Matrices as Tensor Network Diagrams
In many different areas of scientific investigation, purely diagrammatic expressions have begun to be used as mathematical substitutes for the traditional symbolic expressions. These new languages reflect clear operational concepts such as ‘systems’ and ‘processes’, and ‘wires’ are used to represent information flow. Composition is built-in, as the act of connecting wires together. These diagrams are more intuitive, and easier to manipulate, than conventional symbolic models of reasoning.<p>Examples of areas in which these techniques are being used include logic, for the study of propositions and deductions [43]; physics, for the study of space and spacetime [8, 32]; linguistics, to model the interaction of words within a sentence [21]; knowledge, to represent information and belief revision [22]; and computation, for the study of data and computer programs [1, 10]. These diagrams have been provided with a rigorous underpinning using category theory, a powerful branch of mathematics which relates these diagrammatic theories to algebraic structures. More deeply, this approach can also be applied to the study of diagrammatic theories themselves [12, 33]. As such, it is its own ‘metatheory’. One major goal of the project will be to understand to what extent this can serve as a foundation for mathematics itself, as a replacement for set theory. To work towards this, one goal will be to understand how various mathematical structures can be defined from this perspective, a rich line of enquiry for which some answers are already known, but much more is left to still be discovered. An interesting feature of categorical approaches to foundations is that standard limitative theorems, such as the incompleteness theorem of Godel, are no longer set in stone [7, 39]; instead, they become malleable, and can be directly ¨ controlled by changing the categorical universe in which one works.<p>Switching to the field of artificial intelligence and automation — perhaps, so it seems, a world away from the abstraction of higher mathematics — the diagrammatic categories themselves have been implemented as the actual language and deduction mechanism of automated reasoning software packages. These tools make it possible to automatically explore the theories formulated in the diagrammatic language. For the case of the abstract mathematical expressions, the package quantomatic, currently in development at the Computing Laboratory at the University of Oxford, performs this function. For the particular case of diagrammatic models of language and meaning, parsers and tools for corpus exploration are currently under development.<p>The study of each of these ‘diagrammatic theories’, each modelling a separate part of cognitive, physical, or mathematical reality, are currently separate. Our grand vision is to develop a fully integrated framework, in which all of the above can be comprehended and dealt with together. This would be a foundation for mathematics, not in isolation but in direct relation to the cognitive and physical worlds; not only as an exercise in abstract thought, but directly implemented computationally, with tools available for automatic calculation and deduction. The unique perspective given by categorical foundations on Godel-style theorems will allow their relevance to cognition to be rigorously studied. The strong mathematical similarities between the diagrammatic techniques used in these different fields, which have only recently begun to be appreciated, provides a good impetus for this research programme.<p>Source: An integrated diagrammatic universe for knowledge, language and artificial reasoning. Abramsky &amp; Coecke, Oxford.<p><a href="http:&#x2F;&#x2F;www.cs.ox.ac.uk&#x2F;people&#x2F;bob.coecke&#x2F;JTFCats.pdf" rel="nofollow">http:&#x2F;&#x2F;www.cs.ox.ac.uk&#x2F;people&#x2F;bob.coecke&#x2F;JTFCats.pdf</a>