We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Tribonacci-automatic". This class includes, for example, the famous Tribonacci word T = 0102010010201 · · · , the fixed point of the morphism 0 → 01, 1 → 02, 2 → 0. We use it to reprove some old results about the Tribonacci word from the literature, such as assertions about the occurrences in T of squares, cubes, palindromes, and so forth. We also obtain some new 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 = 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.
Abstract. We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered as a "circular word", i.e., the end of the word wraps around to the beginning. We determine the best possible avoidance exponent for alphabet size 2 and 3, and provide a lower bound for larger alphabets.
Self-testing has been a rich area of study in quantum information theory. It allows an experimenter to interact classically with a black box quantum system and to test that a specific entangled state was present and a specific set of measurements were performed. Recently, self-testing has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick \cite{low-degree}, iterated compression by Fitzsimons et al. \cite{iterated-compression}, and NEEXP in MIP* due to Natarajan and Wright \cite{neexp}. The most studied self-test is the CHSH game which features a bipartite system with two isolated devices. This game certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. Most of the self-testing literature has focused on extending these results to self-test for tensor products of EPR states and tensor products of Pauli measurements.In this work, we introduce an algebraic generalization of CHSH by viewing it as a linear constraint system (LCS) game, exhibiting self-testing properties that are qualitatively different. These provide the first example of LCS games that self-test non-Pauli operators resolving an open questions posed by Coladangelo and Stark \cite{RobustRigidityLCS}. Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal \cite{BCSTensor}. Additionally, our games have 1 bit question and logn bit answer lengths making them suitable candidates for complexity theoretic application. This work is the first step towards a general theory of self-testing arbitrary groups. In order to obtain our results, we exploit connections between sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem from approximate representation theory. A crucial part of our analysis is to introduce a sum of squares framework that generalizes the solution group of Cleve, Liu, and Slofstra \cite{BCSCommuting} to the non-pseudo-telepathic regime. Finally, we give the first example of a game that is not a self-test. Our results suggest a richer landscape of self-testing phenomena than previously considered.
We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is 1 or at most 1 2 is complete for the class Σ 1 (i.e., RE). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to 1 is complete for the class Π 1 (i.e., coRE).We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to 1 is complete for Π 2 ; this class is in the second level of the arithmetical hierarchy and corresponds to formulas of the form "∀x ∃y φ(x, y)". This shows that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value (either exactly or approximately).We explain how results about the complexity of nonlocal games all follow in a unified manner from a technique known as compression. At the core of our Π 2 -completeness result is a new "gapless" compression theorem that holds for both quantum and commuting operator strategies. Our compression theorem yields as a byproduct an alternative proof of Slofstra's result that the set of quantum correlations is not closed. We also show how a "gap-preserving" compression theorem for commuting operator strategies would imply that approximating the commuting operator value is complete for Π 1 .
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