The most basic form of type theory is simply typed lambda calculus, which extends the untyped lambda calculus with a type system. In simply typed lambda calculus, every term has a unique type, and the type of a term is determined by its structure. For example, in the term λx.x, the variable x can have any type, and the entire term has the function type A → A, where A is the type of x.
Type theory is used in various areas of computer science, including programming language theory, type systems, and formal verification. It provides a foundation for understanding the semantics of programming languages, and it is used to design and analyze new programming languages and type systems. Additionally, type theory is used in formal verification to prove the correctness of programs and hardware.
One of the key features of type theory is its ability to express and reason about dependent types. In dependent type theory, the type of a term can depend on the value of another term. This allows for more expressive and precise specifications of programs and their properties. For example, in dependent type theory, one can define a type that represents a list of integers of a given length, and one can write a function that takes such a list and returns its length.
Another important aspect of type theory is its ability to express and reason about higher-order types. In higher-order type theory, types can themselves be classified by other types, allowing for more complex and expressive type systems. For example, in higher-order type theory, one can define a type that represents a function from types to types, and one can write a function that takes such a function and applies it to a given type.