do. What’s not to like?

There are two main problems with exceptions:

  1. In real systems, exceptions halt a program’s execution in unpredictable ways. A caller to avg1 may be half-way through doing something else (e.g., it may have opened a file that it intends to close), but the exception causes the call to not finish cleanly, causing the remaining computation to not run, leaving the system in a messy state.

  2. Relatedly, what we presented as a feature should actually be treated as a problem: the contract lies! There’s no indication at all in the contract that an exception might occur. A programmer has to read the whole implementation—which could change at any time—instead of being able to rely on its published contract, when the whole point of contracts was that they saved us from having to read the whole implementation!

Indeed, some modern programming languages designed for large-scale programming (such as Go and Rust) no longer have exception constructs. Therefore, you should not assume that this will continue to be the “standard” way of doing things in the future.

Observe that there is are two kinds of exceptions that can occur. One is as we’ve written above. The other is when we completely ignore (or forget to even think about) the empty list case, and end up getting an error from Pyret, which is also a kind of exception. If Pyret will raise an exception anyway, does it make sense for us to go through the trouble of doing it ourselves?

Yes it does! For several reasons:

  1. First, you get to control where the exception occurs and what it says.

  2. You can document that the exception will occur.

  3. You are less dependent on the behavior of Pyret or whatever underlying programming language, which can change in subtle ways.

  4. You can create an exception that is unique to you, so it can’t be confused with other division-by-zero errors that may lurk in your program.

For these reasons, it’s better to check and raise an exception explicitly than letting it “fall through” to the programming language. Instead, the real problems with this solution are subtler: the lying contract, and the impact on program execution.

22.3 The Option Type

Let’s revisit avg0. The problem with it was that it returned a value that was not distinguishable from an actual answer. So perhaps another approach is to return a value that is guaranteed to be distinguishable! For this, a growing number of languages (including Pyret) have something like this type: 

data Option<T>:
  | none
  | some(value :: T)
end

This is a type we use when we aren’t sure we will have an answer: none means we don’t have an answer, whereas some means we do and value is that answer.

Here’s how our program now looks: 

avg2 :: LoN -> Option<Number>

fun avg2(l):
  cases (List) l:
    | empty => none
    | link(_, _) =>
      s = sum(l)
      c = l.length()
      some(s / c)
  end
end

Now our tests look a bit different: 

check:
  avg2([list: 1]) is some(1)
  avg2([list: 1, 2, 3]) is some(2)
  avg2([list: 1, 2, 3, 10]) is some(4)
end

check:
  avg2(empty) is none
end

The good news is, the contract is now truthful. Just by looking at it, we are reminded that avg0 may not always be able to compute an answer.

Unfortunately, this imposes some cost on every user: they have to use cases to check return values and only use them if they are legitimate. However, this is the same thing we expected in avg0—except we lacked a discipline for making sure we didn’t abuse that value! So this is avg0 done in a principled way.

22.4 Total Domains, Dynamically

All these problems arise because we said that average (like median) is partial. However, it’s only partial if we give the domain as List<Number>; it’s actually a total function on the non-empty list of numbers. But how do we represent that?

In some languages, like Pyret, we can actually express this directly: 

type NeLoND = List<Number>%(is-link)

This says that we’re refining numeric lists to always have a link, i.e., to be non-empty. In Pyret, currently, this check is only done at run-time; in some other programming languages, this can be done by the type-checker itself.

This refinement lets us pretend that we’re dealing with regular lists and reuse all existing list code, while knowing for sure we will never get a divide-by-zero error: 

avg3 :: NeLoND -> Number

fun avg3(l):
  s = sum(l)
  c = l.length()
  s / c
end

check:
  avg3([list: 1]) is 1
  avg3([list: 1, 2, 3]) is 2
  avg3([list: 1, 2, 3, 10]) is 4
end

If we do try passing an empty list, we get an internal exception: 

check:
  avg3(empty) raises ""
end

This is a pretty interesting solution. Our function’s code is clean. We don’t deal with nonsensical values. The interface is truthful! (However, it does require a careful reading to observe that there’s an exception lurking underneath the domain.) And it lets us reuse existing code.

There are two main weaknesses:

  1. Dynamic refinements aren’t found in most languages, so we’d have to do more manual work to obtain the same solution.

  2. We don’t get a static guarantee (i.e., before even running the program) that we’ll never get an exception.

22.5 Total Domains, Statically

How do we make the function total with a static guarantee? That would require that we ensure that we can never construct an empty list! Obviously, this is not possible with the existing lists in Pyret. However, we can construct a new list-like datatype that “bottoms out” not at empty lists but at lists of one element: 

data NeLoN:
  | one(n :: Number)
  | more(n :: Number, r :: NeLoN)
end

Observe that there is simply no way to make an empty list: the smallest list has one element in it. Furthermore, our type checker enforces this for us.

Of course, this is an entirely different datatype than a list of numbers. We can’t, for instance, use the existing sum or length code on it. However, one option is to convert a NeLoN into a LoN, which is always safe, and reuse that code: 

fun nelon-to-lon(nl :: NeLoN):
  cases (NeLoN) nl:
    | one(n) => [list: n]
    | more(n, r) => link(n, nelon-to-lon(r))
  end
end

fun nl-sum(nl :: NeLoN) -> Number:
  sum(nelon-to-lon(nl))
end

fun nl-len(nl :: NeLoN) -> Number:
  nelon-to-lon(nl).length()
end

Now we can write the average in an interesting way: 

fun avg4(nl :: NeLoN) -> Number:
  s = nl-sum(nl)
  c = nl-len(nl)
  s / c
end

Once again, we don’t have to have any logic for dealing with errors. However, it’s not because we’re sloppy or letting Pyret deal with it or getting it checked at runtime or anything else: it’s because there is no way for an empty list to arise. Thus we have both the simplest body and the most truthful interface! But it comes at a cost: we need to do some work to reuse existing functions.

This problem extends to writing tests, which is now more painful: 

check:
  nl1 = one(1)
  nl2 = more(1, more(2, one(3)))
  nl3 = more(1, more(2, more(3, one(10))))

  avg4(nl1) is 1
  avg4(nl2) is 2
  avg4(nl3) is 4
end

That is, we’ve lost our convenient way of writing lists. We can recover that by writing a helper that creates NeLoNs: 

fun lon-to-nelon(l :: LoN) -> NeLoN:
  cases (List) l:
    | empty => raise("can't make an empty NeLoN")
    | link(f, r) =>
      cases (List) r:
        | empty => one(f)
        | else => more(f, lon-to-nelon(r))
      end
  end
end

check:
  avg4(lon-to-nelon([list: 1])) is 1
  avg4(lon-to-nelon([list: 1, 2, 3])) is 2
  avg4(lon-to-nelon([list: 1, 2, 3, 10])) is 4
end

Notice that if we try to use an empty list, we get an exception: 

check:
  avg4(lon-to-nelon(empty)) raises ""
end

However, it’s very important to understand where the error is coming from: the exception is not from avg4, it’s coming from lon-to-nelon, i.e., from the “interface” function. The bad datum never makes it as far as avg4! We can verify this: 

check:
  lon-to-nelon(empty) raises ""
end

Remember, there’s no way to send an empty list to avg4! Nevertheless, this suggests a trade-off: we can either use NeLoN explicitly but with more notational pain, or we can use list but run the risk of some confusion about exceptions. This is a trade-off in general, but there are better options in some languages (A Note on Notation).

So this is actually a very powerful technique: building a datatype that reflects exactly what we want, thereby turning a partial function into a total one. Programmers call this principle making illegal states unrepresentable. It may require writing some procedures to convert to and from other convenient representations for code reuse. Somewhere in those procedures there must be checks that reflect the partiality.

22.6 Summary

In general, there is one non-solution:

  • Return a sentinel value. Do not ever do this unless you’ve first fixed all the security bugs lurking in C programs from the past several decades.

and there are four solutions:

  • Use raise. This is not very good for software engineering in general because exceptions are clunky, semantically complicated, and not compositional.

  • Use a dynamic refinement. Dynamic refinements aren’t in most languages. Also, it’s less good than each of the other solutions, but it’s a decent compromise in many settings.

  • Define a datatype to make illegal states unrepresentable. A bit of work. Pretty sophisticated, invaluable in some places, but not always worth the effort.

  • Use Option. Often the ideal option, because:

    • The type tells us to expect funny business. (raise hides that.)

    • We can’t accidentally misuse the value. (Sentinels hide that.)

    • It’s compositional: we can create functions to help us handle it.

    • It’s much lower overhead than the static totality solution.

    • It’s more statically robust than the dynamic totality solution.

    • It generalizes: in practice, instead of just none and some, a real program will have some for the “normal” case, and a bunch of variants describing the different kinds of errors that are possible, with extra information in each case. For concrete examples of this, see Picking Elements from Sets on sets Combining Answers on queues.

22.7 A Note on Notation

When we wrote above that we can’t get the convenience of writing, say, [list: 1, 2, 3] when using NeLoNs, we were speaking in general. In some languages, we can actually make similar convenient constructors. In Pyret, for instance, there is a protocol for defining custom constructors; in fact, seemingly built-in constructors like list and set are built using this protocol. The code for doing this is a bit ungainly (in part because it’s optimized to save some space and time by making the constructor-writer’s life a little harder), but it only needs to be written once. Here’s a nelon constructor for NeLoNs: 

fun ra-to-nelon(r :: RawArray<Number>) -> NeLoN:
  len = raw-array-length(r)
  fun make-from-index(n :: Number):
    v = raw-array-get(r, n)
    if n == (len - 1):
      one(v)
    else:
      more(v, make-from-index(n + 1))
    end
  end
  make-from-index(0)
end

nelon = {
  make0: {(): raise("can't make an empty NeLoN")},
  make1: {(a1): one(a1)},
  make2: {(a1, a2): more(a1, one(a2))},
  make3: {(a1, a2, a3): more(a1, more(a2, one(a3)))},
  make4: {(a1, a2, a3, a4): more(a1, more(a2, more(a3, one(a4))))},
  make5: {(a1, a2, a3, a4, a5): more(a1, more(a2, more(a3, more(a4, one(a5)))))},
  make: {(args :: RawArray<Number>): ra-to-nelon(args)} }

These tests show that this constructor works very much like the built-in list: 

check:
  [nelon: ] raises "empty"
  [nelon: 1] is one(1)
  [nelon: 1, 2] is more(1, one(2))
  [nelon: 1, 2, 3] is more(1, more(2, one(3)))
  [nelon: 1, 2, 3, 4] is more(1, more(2, more(3, one(4))))
  [nelon: 1, 2, 3, 4, 5] is more(1, more(2, more(3, more(4, one(5)))))
  [nelon: 1, 2, 3, 4, 5, 6] is
  more(1, more(2, more(3, more(4, more(5, one(6))))))
  [nelon: 1, 2, 3, 4, 5, 6, 7] is
  more(1, more(2, more(3, more(4, more(5, more(6, one(7)))))))
end

With this, we can rewrite the tests from Total Domains, Statically very conveniently: 

check:
  avg4([nelon: 1]) is 1
  avg4([nelon: 1, 2, 3]) is 2
  avg4([nelon: 1, 2, 3, 10]) is 4
end

thereby having our cake and eating it too!

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