Types and forgetfulness in categorical linguistics and quantum mechanics

The role of types in categorical models of meaning is investigated. A general scheme for how typed models of meaning may be used to compare sentences, regardless of their grammatical structure is described, and a toy example is used as an illustration. Taking as a starting point the question of whether the evaluation of such a type system 'loses information', we consider the parametrized typing associated with connectives from this viewpoint. The answer to this question implies that, within full categorical models of meaning, the objects associated with types must exhibit a simple but subtle categorical property known as self-similarity. We investigate the category theory behind this, with explicit reference to typed systems, and their monoidal closed structure. We then demonstrate close connections between such self-similar structures and dagger Frobenius algebras. In particular, we demonstrate that the categorical structures implied by the polymorphically typed connectives give rise to a (lax unitless) form of the special forms of Frobenius algebras known as classical structures, used heavily in abstract categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure

Game semantics for first-order logic

We refine HO/N game semantics with an additional notion of pointer (mu-pointers) and extend it to first-order classical logic with completeness results. We use a Church style extension of Parigot's lambda-mu-calculus to represent proofs of first-order classical logic. We present some relations with Krivine's classical realizability and applications to type isomorphisms

Defining determinism

The article puts forward a branching - style framework for the analysis of determinism and indeterminism of scientific theories, starting from the core idea that an indeterministic system is one whose present allows for more than one alternative possible future. We describe how a definition of determinism stated in terms of branching models supplements and improves current treatments of determinism of theories of physics. In these treatments, we identify three main approaches: one based on the study of (differential) equations, one based on mappings between temporal realizations, and one based on branching models. We first give an overview of these approaches and show that current orthodoxy advocates a combination of the mapping- and the equations - based approaches. After giving a detailed formal explication of a branching - based definition of determinism, we consider three concrete applications and end with a formal comparison of the branching- and the mapping-based approach. We conclude that the branching - based definition of determinism most usefully combines formal clarity, connection with an underlying philosophical notion of determinism, and relevance for the practical assessment of theories

A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons

Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group $G$ and its group 4-cocycle $\omega_4 \in \mathcal H^4[G;U(1)]$ up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.Comment: An important new result "Untwisted sector of dimension reduction is the Drinfeld center of E" is added in Sec. IIIC; other minor refinements and improvements; 23 pages, 10 figure