The dagger lambda calculus

The dagger lambda calculus

We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols [4] into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus [6] with a linear negation [1] of ”trivialised” De Morgan duality [5]. Reduction is realised through explicit substitution, based on a symmetric notion of binding of glo...

Infinitary Lambda Calculus Innnitary Lambda Calculus

In a previous paper we have established the theory of transsnite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. From the viewpoint of innnitary rewriting, the BB ohm model of the lambda calculus can be seen as an innnitary term model. In contrast to term rewriting, there are several diierent possible notions of innnite term, whic...

The algebraic lambda calculus

We introduce an extension of the pure lambda-calculus by endowing the set of terms with a structure of vector space, or more generally of module, over a xed set of scalars. Terms are moreover subject to identities similar to usual pointwise de nition of linear combinations of functions with values in a vector space. We then study a natural extension of beta-reduction in this setting: we prove i...

The Lazy Lambda Calculus

The Partial Lambda-Calculus

This thesis investigates various formal systems for reasoning about partial functions or partial elements, with particular emphasis on lambda calculi for partial functions. Beeson's (intuitionistic) logic of partial terms (LPT) is taken as the basic formal system and some of its metamathematical properties are established (for later application). Three di erent avours of Scott's logic of partia...