Infinite Strings and Imaginary Machines
Strings of symbols dwell in so many realms of our lives; we use them to communicate the language and the stories we share with each other every day. For another example, look slightly deeper into the recesses of your computer and you’ll find streams upon streams of symbols from which emerge fascinating websites and programs which allow humans to come together and connect in new and exciting ways. But how far can we take this? These are examples of using symbols to communicate and store information – be it a fairytale or a photograph. By assigning constraints to the process of forming a sequence of symbols, which we call a string or word, we have what mathematicians call a code. These constrains allow us to interpret meaning, in the same way the constraints which we call grammar empower us to communicate through language. In general, this is a way of creating, capturing and communicating information in an abstract form.
However, we can make this even more enjoyable, because mathematics doesn’t like to live in the real world! So we try and expand these notions into the bizarre and the beautiful to see what things we can create if we take reality with a pinch, or more often a bucketful, of salt. For example, beautiful literature is communicated on paper through sequences of finite words, but what could we create if we delve into the worlds of infinite words, and how might we transform and interact with that which we have created?
So for three weeks I’ve been exploring the effect of some imaginary machines, called synchronising transducers, on bi-infinite strings – which are strings that extend infinitely in both directions. In particular I spent the first week finding isomorphisms between the “group” of such synchronising transducers and other objects such as “sliding block codes” and some “Automorphism groups”. It’s easiest to think of an “isomorphism” as a metaphor. In mathematics we create these ‘metaphors’ because an alternative perspective or representation of an object may be illuminating in some way.
I must be brief (so please forgive me!), but a group is essentially an object mathematicians created to abstractly capture and form a structure of the symmetries of an entity. The second isomorphism I created involved a group of sliding block codes, which are a way of encoding information by mapping ‘windows’ of information on to individual points – which intimately reflects the idea of synchronicity explained later. The last and final isomorphism I looked at concerns automorphism groups, which are a set of isomorphisms between an object and itself together with a notion of how they interact with each other. This last is of particular interest, because the particular automorphism group I looked at concerns the isomorphisms of a well known and extensively studied dynamical system. Thus the isomorphism I worked on will allow us to view this classic object in terms of transducers – an entirely new perspective to explore in future weeks.
I’ve since moved on to begin looking at the order question for these synchronising transducers. That is, trying to understand whether successively transforming a Cantor space with a given transducer will ever cause it to return to its original state.
Mathematicians have pondered similar questions before, but not for the transducers which we call synchronising. This beautiful property defines a structure on the machine, where reading certain words almost magically transports you to a certain state in the machine – regardless of what state you start in. This is analogous to a set of directions to Kings Cross Station which work whether you start in Johannesburg or on Jupiter!
And this is where I stand now! I’m really happy to have seven weeks left to explore this inspiring area and couldn’t be more excited to see where it will lead me.