Linear type systems (allow exchange, but neither weakening nor contraction): Every variable is used exactly once. Affine type systems (allow exchange and weakening, but not contraction): Every variable is used at most once.

A linear type system is to a linear logic as a \normal" type system, like the simply-typed -calculus, is to an intuitionistic logic. The overall ff is that every variable must be used exactly once.

Jul 10, 2024 · Linear type theory is the linear logic-version of type theory. In the definition of (Seely 89, prop. 1.5), following , this is the internal language of/has categorical semantics in star-autonomous categories.

In type theory, we elevate the concept of proof to a primary concept. In constructive logic this is important because proofs give rise to computation under reduction, as discussed in the chapter on linear functional programming (Chapter 6). Therefore, our primary judgment has the form M is a proof of A. (M : A).

A simple form of linear types (specifically MLL, the fragment of linear logic without the additive/‘choice’ operators ⊕ and &) can be encoded as a programming language in which each value can be used, and must be used, exactly once, and popularized by Henry Baker in a series of articles about applying them to remove garbage collection ...

Specifically, programming languages can use linear types to get the “best of both worlds” between memory-safe, garbage-collected languages like Java and “memory-unsafe” languages like C and C++ that use manual memory management.

We call a type system that enforces this property linear, after Girard’s linear logic [Gir87]. Even though our ultimate goal is in the runtime system, we start with functional expressions. We say a function is linear in one of its arguments if it uses that argument exactly once.

Mar 24, 2023 · Linear types, also known as linear logic or linear type systems, are a concept in programming language theory and type systems that enforces the constraint that each variable can be used...

I Determine whether type or context can be used in linear setting I un(˝) if and only if ˝= un ˇ. I lin(˝) if and only if ˝= un ˇor ˝= lin ˇ. I q() if and only if for all ( x:˝) 2, we have q(˝). 23/30

Hype for Types Linear Logic and Linear Type Systems February 15, 20227/23 Introduction to Linear Logic In linear logic, we have neither weakening nor contraction.