CSE-321 Programming Languages Simply Typed -Calculus 박성우 POSTECH April 3, 2006

-Calculus Abstract syntax Operational semantics (call-by-value)

Simulating Base Types Booleans Natural numbers Fixed point combinator

Motivation The -calculus is equally expressive as Turing machines. booleans, integers, lists, recursive functions, ... But it is highly inefficient to program in the -calculus. Why not just use instead of ? So we introduce a type system!

Simply Typed -Calculus An extension of the untyped -calculus with types Assumes a fixed set of base types E.g. base type primitive constructs A subset of Standard ML

Outline Introduction V The simply typed -calculus Abstract syntax Operational semantics Type system Type safety

Abstract Syntax

What if there is no base type? No interesting expression!

Outline Introduction V The simply typed -calculus Abstract syntax V Operational semantics Type system Type safety

Simply Untyped -Calculus

Reduction Rules for Booleans

Capture-Avoiding Substitutions Completely analogous

Free Variables Completely analogous

Outline Introduction V The simply typed -calculus Abstract syntax V Operational semantics V Type system Type safety

What is the type of ? Answer:

How to find the type of Assume that the type of x is A. okay Find the type of x. A Build a function type A ! A Need to make assumptions on types of variables!

Type System Typing context Typing judgment

Typing Rules --- Top-down

Typing Rules --- Bottom-up

Typing Rules for Booleans

Typing Derivation

Typing Derivation

Outline Introduction V The simply typed -calculus V Abstract syntax V Operational semantics V Type system V Type safety

Unsafe Operations in C Adding two pointers Subtracting an integer from a string which is okay, but likely to be unintended Null-pointer dereferencing Argh... segmentation fault! Using an integer as a destination address in a function call ...

Huffman Coding in SML Suppose that you spent 10 hours before getting your program to typecheck. 0am: You start. 10am: Your program compiles with no type errors. How many more hours did you spend after that?

Type Safety Slogan "well-typed expressions never go wrong" Two theorems Type preservation: "A well-typed expression reduces to another expression of the same type." Progress: "A well-typed expression is not stuck: either it is a value or reduces to another expression."

Type Preservation + Progress A well-typed expression e: If it is a value, we are finished. If it is not, It reduces to another e' [Progress] e' has the same type as e. [Type preservation]

Type Safety Type preservation Progress

Proof of Type Safety Use the rule induction. In the next lecture!