1. Type Theory Overview

Overview of Type Theory: Historical Context and Motivation

Type Theory is a foundational framework in mathematics and computer science for formalizing the notion of computation and reasoning about types of data.

The roots of Type Theory can be traced back to the early 20th century, with the work of David Hilbert and Alonzo Church.

In the 1930s, Church introduced the concept of the lambda calculus, a mathematical formalism for expressing computation.

Church's work laid the groundwork for Type Theory by providing a basis for understanding computation in terms of functions and types.

In the 1960s, Per Martin-Löf developed Martin-Löf Type Theory (MLTT), which aimed to provide a foundation for constructive mathematics and proof theory.

MLTT introduced dependent types, allowing types to depend on terms, which enhanced its expressive power and facilitated the formalization of mathematics.

In computer science, Type Theory has gained prominence due to its role in the design and analysis of programming languages.

Functional programming languages, such as Haskell and OCaml, are based on Type Theory principles, enabling static type checking and type inference.

Type Theory serves as the basis for formal verification techniques, where programs are rigorously verified to ensure correctness.

Lab Activity: Implementing a Simple Type Checker

Let's implement a simple type checker for a lambda calculus language with basic types.

def type_check(expr):
    # Type checking logic goes here
    pass

def main():
    expr = "(λx:Bool. x)"
    try:
        result = type_check(expr)
        print("Expression is well-typed with type:", result)
    except Exception as e:
        print("Type error:", e)

if __name__ == "__main__":
    main()
  

The `type_check` function takes an expression as input and returns its type if it is well-typed, otherwise raises an error.

We'll define the syntax of our lambda calculus language and specify the typing rules to be implemented.

For example, we might have rules like:

# Typing rules
def type_check(expr):
    if expr.startswith("(λ"):
        # Lambda abstraction
        _, var_type, body = expr.split(":", 2)
        body = body.strip()
        if body.startswith("("):
            # Application
            arg_type, _ = body[1:-1].split(".", 1)
            if var_type.strip() == arg_type.strip():
                return "Bool"  # Type of lambda abstraction
            else:
                raise Exception("Type mismatch in application")
        else:
            if body.strip() == var_type.strip():
                return "Bool"  # Type of lambda abstraction
            else:
                raise Exception("Type mismatch in lambda abstraction")
    else:
        raise Exception("Invalid expression")

def main():
    expr = "(λx:Bool. x)"
    try:
        result = type_check(expr)
        print("Expression is well-typed with type:", result)
    except Exception as e:
        print("Type error:", e)

if __name__ == "__main__":
    main()
  

These rules specify how to infer the type of a lambda abstraction.

Students can then implement these rules in the `type_check` function and test their type checker with various lambda calculus expressions.