Programming in Scala”

Chapter notes 12
Notes on chapter 4: Handling errors without
exceptions
Partial and total functions
A partial function⁴⁷, as opposed to a total function, is a function that is not defined for some of
its inputs. That is, it’s not true for partial functions that every value of the input type maps to a
value of the output type. There are different approaches to representing this kind of function, but
in functional programming we commit to using only total functions. The approach we take in this
chapter is to augment the return type of partial functions to make it “the same size” as the input
type.
For example, the Option[T] type simply adds one data constructor, None, to the underlying type T.
That way we can have a total function of type X => Option[T] represent a partial function from X
to T. Wherever the function is not defined for a given X, we simply map that value to None in the
result type.
Turing completeness and the halting problem
Another possibility in languages that are Turing complete⁴⁸, which includes Scala, is that a function
may “hang” or “run forever” in an infinite loop. Functions that do this are also considered to be
partial. However, in a language like Scala we cannot prevent this kind of partiality nor can we
recover from it in the general case. It is equivalent to the halting problem⁴⁹.
Bottoms
A program that hangs (by running in an infinite loop) or crashes (by throwing an exception) is said
to “evaluate to bottom”. The term “bottom” (sometimes “falsum”) comes from logic, where it is used
to denote a contradiction. The Curry-Howard isomorphism⁵⁰ says that types are propositions, and
a program of a given type is a proof of that proposition. Since a non-terminating program can have
any type, we can say that it’s a proof of every proposition. This is exactly like the situation with
contradictions in logic–if you assume a contradiction, you can prove any proposition.
Total languages
Some programming languages are total languages⁵¹, and do not provide facilities for unbounded
recursion like Scala does. A program in such a language provably terminates, and there are no
⁴⁷http://en.wikipedia.org/wiki/Partial_function
⁴⁸http://en.wikipedia.org/wiki/Turing-complete
⁴⁹http://en.wikipedia.org/wiki/Halting_problem
⁵⁰http://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
⁵¹http://en.wikipedia.org/wiki/Total_functional_programming