As I present this new thesis on the role of topos theory in computing to the world, I have anticipated a number of possible questions and answers ahead of time. I will do my best to answer any questions you may have.
What prior work is closest to yours?
With the help of a commentor on hacker news, I have determined that the Algebraic Structure Theory of Sequential Machines (1966) comes closest my work. Most impressively, they came close to developing the idea of a dataflow relation without using Sierpinski topos theory. Instead of an ordinary algebraic structure theory, I provide an updated theory based instead upon topoi.
What other prior works do you build upon the most?
The first and foremost is probably Robert Goldblatt's Topoi: The Categorial Analysis of Logic. It is with this text that I first discovered the Sierpinski topos which opened up the way for my own further studies in the subject. A second reference text is Sketches of an Elephant – A Topos Theory Compendium. Aside from these a variety of lattice theory textbooks and papers provided extra tidbits of information.
What is the practical basis of applied topos theory?
I believe the answer to this is the principle of locality. The physical universe of which we are all a part is organized around this principle which states that an object is influenced directly only by its immediate surroundings. This creates an organizing principle of relevance to all fields of engineering. In classical physics this motivates the modeling of physical systems based upon their local behaviour. In computing, this motivates our theories of local computation and dataflow.
What use is a topos theory of computing?
The topos theory of computing can be used to model information loss and information flow. The first is relevant because the second law of thermodynamics states that the entropy of a logical system cannot decrease - which implies that information loss leads to heat dissipitation. The second is relevant because the principal of locality necessitates that computer systems should be modeled in terms of information flows and local effects.
What is this topos theory of computing?
I would like to explain it by analogy to physics. In physics we study the movement of particles from place to place. In this theory of computing, we instead study the movement of packets of bits from place to place. In order to describe these information flows I have presented a new formalism based upon Sierpinski topos theory.
No comments:
Post a Comment