Algebra Seminar

Monday, November 8, 2021 10:00 am - 10:00 am EST (GMT -05:00)

Alexi Block Gorman, Fields Institute

"Fractal Dimensions and Definability from Büchi Automata"

Büchi automata are the natural extension of finite automata, also called finite-state machines, to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is r-regular if there is a Büchi automaton that accepts (one of) the base-r representations of every element of X, and rejects the base-r representations of each element in its complement. We can analogously define r-regular subsets of higher arities, and these sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. There are compelling connections between fractal geometry, r-regular subsets of the reals, and the directed graph structure of the automata that witness regularity. For an r-regular subset of the unit box in n-dimensional Euclidean space, we will describe how to obtain Hausdorff dimension, Box counting dimension, and Hausdorff measure (for the appropriate dimension) in terms of a certain variation of induced sub-automata. We will also see how this gives us a characterization for when reducts of a relevant first-order structure, one expanding the reals as an ordered additive group, have definable unary sets whose Hausdorff dimension and Boxing counting dimension disagree. This is joint work with Christian Schulz.

MC 5417