12. Introduction to Martin-Löf Type Theory (MLTT)

Introduction to Martin-Löf Type Theory (MLTT)

Martin-Löf Type Theory (MLTT) is a foundational theory in the field of constructive mathematics and computer science, proposed by Per Martin-Löf in the 1970s.

MLTT is based on the idea of constructive logic, where logical propositions are viewed as types, and proofs are viewed as inhabitants of these types.

Key concepts in MLTT include:

• Types: In MLTT, types are fundamental entities that classify values. Types can themselves be values of higher types.
• Dependent Types: MLTT supports dependent types, where the type of an expression can depend on the value of another expression.
• Universes: MLTT includes a hierarchy of universes, or type levels, where types in one universe can be members of types in a higher universe.
• Functions: Functions in MLTT are first-class citizens and can be used as values. MLTT supports dependent function types, allowing functions to take types or values as arguments and return types or values as results.
• Equality: MLTT has a rich notion of equality, where types can be equal, and values of equal types can be equal.

MLTT has applications in various areas such as programming language theory, formal verification, and proof assistants.

Lab Activity: Basic Concepts of MLTT

Let's implement a simple example in Python to demonstrate basic concepts of MLTT.

class Type:
    def __init__(self, name):
        self.name = name

class Term:
    def __init__(self, term):
        self.term = term

def identity(A):
    return lambda x: x

def main():
    int_type = Type("Int")
    identity_int = identity(int_type)
    result = identity_int(42)
    print("Result of identity function:", result)

if __name__ == "__main__":
    main()
  

In the lab activity, we define a `Type` class to represent types and a `Term` class to represent terms in MLTT.

We then implement the `identity` function, which returns a function that takes an argument and returns it unchanged.

We demonstrate the implementation by creating an identity function for the integer type and applying it to an integer value.