A sequence is k-automatic if its elements are obtained when the base-k representations of integers are successively fed into a given finite automaton. For example, the famous Thue-Morse sequence abbabaabbaab... is 2-automatic: symbol a occurs at position n (indexing from 0) if and only if the base-2 representation of n contains an even number of 1's (this is checkable by a simple automaton). These k-automatic sequences are a well-studied object in combinatorics on words, and they have many good properties. One important feature is that any property of a k-automatic word that is expressible in a certain first-order logic is decidable. Moreover, these sequences satisfy many closure properties. For instance, taking the symbols whose indices are in an arithmetic progression in a k-automatic sequence is again k-automatic.

Using a more exotic way of representing integers than the usual base-k representation, this notion of a k-automatic word can be generalized. In the talk, I will introduce some unusual numeration systems, like Pisot numeration systems and Parry numeration systems. I will compare the corresponding automatic sequences with k-automatic sequences, and sketch ideas that show that the Pisot case is similar to the k-automatic case while the more general Parry-automatic sequences break many of the good properties (like decidability and closure properties mentioned above) of k-automatic sequences.

I will give definitions of automatic sequences and numeration systems, so no prior knowledge of them is needed.

Measure-Once Quantum Automata, also known as Moore-Crutchfield QFA, are the most straightforward quantum variants of probabilistic finite automata: They are obtained by replacing the stochastic matrices by unitary ones, and L1-norm by L2-norm. It turns out that the expressive power of MO-QFA is weaker than that of PFA, but sometimes the MO-QFA may be exponentially more compact than their classical counterparts.

It may appear surprising that determining the emptiness of a strict cut-point languages for MO-QFA is decidable, whereas the same question for PFA is undecidable. On the other hand, removing the attribute "strict" yields an undecidable problem for MO-QFA again. In this talk, I will demonstrate that the ambiguity problem of the acceptance probability for MO-QFA is undecidable, if some radical irrational amplitudes are allowed. It turns out that, conditional to Bombieri-Lang conjecture (or to a speculation by Don Zagier), the problem remains undecidable even if only rational amplitudes are allowed.

A two-dimensional configuration over the alphabet B={0,1} is a binary coloring Z^2 -> B of the infinite grid. For a finite subset D of Z^2, the D-patterns of a configuration are the patterns of shape D that appear in the configuration. The algebraic subshift defined by shape D is the set of the configurations whose D-patterns all contain an even number of 1s. For example, the Ledrappier subshift consists of those configurations where each cell is colored by the modulo 2 sum of the colors on its north and north-east neighbors. We say a configuration has low complexity if it has at most |C| different C-patterns, for some finite C. Every low complexity configuration is in some algebraic subshift so to understand the periodicity properties of general low complexity configurations it is enough to consider elements of algebraic subshifts. It turns out that in the Ledrappier subshift all low complexity configurations are periodic, and the same is true for any algebraic subshift whose defining shape D has line factors in at most one direction.