Girard described two translations of intuitionistic logic into linear logic, one where A -> B maps to (!A) -o B, and another where it maps to !(A -o B). We detail the action of these translations on terms, and show that the first corresponds to a call-by-name calculus, while the second corresponds to call-by-value. We further show that if the target of the translation is taken to be an affine calculus, where ! controls contraction but weakening is allowed everywhere, then the second translation corresponds to a call-by-need calculus, as recently defined by Ariola, Felleisen, Maraist, Odersky, and Wadler. Thus the different calling mechanisms can be explained in terms of logical translations, bringing them into the scope of the Curry-Howard isomorphism.

. We present a separated-linear lambda calculus of resource consumption based on a refinement of linear logic which allows separate control of weakening and contraction. The calculus satisfies subject reduction and confluence, and inherits previous results on the relationship of Girard's two translations from minimal intuitionistic logic to linear logic with call-by-name and call-by-value. We construct a hybrid translation from Girard's two which is sound and complete for mapping types and reduction sequences from call-by-need into separatedlinear . This treatment of call-by-need is more satisfying than in previous work, allowing a contrasting of all three reduction strategies in the manner (for example) that the CPS translations allow for call-by-name and call-by-value. H OW can we explain the differences between parameter-passing styles? With the continuation-passing style (CPS) transforms [24, 25], one makes the flow of control explicit. Each parameter-passing style is associated ...

Many predominantly lazy languages now incorporate strictness enforcing primitives, for example a strict let or sequential composition seq. Reasons for doing this include gains in time or space efficiencies, or control of parallel evaluation. This thesis studies how to prove equivalences between programs in languages with selective strictness, specifically, we use a restricted core lazy functional language with a selective strictness operator seq whose operational semantics is a variant of one considered by van Eckelen and de Mol, which itself was derived from Launchbury’s natural semantics for lazy evaluation. The main research contributions are as follows: We establish some of the first ever equivalences between programs with selective strictness. We do this by manipulating operational semantics derivations, in

This thesis studies how to prove equivalences between programs in languages with selective strictness, specifically, we use a restricted core lazy functional language with a selective strictness operator seq. We establish some of the first ever equivalences between lazy programs with selective strictness by manipulating operational semantics derivations. Our operational semantics is similar to that used by van Eekelen and De Mol, though we introduce a ‘garbage-collecting’ rule for (let) which turns out to cause expressiveness restrictions. For example, arguably reasonable lazy programs such as let y = λz.z in λx.y do not reduce in our operational semantics. We prove some properties of seq, including associativity, idempotence, and left-commutativity. The proofs use our three notions of program equivalence defined