We present System F ◦ , an extension of System F that uses kinds to distinguish between linear and unrestricted types, simplifying the use of linearity for general-purpose programming. We demonstrate through examples how System F ◦ can elegantly express many useful protocols, and we prove that any protocol representable as a DFA can be encoded as an F ◦ type. We supply mechanized proofs of System F ◦ ’s soundness and parametricity properties, along with a nonstandard operational semantics that formalizes common intuitions about linearity and aids in reasoning about protocols. We compare System F ◦ to other linear systems, noting that the simplicity of our kind-based approach leads to a more explicit account of what linearity is meant to capture, allowing otherwiseconflicting interpretations of linearity (in particular, restrictions on aliasing versus restrictions on resource usage) to coexist peacefully. We also discuss extensions to System F ◦ aimed at making the core language more practical, including the additive fragment of linear logic, algebraic datatypes, and recursion.

Static program analysis and compile time program optimizations are important aspects of functional language compilers. Strictness optimization is a big part of a Haskell compiler, it is interesting from a research perspective, as well as being needed for practical performance issues. In this thesis we explore a new, typed intermediate language designed for doing static program analysis. This language is well suited to implementing optimizing transformations. We implement strictness optimization for first-order functions, and see how we can safely combine this optimization with other optimizations. Finally, we discuss how we can extend the language to implement strictness optimization for higher-order functions. 3 4