Luke Schaeffer

Avoiding Three Consecutive Blocks of the Same Size and Same Sum

We show that there exists an infinite word over the alphabet {0, 1, 3, 4} containing no three consecutive blocks of the same size and the same sum. This answers an open problem of Pirillo and Varricchio from 1994.

The Critical Exponent is Computable for Automatic Sequences

The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence.

Decision algorithms for Fibonacci-automatic Words, I: Basic results

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) “Fibonacci-automatic”. This class includes, for example, the famous Fibonacci word f = f 0 f 1 f 2 ··· = 01001010··· , the fixed point of the morphism 0 → 01 and 1 → 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth.

Decision Algorithms for Fibonacci-Automatic Words, III: Enumeration and Abelian Properties

We continue our study of the class of Fibonacci-automatic words. These are infinite words whose nth term is defined in terms of a finite-state function of the Fibonacci representation of n. In this paper, we show how enumeration questions (such as counting the number of squares of length n in the Fibonacci word) can be decided purely mechanically, using a decision procedure. We reprove some known results, in a unified way, using our technique, and we prove some new results. We also examine abelian properties of these words. As a consequence of our results on abelian properties, we get the result that every nontrivial morphic image of the Fibonacci word is Fibonacci-automatic.

Subword Complexity and k-Synchronization

We show that the subword complexity function ρ x (n), which counts the number of distinct factors of length n of a sequence x, is k-synchronized in the sense of Carpi if x is k-automatic. As an application, we generalize recent results of Goldstein. We give analogous results for the number of distinct factors of length n that are primitive words or powers. In contrast, we show that the function that counts the number of unbordered factors of length n is not necessarily k-synchronized for k-automatic sequences.

A Physically Universal Cellular Automaton

Several cellular automata (CA) are known to be universal in the sense that one can simulate arbitrary computations (e.g., circuits or Turing machines) by carefully encoding the computational device and its input into the cells of the CA. In this paper, we consider a different kind of universality proposed by Janzing. A cellular automaton is physically universal if it is possible to implement any transformation on a finite region of the CA by initializing the complement of the region and letting the system evolve. We give the first known example of a physically universal CA, answering an open problem of Janzing and opening the way for further research in this area.

A Physically Universal Quantum Cellular Automaton

We explore a quantum version of Janzing’s “physical universality”, a notion of computational universality for cellular automata which requires computations to be done directly on the cells. We discuss physical universality in general, the issues specific to the quantum setting, and give an example of a quantum cellular automaton achieving a quantum definition of physical universality.

Ostrowski Numeration and the Local Period of Sturmian Words

We show that the local period at position n in a characteristic Sturmian word can be given in terms of the Ostrowski representation for n + 1.

New hardness results for the permanent using linear optics

In 2011, Aaronson gave a striking proof, based on quantum linear optics, showing that the problem of computing the permanent of a matrix is #P-hard. Aaronsons proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiants seminal proof of the same fact in 1979. Nevertheless, it did not prove that computing the permanent was #P-hard for any class of matrices which was not previously known. In this paper, we present a collection of new results about matrix permanents that are derived primarily via these linear optical techniques. First, we show that the problem of computing the permanent of a real orthogonal matrix is #P-hard. Much like Aaronsons original proof, this will show that even a multiplicative approximation remains #P-hard to compute. The hardness result even translates to permanents over finite fields, where the problem of computing the permanent of an orthogonal matrix is ModpP-hard in the finite field F_{p^4} for all primes p not equal to 2 or 3. Interestingly, this characterization is tight: in fields of characteristic 2, the permanent coincides with the determinant; in fields of characteristic 3, one can efficiently compute the permanent of an orthogonal matrix by a nontrivial result of Kogan. Finally, we use more elementary arguments to prove #P-hardness for the permanent of a positive semidefinite matrix, which shows that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly despite the fact that they can be efficiently sampled by a classical computer.

Game Values and Computational Complexity: An Analysis via Black-White Combinatorial Games

A black-white combinatorial game is a two-person game in which the pieces are colored either black or white. The players alternate moving or taking elements of a specific color designated to them before the game begins. A player loses the game if there is no legal move available for his color on his turn.