Dan Marsden

Interacting Conceptual Spaces

We propose applying the categorical compositional scheme of [6] to conceptual space models of cognition. In order to do this we introduce the category of convex relations as a new setting for categorical compositional semantics, emphasizing the convex structure important to conceptual space applications. We show how conceptual spaces for composite types such as adjectives and verbs can be constructed. We illustrate this new model on detailed examples.

A Graph Theoretic Perspective on CPM(Rel)

This volume contains the proceedings of the 12th International Workshop on Quantum Physics and Logic (QPL 2015), which was held July 15-17, 2015 at Oxford University. The goal of this workshop series is to bring together researchers working on mathematical foundations of quantum physics, quantum computing, spatio-temporal causal structures, and related areas such as computational linguistics. Of particular interest are topics that use logical tools, ordered algebraic and category-theoretic structures, formal languages, semantical methods and other computer science methods for the study of physical behaviour in general.Mixed states are of interest in quantum mechanics for modelling partial information. More recently categorical approaches to linguistics have also exploited the idea of mixed states to describe ambiguity and hyponym / hypernym relationships. In both these application areas the category Rel of sets and binary relations is often used as an alternative model. Selinger’s CPM construction provides the setting for mixed states in Hilbert space based categorical quantum mechanics. By analogy, applying the CPM construction to Rel is seen as introducing mixing into a relational setting. We investigate the category CPM(Rel) of completely positive maps in Rel. We show that the states of an object in CPM(Rel) are in bijective correspondence with certain graphs. Via map-state duality this then allows us provide a graph theoretic characterization of the morphisms in CPM(Rel). By identifying an appropriate composition operation on graphs, we then show that CPM(Rel) is isomorphic to a category of sets and graphs between them. This isomorphism then leads to a graph based description of the complete join semilattice enriched †-compact structure of CPM(Rel). These results allow us to reason about CPM(Rel) entirely in terms of graphs. We exploit these techniques in several examples. We give a closed form expression for the number of states of a finite set in CPM(Rel). The pure states are characterized in graph theoretic terms. We also demonstrate the possibly surprising phenomenon of a pure state that can be given as a mixture of two mixed states.

Equational reasoning with lollipops, forks, cups, caps, snakes, and speedometers

Abstract Proofs in elementary category theory typically involve either the pasting together of commuting diagrams or calculational reasoning using chains of equalities. Much of the effort in these styles can be consumed with trivial administrative steps involving functoriality, naturality, and the handling of identities. As an alternative, we advocate the use of string diagrams, a two-dimensional form of notation which silently deals with these distracting bookkeeping steps. While originally developed to reason about functors and natural transformations, we show that string diagrams can also easily accommodate objects and arrows. Throughout the paper we emphasize how the topological freedom inherent in the notation can be exploited to aid the use of geometric intuition in the development of proofs. Indeed, drawing string diagrams is a bit of an art: good diagrammatic choices can make all the difference.

Generalized Relations in Linguistics and Cognition

Categorical compositional models of natural language exploit grammatical structure to calculate the meaning of sentences from the meanings of individual words. This approach outperforms conventional techniques for some standard NLP tasks. More recently, similar compositional techniques have been applied to conceptual space models of cognition.

Coalgebras with Symmetries and Modelling Quantum Systems

This paper describes a generalization of the usual category in which coalgebras are considered, and its application to modelling quantum systems and their physical symmetries. Following the programme of work initiated in [1], [2], we aim to model systems described by the laws of quantum physics using coalgebraic techniques. A broader notion of the morphisms of coalgebras is given, in which diagrams are allowed to commute only up to appropriate natural isomorphism. This relaxed setting is then shown to have analogues of coalgebraic notions such as bisimulations, with properties that parallel the usual coalgebraic ones closely. This new setting is then exploited to give coalgebraic models of quantum systems in which the conceptually important physical symmetries are given as automorphisms of a suitable coalgebra.

Generalized relations in linguistics & cognition

Categorical compositional models of natural language exploit grammatical structure to calculate the meaning of sentences from the meanings of individual words. This approach outperforms conventional techniques for some standard NLP tasks. More recently, similar compositional techniques have been applied to conceptual space models of cognition.

Dragging Proofs Out of Pictures

String diagrams provide category theory with a different and very distinctive visual flavour. We demonstrate that they are an effective tool for equational reasoning using a variety of examples evolving around the composition of monads. A deductive approach is followed, discovering necessary concepts as we go along. In particular, we show that the Yang–Baxter equation arises naturally when composing three monads. We are also concerned with the pragmatics of string diagrams. Diagrams are carefully arranged to provide geometric intution for the proof steps being followed. We introduce thick wires to distinguish composite functors and suggest a two-dimensional analogue of parenthesis to partition diagrams.

Fibred Coalgebraic Logic and Quantum Protocols

This volume contains the proceedings of the 10th International Workshop on Quantum Physics and Logic (QPL X), which was held July 17-19, 2013 at ICFO in Castelldefels (Barcelona), Spain. The goal of this workshop series is to bring together researchers working on mathematical foundations of quantum physics, quantum computing and spatio-temporal causal structures, and in particular those that use logical tools, ordered algebraic and category-theoretic structures, formal languages, semantic methods and other computer science methods for the study of physical behavior in general. Over the past few years, there has been growing activity in these foundational approaches, together with a renewed interest in the foundations of quantum theory, which complement the more mainstream research in quantum computation.