## August 8, 2012

### Categorical Unification

#### Posted by David Corfield

A couple of postdoc positions have been advertised to work with Bob Coecke and Samson Abramsky on a research project – Categorical unification: where foundational physics, natural language and rational processes meet – in the Department of Computer Science in Oxford. In view of the Executive Summary, I’d be interested to see the full description of the project.

Recent advances show that category theory can be used to capture the essential behavioural properties of many complex systems, and provides the right language to study their foundational concepts across a broad range of disciplines, including the physical world, logical and deductive systems, the way that meaning is encoded into a sentence, and closely related, cognition.

Essential mathematical components of this work include monoidal categories, sheaf theory and coalgebra. The mathematical study of these is currently separate, but the common mathematical formalism underlying them suggests that they should be studied together, as a part of a whole. The ultimate aim is to develop a fully-integrated mathematical formalism for modelling the physical world, making deductions about it, and communicating those deductions linguistically – in short, a mathematical formalism for intelligent reasoning.

These developments would go hand-in-hand with ongoing projects which aim to automate reasoning about linguistic meaning and quantum processes, by exploiting the logical content carried by the graphical languages which describe these areas.

This project addresses both of the Big Questions of this call, by proposing new paradigms for knowledge representation, language and reasoning, as well as for the artificial implementation of these. The models of meaning moreover model the cognitive mind in a similar manner as we model abstract mathematical reasoning.

In particular, I wonder what the second paragraph is hinting at.

Posted at August 8, 2012 4:31 PM UTC

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### Re: Categorical Unification

In particular, I wonder what the second paragraph is hinting at.

Maybe I can trigger a reaction from somebody who knows more by making a critical comment: given that I know a bit of the Coecke-Abramsky program – which is all great as far as it goes – I find these suggestions a bit strong.

A while back we had a speaker in our colloquium introducing a talk on related matters with advertizements of how this program provides a foundation for all of quantum mechanics. But the formalism really formalizes just quantum kinematics and just for systems with finitely many degrees of freedom. This is of course precisely the case that quantum information theory is concerned with, where it is all about shuffling a bunch of qbits around. But it excludes already simple things like the particle on the line and it is a long way from there to a foundation of quantum physics as a whole.

Not that the necessary generalization are not possible and have not been considered: Stephan Stolz and Peter Teichner have written out many of the details as a low-dimensional example in the context of their work of quantum field theories, where quantum mechanics is formalized as a monoidal functor on a category of Riemannian 1-dimensional cobordisms with values in a suitable monoidal category of topological vector spaces.

This has received less attention than it deserves. I would enjoy seeing the thrust of the Coecke-Abramsky school be expanded to this general context of quantum physics.

Posted by: Urs Schreiber on August 13, 2012 9:11 AM | Permalink | Reply to this

### Re: Categorical Unification

Recent advances show that category theory can be used to capture the essential behavioural properties of many complex systems, and provides the right language to study their foundational concepts across a broad range of disciplines, including the physical world, logical and deductive systems, the way that meaning is encoded into a sentence, and closely related, cognition.

I wonder if it might become less plausible as one expands into the “general context of quantum physics”, as you suggest, that there are to be found common structures between the physical world and cognition.

Unless, perhaps, one takes physics to capture not the world as it is in itself, but our cognitive dealings with it. We had a discussion about such matters in the early days of the Café.

Also, I meant to see if I could make some sense of Liang Kong’s remark

…our physical space is nothing but a network of structured stacks of information, from which spacetime can emerge.

mentioned here.

Posted by: David Corfield on August 13, 2012 9:31 AM | Permalink | Reply to this

### Re: Categorical Unification

I wonder if it might become less plausible as one expands into the “general context of quantum physics”, as you suggest, that there are to be found common structures between the physical world and cognition.

By the way, in case you or other readers here haven’t seen it yet: that part of the proposal probably arises as an extrapolation of this work:

Bob Coecke, Mehrnoosh Sadrzadeh, Stephen Clark, Mathematical Foundations for a Compositional Distributional Model of Meaning (arXiv:1003.4394)

Abstract We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are “lifted” to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the inner-product can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our `categorical model’ which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montague-style Boolean-valued semantics.

A New Scientist article on this stuff is here.

Posted by: Urs Schreiber on August 13, 2012 4:01 PM | Permalink | Reply to this

### Re: Categorical Unification

In particular, I wonder what the second paragraph is hinting at.

I am also wondering what the first paragraph is hinting at:

Recent advances show that category theory can be used to capture the essential behavioural properties of many complex systems, and provides the right language to study their foundational concepts across a broad range of disciplines

In particular I am wondering what is meant by a “complex system” and “behavioural properties.” Also, how does one decide which behavioural properties are “essential” and which ones are not?

This may sound like I am nitpicking, but this is really not my intent. I have a pretty good sense of what a complex continuous time dynamical system could mean, how such systems would form a category and why this category would fiber over a category opposite to that of a certain subcategory of directed graphs… But I would be at a loss to define a complex system in general.

Posted by: Eugene on August 13, 2012 4:51 PM | Permalink | Reply to this

### Re: Categorical Unification

The first sentence of the second paragraph clarifies, I think: “Essential mathematical components of this work include monoidal categories, sheaf theory and coalgebra.”

All of these are used heavily in computer science as abstractions for modelling dynamical systems.

1. Monoidal categories give models of linear and other substructural logics, which have proof-theoretic readings in terms of state change.

2. Sheaves arise as categorifications of Kripke models. By considering the internal set theory of sheaf categories, we can extend traditional Kripke models to talk about (say) variable sets, and not merely variable propositions.

3. Coalgebra let us model systems in terms of their observable behavior – a class of behaviors is described by the final coalgebra for a functor $F$, and a particular system is a (non-final) coalgebra for this functor $c : A \to F(A)$. We can equate two systems $c : A \to F(A)$ and $d : B \to F(B)$ just when they are bisimilar.

Posted by: Neel Krishnaswami on August 15, 2012 10:35 AM | Permalink | Reply to this

### Re: Categorical Unification

Modal logic seems to provide ideas for potential glue between 2 and 3, as we discussed here.

Eugene and I discussed before coalgebra and dynamical systems.

Hmm, I wonder if there was something worth exploring in that post and first comment.

Posted by: David Corfield on August 15, 2012 11:01 AM | Permalink | Reply to this
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