Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory

We describe categorical models of a circuit-based (quantum) functional programming language. We show that enriched categories play a crucial role. Following earlier work on QWire by Paykin et al., we consider both a simple first-order linear language for circuits, and a more powerful host language, such that the circuit language is embedded inside the host language. Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads. We interpret the circuit language not in an ordinary category, but in a category that is enriched in the host category. We show that this structure is also related to linear/non-linear models. As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types

Semantics for first-order affine inductive data types via slice categories

Affine type systems are substructural type systems where copying of information is restricted, but discarding of information is permissible at all types. Such type systems are well-suited for describing quantum programming languages, because copying of quantum information violates the laws of quantum mechanics. In this paper, we consider a first-order affine type system with inductive data types and present a novel categorical semantics for it. The most challenging aspect of this interpretation comes from the requirement to construct appropriate discarding maps for our data types which might be defined by mutual/nested recursion. We show how to achieve this for all types by taking models of a first-order linear type system whose atomic types are discardable and then presenting an additional affine interpretation of types within the slice category of the model with the tensor unit. We present some concrete categorical models for the language ranging from classical to quantum. Finally, we discuss potential ways of dualising and extending our methods and using them for interpreting coalgebraic and lazy data types

Presenting dcpos and dcpo algebras

Dcpos can be presented by preorders of generators and inequational relations expressed as covers. Algebraic operations on the generators (possibly with their results being ideals of generators) can be extended to the dcpo presented, provided the covers are “stable” for the operations. The resulting dcpo algebra has a natural universal characterization and satisfies all the inequational laws satisfied by the generating algebra. Applications include known “coverage theorems” from locale theory

From non-commutative diagrams to anti-elementary classes

Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form L $\infty$$\lambda$. We prove that many naturally defined classes are anti-elementary, including the following: $\bullet$ the class of all lattices of finitely generated convex {\ell}-subgroups of members of any class of {\ell}-groups containing all Archimedean {\ell}-groups; $\bullet$ the class of all semilattices of finitely generated {\ell}-ideals of members of any nontrivial quasivariety of {\ell}-groups; $\bullet$ the class of all Stone duals of spectra of MV-algebras-this yields a negative solution for the MV-spectrum Problem; $\bullet$ the class of all semilattices of finitely generated two-sided ideals of rings; $\bullet$ the class of all semilattices of finitely generated submodules of modules; $\bullet$ the class of all monoids encoding the nonstable $K_0$-theory of von Neumann regular rings, respectively C*-algebras of real rank zero; $\bullet$ (assuming arbitrarily large Erd"os cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large 4-frame. The main underlying principle is that under quite general conditions, for a functor $\Phi$ : A $\rightarrow$ B, if there exists a non-commutative diagram D of A, indexed by a common sort of poset called an almost join-semilattice, such that $\bullet$ $\Phi$ D^I is a commutative diagram for every set I, $\bullet$ $\Phi$ D is not isomorphic to $\Phi$ X for any commutative diagram X in A, then the range of $\Phi$ is anti-elementary.Comment: 49 pages. Journal of Mathematical Logic, World Scientific Publishing, In pres