That wasn't readily explained in online resources. Then fresh, separate copies of the type of id can be substituted into the constraints for each place id is used, for example a2 -> a2 and a3 -> a3. The key is to solve constraints in a particular order: first solve constraints for the definition of id. The first is of type (a -> a) -> (a -> a). The second id is a function of type a -> a, where a can be anything. The problem is the type of the function's parameter x is different on each usage of id. Here, we've bound id to a function that returns whatever you've passed in, otherwise known as the identity function. Where things get nasty, and what I couldn't understand for a while, is dealing with let: (let ((id (lambda (x) x))) Since we know 1 and 2 are numbers, by substitution, we know the if statement is a number. Here, we can say that the condition of the if statement must be Boolean, and that the type of the if statement is the same as the type of its then and else clauses. A node that applies a function has the same type as the result of the function, and so on.įor a language without let, you can blindly solve the above constraints by substitution. For example, if the node is a + operator, the operands and the results must all be numbers. To generate constraints, you recurse through the syntax tree and generate one or more constraints on each node. First, type inference can be thought of generating and then solving constraints. I'll try to summarize the abstract idea followed by details that I did not find immediately obvious. I got my first introduction to the subject from Programming Languages: Applications and Interpretation It's unfortunate that much of the literature on the subject is very dense. Polymorphic, Existential, Recursive Types (PDF) These assignments are from a more advanced course: PCF Type Inference ( a solution) to implement algorithm W for Hindley-Milner type inference.PCF Interpreter ( a solution) to warm up.I recommend these two accessible homeworks in ML, which you can both complete in less than a day: However, since the best way to learn is to do, I strongly suggest implementing type inference for a toy functional language by working through a homework assignment of a programming languages course. Closure compiler's TypeInference is an example of the data-flow analysis approach to type inference, which is better suited to dynamic languages that the Hindler Milner approach.Chapter 13 (Type Reconstruction) of the new book DCPL, Design Concepts in Programming Languages.Chapter 22 (Type Reconstruction) of the book TAPL, Types and Programming Languages, and the corresponding OCaml implementations recon and fullrecon.Chapter 7 (Types) of the book EOPL, Essentials of Programming Languages.The summer course Interpreting types as abstract values presents elegant evaluators, type checkers, type reconstructors and inferencers using Haskell as a metalanguage.Chapter 30 (Type Inference) of the freely available book PLAI, Programming Languages: Application and Interpretation, sketches unification-based type inference.Important differences between Python 2.x and Python 3.I found the following resources helpful for understanding type inference, in order of increasing difficulty:.Python program to convert a list to string.
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