We continue with my excessively detailed explanation of the seminal paper on the ML type inference system…

1 Introduction This paper is concerned with the polymorphic type discipline of ML, which is a general purpose functional programming language, although it was first introduced as a metalanguage (whence its name) for constructing proofs in the LCF proof system. [4]

These bracketed numbers are not footnotes; they are references to other papers. See the original paper’s list of references if you want to follow up on these references.

By “type discipline” we basically mean the same thing as “type system”. That is, there is some way of determining the “type” of the expressions in a programming language, and detecting when the programs are erroneous due to violations of the typing rules.

By “polymorphic” here we mean what a C# programmer would mean by “generic”. There are many kinds of polymorphism in computer programming. “Subtype polymorphism” is what object-oriented programmers typically think of when someone says “polymorphism”:

void M(Animal a) { ... } ... Giraffe g = new Giraffe(); M(g); // No problem.

That’s not the kind of polymorphism we’re talking about here. Rather, we’re talking about:

void N<T>(List<T> list) { ... } ... List<int> g = ...; N(g); // No problem.

This is often called “parametric polymorphism”, because the method takes a “type parameter”.

ML has parametric polymorphism, not subtype polymorphism.

A metalanguage is a language used to implement or describe another language. In this case, ML was first created to be the metalanguage for LCF, “Logic for Computable Functions”. The purpose of LCF was to automatically find proofs for mathematical theorems; you could write little programs in ML that described to the LCF system various strategies for trying to get a proof for a particular theorem from a set of premises. But as the paper notes, ML and its descendants are now general-purpose programming languages in their own right, not just implementation details of another language.

The type discipline was studied in [5] where it was shown to be semantically sound, in a sense made precise below, but where one important question was left open: does the type-checking algorithm — or more precisely the type assignment algorithm (since types are assigned by the compiler, and need not be mentioned by the programmer) — find the most general type possible for every expression and declaration?

As the paper notes, we’ll more formally define “sound” later. But the basic idea of soundness comes from logic. A logical deduction is *valid* if every conclusion follows logically from a premise. But an argument can be valid and come to a false conclusion. For example “All men are immortal; Socrates is a man; therefore Socrates is immortal” is valid, but not sound. The conclusion follows logically, but it follows logically from an incorrect premise. And of course an invalid argument can still reach a true conclusion that does not follow from the premises. A valid deduction with all true premises is *sound*.

Type systems are essentially systems of logical deduction. We would like to know that if a deduction is made about the type of an expression on the basis of some premises and logical rules, that we can rely on the soundness of those deductions.

The paper draws a bit of a hair-splitting distinction between a type checking algorithm and a type assignment algorithm. A type checking algorithm verifies that a program does not violate any of the rules of the type system, but does not say whether the types were added by the programmer as annotations, or whether the compiler deduced them. In ML, all types are deduced by the compiler using a type assignment algorithm. The question is whether the type assignment algorithm that the authors have in mind finds the most general type for expressions and function declarations.

By “most general” we mean that we want to avoid situations where, say, the compiler deduces “oh, this is a method that takes a list of integers and returns a list of integers”, when it should be deducing “this is a method that takes a list of T and returns a list of T for any T”. The latter is more general.

Why is this important? Well, suppose we deduce that a method takes a list of int when in fact it would be just as correct to say that it takes a list of T. If we deduce the former then a program which passes a list of strings is an error; if we deduce the latter then it is legal. We would like to make deductions that are not just sound, but also that allow the greatest possible number of correct programs to get the stamp of approval from the type system.

Here we answer the question in the affirmative, for the purely applicative part of ML. It follows immediately that it is decidable whether a program is well-typed, in contrast with the elegant and slightly more permissive type discipline of Coppo. [1]

This is a complicated one; we’ll deal with this paragraph in the next episode!